Some fear the volatility and never invest in stocks. The real returns that can be earned on fixed income investments, however, are also risky. This is after taking into account the uncertainty about future inflation. And there are further risks. These include the unknown returns at which these issues can be rolled over or liquidated, and the unknown returns at which additional contributions can be invested. These latter risks also apply to issues whose returns are inflation protected.
In view of these uncertainties, this paper assumes that the annual real returns available on fixed income investments can be represented by those actually realized in the past on intermediate term U. S. government issues. These returns can be replicated by drawings from probability distributions chosen so as to match the observed annual real returns. The annual real returns on stocks can be represented by drawings from probability distributions chosen so as to replicate those actually observed on the S&P 500.
The objective is to determine the most that can be allocated to stocks each year. This is without accepting more risk for terminal wealth each year than if the portfolio were entirely invested in the intermediate term fixed income issues. This risk is measured by the lowest possible 5 percent of terminal values.
Presumably, almost everyone is willing to accept as much risk as investing in intermediate term fixed income issues. Allocating at least as much to stocks as has this risk is efficient because it provides as much terminal wealth as possible for accepting this risk. This paper also looks at the menu of increasing rewards and risks when the risk is uniformly increased each year above that when investing in just the fixed income issues.
The longer the accumulation period, the more that can initially be allocated to stocks without more risk than investing in just the fixed income issues. The reason is that hedging over time reduces risk. And there is more hedging over time the longer the accumulation period. Likewise, the risk for terminal wealth increases as the accumulation period declines for a given stock allocation. Consequently, target date funds reduce their allocation to stocks as the target date approaches to avoid increasing the risk for terminal wealth.
Besides the length of the accumulation period another key factor affecting the size of the stock allocations for the same risk is the size of any future contributions to the portfolio. The larger are future contributions relative to the current value of the portfolio, the larger are the stock allocations that provide the same risk as investing just in the fixed income issues. The reason is that future contributions absorb some of the volatility of earlier investment returns. This dampening effect of future contributions allows larger stock allocations beforehand without increasing risk.
The effect of future contributions can be substantial. Consider a 10 year accumulation period, and assume the base case probability distributions for investment returns that are used in this paper. Suppose first that there are not any additional contributions before the end of the period.
Later analysis shows that the initial stock allocation for the same risk in this case is 64 percent. This allocation declines by 3 percentage points each year over the following five years until it is 49 percent at the beginning of the last five years. Over the last five years the allocation declines by 4 percentage points over each of the first two years and then by 6 and 7 percentage points until it is 28 percent over the last year. These stock allocations increase the median terminal value by 18 percent over investing in just the fixed incomes.
Now instead of not having any additional contributions over the period suppose that the contributions are flat. At the end of each year there is an additional contribution equal in real terms to the initial value of the portfolio. In this case, it is not possible at the beginning of the 10 years to allocate enough to stocks to provide as much risk as investing just in the fixed income issues. Even when the entire initial portfolio is invested in stocks there is less risk than investing just in the fixed incomes. The stock allocations in the future are reduced to those that provide the same risk as investing just in the fixed income issues when that becomes possible. And it is assumed that this will be done when allocating the initial portfolio entirely to stocks, and saying that doing so at that time has less risk than investing just in the fixed incomes.
For the flat contributions, the future stock allocations cannot be determined initially because they depend on the ratio of the annual contribution to the value of the portfolio at future times. And future portfolio values depend on the uncertain investment returns earned beforehand. Simulations show that after one year the allocation is still likely to be entirely in stocks. After two years, however, as the value of the portfolio increases, the chance that the allocation will be this high declines to 0.30. After three years there is virtually no chance that the stock allocation will be this high. After three years, the stock allocations will be those that have the same risk as investing just in the fixed income issues.
The analysis in this paper starts by considering the case when only one more year remains in the accumulation period. Due to the assumed normal distributions for the annual returns on the investments an exact solution is possible for this case. This is for the stock allocation that has the same 5 percent risk for low values of terminal wealth as investing just in the fixed income issues. The sensitivity of this solution is examined with respect to changes in the correlation of the returns on the two investments, and the spreads of their mean returns and volatilities.
The next step is to assume that two more years remain in the accumulation period. And there are not any additional future contributions to the portfolio. In this case, simulations are required to determine the distribution of terminal wealth for different allocations of the initial portfolio.
The stock allocation that gives the same risk as investing in just the fixed incomes has already been determined for the second year of a two year period. Simulations are used to determine the stock allocation for the first year of the two year period, given this known allocation for the second year. Making use of the allocations that have just been determined for the last two years, the allocation can next be determined for the first year of a three year period. Continuing sequentially in this manner the desired allocations can be determined for any number of years.
Analysis then shifts to cases where there are future contributions to the portfolio. The same sequential process is used for these cases starting with a two year period. Now, however, the stock allocation depends on the ratio of the annual contribution to the value of the portfolio each year. Obtaining the results is facilitated by the linear nature of this realationship.
One More Year
Efficient portfolio allocation for a one period investment has been well established ever since the path breaking work of Harry Markowitz over 50 years ago. For the case considered here there are two investments, stocks and fixed incomes. The probability distribution for the real returns has a much higher mean and standard deviation for the stocks.
Consider the mean and standard deviation for the return on a portfolio of the two investments. Start with the portfolio allocated just to the fixed incomes. Consider what happens when a small allocation of stocks is added to the portfolio and gradually increased.
The mean return of the portfolio, of course, increases due to the higher mean return of the stocks. The returns on the stocks, however, are more volatile and have a much higher standard deviation. Nevertheless, as stocks are added, at first there is a decline in the standard deviation of the returns on the portfolio.
Risk is decreasing. At first, adding stocks is both increasing the reward and reducing risk. Risk is declining due to the benefits of diversification. The possibility of high returns on one investment is hedging against the possibility of low returns on the other. The value of this hedge depends on the correlation between the returns on the two investments.
Everyone will clearly want to add more stocks to the portfolio as long as doing so reduces risk. As more stocks are added, however, eventually the standard deviation of the portfolio begins to increase. Risk is now increasing, but the standard deviation of the returns on the portfolio is still less than that of the fixed incomes. This paper assumes that virtually everyone will want to continue increasing the allocation to stocks until the risk is at least equal to that of investing in just the fixed incomes.
The original work of Markowitz measured risk in terms of the standard deviation of the returns on a portfolio. Suppose, however, that the returns on the investments in the portfolio are normally distributed. In this case, the probability distribution of the return on the portfolio is also normally distributed. And it is possible to calculate risk in terms of the lowest returns that are possible with a given small probability.
Many may find this a more helpful way of assessing risk. And this measure of risk is readily calculated when simulations are used to determine the distribution of terminal value. This paper assumes risk is measured by the lowest 5 percent of terminal values.
When there is only one more year the probability distribution of terminal wealth is known under the base case assumptions for the stocks and fixed incomes. This is given the value of the portfolio at the beginning of the year and its allocation. Under these conditions terminal wealth is normally distributed.
To see that this is so, note the base case assumes that the annual real returns on the stocks and the fixed income issues are normally distributed. For the stocks the mean real return is .07 and the standard deviation is .18. For the fixed incomes the mean is .03 and the standard deviation is .07. These values assume normal valuations at the beginning of a year. Justification based on historical real returns is provided in the author’s, The Retrenchment Rule (2012). The effect of variation in these values is considered in the next section of this paper.
In this section it is assumed that the annual real returns for the stocks and the fixed incomes are independently distributed. In the next section, the real return on the stocks for a year is the sum of a fixed proportion of the return on the fixed incomes for that year and an independently and normally distributed variable. The mean and standard deviation of this independent variable are determined so as to keep the annual mean return on the stocks equal to .07 and the standard deviation equal to .18 depending on the value of the fixed proportion.
In any case, given the allocation, the return on the portfolio each year is a fixed weighted sum of independently and normally distributed variables. And such a sum is itself normally distributed. The mean and standard deviation for the portfolio can be calculated based on the allocation and other given parameters.
Suppose for a given allocation that the risk is desired based on the lowest 5 percent of possible real returns on the portfolio. The upper limit on these returns can be calculated using the corresponding percentage point for the cumulative normal distribution, and the mean and standard deviation of the portfolio with the given allocation. Limits can similarly be calculated for the highest 50, 25, and 5 percent of values.
Chart 1 shows how these limits vary with the allocation to stocks for the portfolio when the returns on the stocks and fixed incomes are independently distributed. The lower line shows the upper limit on the lowest 5 percent of real returns on the portfolio. The upper three lines show the lower limits on the highest 50, 25, and 5 percent of real returns. These real returns are easily transformed into terminal real wealth. This is using the value of the portfolio at the beginning of the year and any contribution that is made at the end of the year.
Of particular interest is the variation of the risk, shown by the bottom line, with the allocation to stocks. This risk changes in the same way as just discussed using the standard deviation of the return on the portfolio as the measure of risk. As stocks are added to the portfolio the upper limit on the lowest 5 percent of returns increases. This increase shows that risk at first declines as stocks are added to the portfolio. When the allocation to stocks increases beyond 17 percent, however, risk begins to increase. When the allocation reaches 36 percent the risk is no longer less than when the portfolio is invested just in the fixed income issues
Chart 1. Probability of the real return on a portfolio over a year based on the proportion allocated to stocks at the beginning of that year. The two choices available are stocks and fixed income issues. The real returns attempt to replicate those realized in the past when valuations are normal at the beginning of a year. For the stocks these are the returns on the S&P 500 and for the fixed incomes the returns on intermediate term U. S. government issues. For this chart these returns are independently distributed. The bottom line shows the upper limit on the lowest 5% of returns. The upper three lines show the lower limits on the highest 50%, 25%, and 5% of returns.
Sensitivity
Based on the assumptions for Chart 1, as much as 35 percent can be allocated to stocks with less risk than investing just in the fixed income issues. But how sensitive to these assumptions is this allocation to stocks?One critical assumption is the correlation between the returns on the stocks and the fixed income issues.
Suppose these returns have the high positive correlation that was evident over the period from 1970 to 1989. In this case, only 13 percent of the portfolio can be invested in stocks with less risk than investing just in fixed incomes. On the other hand, suppose that the returns have the negative correlation that was evident over the period from 1990 to 2011. In this case, 44 percent of the portfolio can be invested in stocks with less risk than investing just in the fixed incomes.
These historical correlations indicate that a surge in inflation is likely to cause significant positive correlation between the real returns on stocks and fixed income issues. The extremely high positive correlation over 1970 to 1989 coincided with the unprecedented surge in inflation in the 1970s, and recovery from that inflation in the 1980s. Over the 1990 to 2011 period, on the other hand, general inflation was not a significant problem. There were two major bear markets for stocks over this period, but they were not related to inflation.
It appears that a best guess for the future correlation of the returns on stocks and fixed income issues should average together the correlations observed over the last 50 years. Thus, the effect of the correlation evident over the period from 1960 to 2011 is shown in Chart 2. This correlation reduces from 35 to 28 percent the largest stock allocation that has less risk than investing just in the fixed incomes.
Chart 2. Dashed lines show how Chart 1 changes when the real returns on the stocks and fixed incomes issues are positively correlated. The assumed positive correlation in the chart is that evident in the realized returns over the period from 1960 to 2011. The largest stock allocation with less than the 5% risk of the fixed incomes declines from 35 to 28 percent.
The correlation between the returns was measured by regressing the annual real return for the stocks on the real return for the fixed incomes. Over the 1960 to 2011 period this coefficient was .40. To make the simulations, the return for the stocks for a year was set equal to .40 of the return on the fixed incomes for that year plus an independently and normally distributed variable. The mean and standard deviation of this independent variable were determined so that the mean and standard deviation of the annual return on the stocks remained equal to the base case values of .07 and .18. These values are respectively .058 and .1778.
The regression coefficients over the 1970 to 1989 and the 1990 to 2011 periods were respectively .97 and −.55. Following the same procedure for these cases gives stock allocations respectively of 13 and 44 percent. The stock allocation for the same risk changes by about .02 for a .1 change in the regression coefficient.
Another critical assumption is the spread between the mean returns on the stocks and the fixed incomes. These mean returns may move up and down together over an interest rate cycle. Such concurrent variation, however, has little effect on the stock allocation for the same risk.
A change in the spread between the mean returns on the stocks and fixed incomes, however, does have an effect. A narrowing in the spread by .01 causes a .03 reduction in the stock allocation for the same risk. Anticipating a spread between the mean returns of only .02 instead of the base case .04 reduces the stock allocation for the same risk from 28 to 22 percent.
When the volatilities for the returns on the stocks and the fixed incomes move up and down together there is also little change. If the standard deviation of the return on the stocks is increased to .19 instead of .18, however, the stock allocation for the same risk declines from 28 to 25 percent. And the same decline in the stock allocation occurs if the standard deviation for the returns on the fixed incomes declines from .07 to .065.
Two More Years
Efficient portfolio allocation for a one period investment has been known for over 50 years. And the probability distribution of the return on the portfolio is easily calculated when the returns on the underlying investments are normally distributed. When the returns on such a portfolio are compounded over two or more periods, however, the probability distribution of the terminal returns can no longer be readily calculated.
For two or more periods, however, the terminal returns can readily be simulated on a computer. And with enough iterations for each simulation, sufficient accuracy is achievable with respect to the distribution of returns to make the same assessments as for a one period case. It is possible to determine with sufficient precision how much can be allocated to stocks each period without exceeding the risk of investing in just fixed income issues.
The process for making this assessment uses the well known procedure in financial analysis of backwards recursion. Consider first a two year period and assume that there are not future contributions. First simulate the value that will be obtained after two years when the portfolio each year is invested just in the fixed income issues.
The value obtained at the end of the first year is given by a drawing from a normal distribution with a mean real return of .03 and a standard deviation of .07. This value is reinvested over the second year. Another drawing is made from the same distribution to get the value at the end of the second year, which in this case is the terminal value.
To get a reliable indication of the distribution of terminal values for this paper this process was repeated one million times. And the upper limit on the lowest 5 percent of values was determined. This value provides the desired measure of risk. This same simulation was made a total of 10 times. As will be discussed shortly these repetitions are needed to reliably determine the stock allocations that have the same risk as investing just in the fixed incomes.
The next step is to repeat the simulation of terminal value when a portion of the portfolio at the beginning of each year is allocated to stocks. The real return on the stocks for each year is obtained as discussed in the last section. It is equal to a portion of the drawing for that year for the real return on the fixed incomes plus an independent drawing from another distribution. The latter is a normal distribution with a mean and standard deviation calculated so that the real return on the stocks each year is a drawing from the distribution specified earlier in this paper. This is a distribution with a mean of .07 and a standard deviation of .18.
The stock allocation for the second year of the two year period that has the same risk as investing just in the fixed incomes was obtained in the last section, and is 28 percent. The objective now is to find the stock allocation at the beginning of the first year that has the same 5 percent risk for terminal value over two years as investing just in the fixed incomes. This is using the stock allocation of 28 percent that satisfies this condition for one year as the allocation at the beginning of the second year.
Determining the allocation for the first year of a two year period that satisfies this condition is a trial and error process. Guess a stock allocation for the first year in whole percentage points. Next, run a simulation with a million iterations using this guess as the stock allocation for the first year and 28 percent as the allocation for the second year. Determine the upper limit on the lowest 5 percent of terminal values for these stock allocations.
Suppose this 5 percent value is less than that for any of the 10 simulations when investing just in the fixed incomes. In this case, the initial guess for the stock allocation for the first year is rejected. This allocation is deemed not to have less risk than investing in just the fixed incomes over the two year period. The initial guess is also rejected if the 5 percent value is lower for any one of a total of 10 simulations when the simulation is repeated. If the initial guess is rejected a new lower guess for the stock allocation is tested.
Suppose after 10 trials that a guess is not rejected. This guess is not accepted, however, unless the next higher stock allocation in whole percentage points is rejected by the same process. In view of the latter test, the initial guess is increased for the next simulation if the 5 percent value for the first simulation appears to be too high for the initial guess to be accepted.
Increasing the number of iterations in each simulation reduces the range of the 5 percent values for the simulations. Reducing this range in general reduces the number of simulations required to complete the trial and error process. On the other hand, more iterations, of course, require more time for each simulation.
Using one million iterations for a simulation provides a good balance between the number of simulations and the time for each simulation. The simulations for this paper were made using @Risk 6.2 software and an AMD A10−5745M processor. For a two period accumulation period the run time for a simulation was a little over a minute. The run times increase, of course, with the length of the accumulation period and were over four minutes for 20 periods.
Following the trial and error process just discussed a stock allocation of 36 percent for the first year was easily rejected. It gives a lower 5 percent terminal value than investing just in the fixed incomes. On the other hand, a stock allocation of 35 percent gave a range of the 5 percent terminal values for 10 simulations that was above the range of these values for 10 simulations when investing in just the fixed incomes.
These tests show that the highest stock allocations in whole percentage points that provide less risk than investing in just the fixed incomes over two years are respectively 35 and 28 percent. This is when this risk is viewed from the beginning of either of the two years. The stock allocation for the first year is significantly higher than for the second year.
The stock allocation for the same risk is higher for the first year because of the benefits of hedging over time. The possibility of high returns in one year is hedging against the possibility of low returns in the other year. This hedging reduces risk allowing a larger allocation to stocks to provide the same risk as investing in just fixed incomes. The hedging effect is similar to the hedging for stocks and fixed incomes for a one period investment. This hedging allows an allocation to stocks to have the same risk as investing in just fixed incomes.
The next section shows that adding more periods increases the hedging. This allows larger increases in the allocation to stocks without more risk than investing in just fixed incomes. The benefit of adding more periods, however, gradually diminishes as more periods are added. This diminishing benefit is similar to the diminishing benefit when adding more investments to the portfolio for a one period investment.
Besides the risk, the charts for the one year case also show the improvement in the upper 50, 25, and 5 percent terminal values. The simulations for the two year case also show this improvement. For stock allocations of 35 and 28 percent these values increase respectively by 2.4, 3.2, and 4.4 percent over investing in just the fixed incomes. The improvements become more significant, of course, as the accumulation period increases.
More Than Two More Years
The solution for the two year case indicates how to find the stock allocations for any number of years. For three years, the stock allocations for the last two years are known from the solution for the two year case. And the allocation for the first year of the three years is obtained in a similar way to the first year of the two year case, given the known solution in that case for the last year.
The allocation for the first year of a four year case can then be determined using the known allocations for the last three years. And the allocations can successively be determined in this way for any number of years. Successively carrying out this process for three, four, and five years gives allocations over five years of 49, 45, 41, 35, and 28 percent.
When adding another year or two to the five years the stock allocation at the beginning of the longer period increases by 3 percentage points for each year. To reduce the number of simulations, however, for periods over five years the simulations for longer periods were made in five year segments. And over these five year segments, the stock allocations were assumed in each segment to increase by the same number of percentage points each year. These changes from year to year could vary in terms of half or full percentage points.
With these assumptions, the largest stock allocation at the beginning of 10 years that has no more risk than investing in just fixed incomes is 64 percent. From 64 percent the stock allocations decline by 3 percentage points each year over the following five years. With these declines the allocation becomes 49 percent at the beginning of the last five years as required by the earlier results.
Following the same procedure for the first five years of a 15 year period, the stock allocation at the beginning of the 15 years is 76.5 percent. This allocation declines by 2.5 percentage points each year until it is 64 percent at the beginning of the last 10 years as required by the earlier results. At the beginning of a 20 year period the stock allocation is 84 percent. This allocation declines by 1.5 percentage points a year until it is 76.5 percent at the beginning of the last 15 years as required.
These results illustrate some important general principles with respect to investing over time. The longer the accumulation period, the larger the allocation that can be made to stocks without more risk than investing in just fixed income issues. The size of these increases in stock allocation, however, becomes smaller as the accumulation period becomes longer. This risk refers to the lowest possible 5 percent of values at the end of the accumulation period as viewed from any remaining year in the period.
These principles are explained as the result of hedging over time. Adding an additional year to the accumulation period allows additional hedging of possible low and high returns each year. This increase in diversification allows a larger allocation to stocks without more risk than investing in just fixed income issues. This diversification effect is similar to the hedging that occurs in a single year by combining stocks and fixed income issues in a portfolio.
After more years have been added, however, adding an additional year has a smaller marginal benefit. This declining marginal benefit of hedging over time is also similar to the hedging in a single period investment. After more different investments have been added to the portfolio, there is less marginal benefit in risk reduction by adding an additional different investment.
While these stock allocations do not increase the risk at any time, versus investing in just fixed incomes, they do, of course, significantly increase the wealth accumulated. Compared to investing in just the fixed incomes over 5, 10, 15, and 20 years the median value accumulated increases by 7, 18, 32, and 48 percent respectively. The upper 25 percent of value accumulated increases by 10, 26, 46,and 72 percent. And the upper 5 percent of value increases by 14, 37, 69, and 111 percent.
Future Contributions
The next step is to consider the effect of making future contributions to the portfolio. Future contributions increase the allocations that can be made to stocks without exceeding the risk of investing in just the fixed income issues. The reason is that the future contributions absorb some of the increase in volatility caused by earlier stock allocations.
To facilitate the analysis, it is assumed that the contributions are added at the end of each year, and are constant in real terms. In this case, the effect of the contributions on increasing the allocation to stocks is directly proportional to the size of the annual contribution relative to the current size of the portfolio. This ratio will be referred to as the Contribution Portfolio Ratio (CPR). The proportion of the CPR that adds to the stock allocation increases as the remaining number of years increases in the accumulation period.
This linear relationship substantially facilitates dealing with future contributions in the simulations. The proportion of the CPR for a given accumulation period can be determined with a series of simulations for a single value of the CPR. And this proportion gives the appropriate increase in the stock allocation for this accumulation period whatever the value of the portfolio turns out to be for that period.
These proportions are determined for successively longer periods using a similar backwards recursion process to that used earlier when there are not any contributions. The allocation for a one year period is unaffected by a contribution and remains equal to 28 percent. Next, find the allocation for a 2 year period when there is a future contribution equal in real value to the initial value of the portfolio so that the CPR is 1.0.
The first step is finding the lowest 5 percent of terminal values for the specified contribution when investing just in the fixed incomes. Next, use the same trial and error process as before to find the stock allocation for the first year given the allocation of 28 percent for the second year. For the first year this is the largest stock allocation in whole percentage points that gives 5 percent terminal values that are higher than those obtained when invested just in the fixed incomes.
With an annual contribution equal to the initial value of the portfolio, a stock allocation of 46 percent for the first year is rejected. An allocation of 45 percent, however, gives a range of 5 percent terminal values above the range for investing in just the fixed incomes. Without the contributions, the earlier tests gave a stock allocation for the first year of 35 percent.
Thus, the contributions increase the stock allocation by 10 percentage points. This increase is equal to 0.1 of the ratio of the annual contribution to the current portfolio value, which is 1.0. Running the same test when the contribution is half the initial portfolio reduces the increase in the stock allocation by a half. These tests confirm the linearity of the relationship.
For the first year of a two year period, the stock allocation is 35 percent plus 10 percent of the initial CPR. For a three year period, the stock allocation for the second year is given by this relationship and depends on the value of the CPR at the beginning of the last two years. The stock allocation for the first year of the three year period is based on simulations that take into account an assumed value of the CPR at the beginning of the three years.
The stock allocation for the first year of a three year period is 41 percent plus 22 percent of the CPR at the beginning of the three years. Successively increasing the accumulation period by one year each time, the proportion of the CPR to add can be determined each year in this same way for a period of any number of years. The proportions for accumulation periods from 1 to 10 years are respectively 0, .10, .22, .38, .56, .76, 1.00, 1.26, 1.54, and 1.84.
A stock allocation can be no more than 100 percent in this analysis as borrowing is not allowed. For a CPR of 1.0, the limit on the stock allocation of 100 percent is applicable for any accumulation period of five years or more. In these cases it is not possible to invest enough in stocks initially to provide as much risk as investing just in the fixed income issues. This last statement assumes that the stock allocations will be reduced below the limit of 100 percent when that becomes possible in the future.
The CPR tends to decline over time because the contribution is fixed and the portfolio is increasing. Eventually, the stock allocations determined earlier will necessarily become applicable. For a five year period when the CPR is 1.0, for instance, the stock allocation at the beginning of the next year is sure to be less than 100 percent.
Contributions reduce the effective length of the accumulation period. Thus, it is not surprising that contributions reduce the improvement in terminal value. This is when investing as much as possible in stocks without more risk than investing just in fixed income issues. Consider a 10 year period and assume that the contributions are equal to the initial value of the portfolio so that the CPR is 1.0.. The median value increases by 10.5 percent in this case compared to investing just in the fixed incomes. Without any contributions the increase is 18 percent.
Including contributions tends to increase the variability of the 5 percent terminal values obtained with the simulations. For the contributions, therefore, the number of iterations was increased from one to two million. And the number of simulations for each case to establish rejection was reduced from 10 to 5. For a 10 year period with 2 million iterations the run time for a simulation is about six minutes with the processor given earlier.
Menu for Increasing Rewards and Risks
The focus so far has been on the largest stock allocations over an accumulation period that have less risk each year than investing just in fixed income issues. Presumably, almost everyone will be willing to accept at least this much risk to get terminal values for wealth that are as large as possible. Some, however, will be willing to accept more risk to increase their chances of getting higher values.
When increasing risk, it seems reasonable that risk should be increased uniformly over each of the years of the accumulation period. A uniform increase can be approximately achieved by increasing the allocations obtained earlier by the same number of percentage points each year. Gradually raise these sets of increases. Simulating terminal wealth for each set of these increases generates a menu of increasing rewards and risks.
Such a menu is shown in Chart 3. This chart assumes a 10 year accumulation period and contributions at the end of each year equal in real terms to 10 percent of initial portfolio value. The vertical axis shows terminal wealth in real terms as a percentage of initial portfolio value. The horizontal axis shows the percentage points by which the stock allocation has been increased for each year. This is an increase above the allocation that provides the same risk of terminal wealth for each year as investing just in the fixed income issues. The increases are limited, however, in cases where they would put a stock allocation over 100 percent.
Chart 3. Increasing rewards and risks as the percentage points shown on the horizontal axis increase the stock allocations for each year. The increases are limited, however, when the allocation would become more than 100%. Vertical axis shows terminal wealth in real terms as a percent of initial portfolio value for accumulation over 10 years. This is when a contribution in real terms is made at the end of each year equal to 10% of the initial portfolio. The rewards and risks on the far left are for allocating to stocks each year so that the risk is the same as investing in the fixed incomes. The lower solid line shows the upper limit on the lowest 5% of terminal wealth. The upper three solid lines show the lower limits on the highest 5%, 25%, and 50% of terminal wealth. The dashed lines show these limits when investing in just the fixed income issues.
The total contributed value in terminal wealth is 200. This includes the 100 of initial portfolio value and a total of 100 for the contributions of 10 at the end of each of the 10 years. Note on the far left of the chart that there is a .05 chance that terminal wealth will be slightly less than the total contributed value of 200. This is irrespective of whether investing in just the fixed income issues or using the stock allocations over the 10 years that have the same risk and were obtained in the last section.
When investing in just the fixed incomes the lower limits on the upper 50, 25, aand 5 percent of terminal values are shown by the short dashed lines in the chart. As expected, when using the stock allocations obtained in the last section the lower limits on the upper 50, 25, and 5 percent of terminal values are substantially higher. These values are given by the intersection of the vertical axis with the three solid upper lines.
Moving to the right, the solid lines show the increasing risk and rewards when the stock allocations are increased uniformly over each of the 10 years. There is a small increase in risk shown by the small decline in the lower line. The median values show a comparable small increase to the small increase in risk. The upper 25 and 5 percent of values show larger improvements. Those accumulating wealth will have to decide whether this additional improvement is worth the added risk.
Conclusions
When investing over time hedging occurs as low returns in one year can be offset by high returns in another. This diversification over time allows higher allocations to stocks at the beginning of longer periods. This is without more risk of low values of terminal wealth than occurs when investing in just fixed income issues over the same period.
This paper has used simulated investment returns to make quantitative estimates of how much stock allocations can be increased without increasing such risk. This is as the length of the accumulation period increases. The simulations also show the increased terminal wealth obtained as a result of the higher stock allocations.
Another factor that increases the stock allocations that can be made without increasing risk is the size of future contributions to a portfolio. The larger are future contributions relative to the current value of the portfolio, the larger are the stock allocations that can be made without increasing risk. The reason is that future contributions absorb some of the volatility caused by an increase in earlier stock allocations. Simulations in this paper have estimated the size of this effect and show that it can be very large.
Some investors are willing to accept more risk than investing in just fixed income issues to further improve their prospects. Simulations in this paper also show the increase in terminal wealth obtained by accepting more risk. This is more risk for low values of terminal wealth than when investing in just fixed income issues as viewed from each year of the accumulation period.
Posted May 2014
Addendum*
Addendum** of Sensitivity introduces a set of low return assumptions consistent with the real returns realized on intermediate term U.S. government issues since 2008. These low returns reduce from 3.5% to 2.5% the initial disbursement that provides a sustainable stream of disbursements over retirement under the conditions otherwise generally assumed to illustrate disbursements at this site. Reducing the initial disbursement from 3.5% to 2.5% means that a 40% larger portfolio will be required to get a spending stream of the same size with the low returns. This addendum shows that almost 40% more will also have to be saved over a 20 year period to accumulate a given amount of funds with these low returns. As 40% more savings are required to provide 40% more funds, these effects are compounded. Almost twice as much savings will therefore be required for retirement with the low returns.
Suppose a stream of savings for retirement is assumed to be a stream of contributions for 20 years each of which is equal in real terms to the initial value of the portfolio. The text obtains the stock allocations for the last 10 years of such a 20 year stream using two backwards recursive processes. One of these finds the stock allocation each year when there are not any contributions. The other finds the multiple of the ratio of the contribution to portfolio value (CPR) each year that is added to the stock allocation when there are contributions. These processes are described in detail in the text. The stock allocations for a 20 year stream are obtained by simply continuing these backwards recursive processes for 10 more years. The stock allocations for the low returns are obtained by simply repeating these backwards recursive processes using the low returns. (1)
In the early years for a flat stream of contributions the stock allocations given by the recursive processes can be over 100%. In such cases, the stock allocations are limited to 100% as leverage using the intermediates is not allowed. Future contributions in these cases are absorbing sufficient volatility to make stocks less risky than intermediates due to the much higher expected returns of the stocks. In the future, the value of the portfolio increases relative to the value of the contribution. Eventually, the contributions are no longer large enough relative to the size of the portfolio to absorb enough volatility, and the stock allocations given by the recursive processes decline below 100%.
For the low returns, the stock allocations are smaller than for the higher returns when only a few more years remain for accumulation. These smaller allocations occur because the premium expected return for the stocks over the intermediates is assumed to be the same for the low returns, but the volatility of the return on the stocks declines relatively less making the stocks relatively less valuable. In particular, for the low returns the expected return for the stocks declines from .07 to .04, and for the intermediates by the same amount from .03 to .00. In contrast, the standard deviation for the stocks declines from .18 to .14, whereas for the intermediates the decline is relatively more being from .07 to 04.
On the other hand, the increase in the stock allocations for the low returns is stronger as the time for accumulation increases. Eventually, when the time for accumulation becomes sufficiently long, the stock allocations become higher for the low returns. Note for either the stocks or intermediates that volatility increases relative to expected return for the low returns. Thus, the volatility of the portfolio relative to expected return is higher for the low returns. With volatility higher relative to expected return, the reduction in volatility provided by hedging over time and by future contributions makes the higher expected returns of the stocks more desirable, increasing the allocation to stocks.
Given the stock allocations when saving over 20 years for the low and higher returns the value accumulated in either case can be simulated. The cumulative probabilities of the funds after 20 years as a percent of the total funds contributed are shown in Chart 4. For the higher returns, the median value increases from 125% to 171% of contributed value, or by 37%. To accumulate the same median value for a retirement portfolio, 37% more will have to be saved with the low returns. For the lowest 20th percentile, 25% more will have to be saved, and for the highest 20th percentile, 49% more will have to be saved. Thus, the central tendency is that an increase in savings is required that is almost equal to the 40% increase in funds needed for the same sustainable stream of spending with the low returns. Compounding these two effects, the low returns almost double the savings over a 20 year period required to fund a sustainable stream of spending in retirement.
Note
As discussed in the text, a trial and error process is used to determine for any year the largest stock allocation in whole percentage points that has a 5 percentile terminal value that exceeds the 5 percentile terminal value when the portfolio is entirely invested in intermediates. For the new results, this condition is considered to be satisfied when it is true for each of five simulations with 1 million iterations. As assumed in the text, before the last five years when there are not any contributions, trials are conducted for five year segments with changes in terms of either half or full percentage points. When determining the multiples for the CPR with more than 10 years remaining for accumulation, the multiples were determined for two years at a time assuming the intervening multiple was half way in between. For the higher returns when starting initially with 20 years remaining and using the stock allocations determined by the backwards recursive processes, the stock allocation is limited to 100% for every iteration for the first five years. The allocation is limited to 100% for at least one iteration until after 10 years. For the low returns, the stock allocation is limited to 100% for every iteration for the first seven years, and for at least one iteration until after 11 years.
*Addendum posted December, 2020. Combines and simplifies Addendum* posted August, 2020, and Addendum** posted October, 2020.
Accumulation
Efficient Wealth Accumulation
Some fear the volatility and never invest in stocks. The real returns that can be earned on fixed income investments, however, are also risky. This is after taking into account the uncertainty about future inflation. And there are further risks. These include the unknown returns at which these issues can be rolled over or liquidated, and the unknown returns at which additional contributions can be invested. These latter risks also apply to issues whose returns are inflation protected.
In view of these uncertainties, this paper assumes that the annual real returns available on fixed income investments can be represented by those actually realized in the past on intermediate term U. S. government issues. These returns can be replicated by drawings from probability distributions chosen so as to match the observed annual real returns. The annual real returns on stocks can be represented by drawings from probability distributions chosen so as to replicate those actually observed on the S&P 500.
The objective is to determine the most that can be allocated to stocks each year. This is without accepting more risk for terminal wealth each year than if the portfolio were entirely invested in the intermediate term fixed income issues. This risk is measured by the lowest possible 5 percent of terminal values.
Presumably, almost everyone is willing to accept as much risk as investing in intermediate term fixed income issues. Allocating at least as much to stocks as has this risk is efficient because it provides as much terminal wealth as possible for accepting this risk. This paper also looks at the menu of increasing rewards and risks when the risk is uniformly increased each year above that when investing in just the fixed income issues.
The longer the accumulation period, the more that can initially be allocated to stocks without more risk than investing in just the fixed income issues. The reason is that hedging over time reduces risk. And there is more hedging over time the longer the accumulation period. Likewise, the risk for terminal wealth increases as the accumulation period declines for a given stock allocation. Consequently, target date funds reduce their allocation to stocks as the target date approaches to avoid increasing the risk for terminal wealth.
Besides the length of the accumulation period another key factor affecting the size of the stock allocations for the same risk is the size of any future contributions to the portfolio. The larger are future contributions relative to the current value of the portfolio, the larger are the stock allocations that provide the same risk as investing just in the fixed income issues. The reason is that future contributions absorb some of the volatility of earlier investment returns. This dampening effect of future contributions allows larger stock allocations beforehand without increasing risk.
The effect of future contributions can be substantial. Consider a 10 year accumulation period, and assume the base case probability distributions for investment returns that are used in this paper. Suppose first that there are not any additional contributions before the end of the period.
Later analysis shows that the initial stock allocation for the same risk in this case is 64 percent. This allocation declines by 3 percentage points each year over the following five years until it is 49 percent at the beginning of the last five years. Over the last five years the allocation declines by 4 percentage points over each of the first two years and then by 6 and 7 percentage points until it is 28 percent over the last year. These stock allocations increase the median terminal value by 18 percent over investing in just the fixed incomes.
Now instead of not having any additional contributions over the period suppose that the contributions are flat. At the end of each year there is an additional contribution equal in real terms to the initial value of the portfolio. In this case, it is not possible at the beginning of the 10 years to allocate enough to stocks to provide as much risk as investing just in the fixed income issues. Even when the entire initial portfolio is invested in stocks there is less risk than investing just in the fixed incomes. The stock allocations in the future are reduced to those that provide the same risk as investing just in the fixed income issues when that becomes possible. And it is assumed that this will be done when allocating the initial portfolio entirely to stocks, and saying that doing so at that time has less risk than investing just in the fixed incomes.
For the flat contributions, the future stock allocations cannot be determined initially because they depend on the ratio of the annual contribution to the value of the portfolio at future times. And future portfolio values depend on the uncertain investment returns earned beforehand. Simulations show that after one year the allocation is still likely to be entirely in stocks. After two years, however, as the value of the portfolio increases, the chance that the allocation will be this high declines to 0.30. After three years there is virtually no chance that the stock allocation will be this high. After three years, the stock allocations will be those that have the same risk as investing just in the fixed income issues.
The analysis in this paper starts by considering the case when only one more year remains in the accumulation period. Due to the assumed normal distributions for the annual returns on the investments an exact solution is possible for this case. This is for the stock allocation that has the same 5 percent risk for low values of terminal wealth as investing just in the fixed income issues. The sensitivity of this solution is examined with respect to changes in the correlation of the returns on the two investments, and the spreads of their mean returns and volatilities.
The next step is to assume that two more years remain in the accumulation period. And there are not any additional future contributions to the portfolio. In this case, simulations are required to determine the distribution of terminal wealth for different allocations of the initial portfolio.
The stock allocation that gives the same risk as investing in just the fixed incomes has already been determined for the second year of a two year period. Simulations are used to determine the stock allocation for the first year of the two year period, given this known allocation for the second year. Making use of the allocations that have just been determined for the last two years, the allocation can next be determined for the first year of a three year period. Continuing sequentially in this manner the desired allocations can be determined for any number of years.
Analysis then shifts to cases where there are future contributions to the portfolio. The same sequential process is used for these cases starting with a two year period. Now, however, the stock allocation depends on the ratio of the annual contribution to the value of the portfolio each year. Obtaining the results is facilitated by the linear nature of this realationship.
One More Year
Efficient portfolio allocation for a one period investment has been well established ever since the path breaking work of Harry Markowitz over 50 years ago. For the case considered here there are two investments, stocks and fixed incomes. The probability distribution for the real returns has a much higher mean and standard deviation for the stocks.
Consider the mean and standard deviation for the return on a portfolio of the two investments. Start with the portfolio allocated just to the fixed incomes. Consider what happens when a small allocation of stocks is added to the portfolio and gradually increased.
The mean return of the portfolio, of course, increases due to the higher mean return of the stocks. The returns on the stocks, however, are more volatile and have a much higher standard deviation. Nevertheless, as stocks are added, at first there is a decline in the standard deviation of the returns on the portfolio.
Risk is decreasing. At first, adding stocks is both increasing the reward and reducing risk. Risk is declining due to the benefits of diversification. The possibility of high returns on one investment is hedging against the possibility of low returns on the other. The value of this hedge depends on the correlation between the returns on the two investments.
Everyone will clearly want to add more stocks to the portfolio as long as doing so reduces risk. As more stocks are added, however, eventually the standard deviation of the portfolio begins to increase. Risk is now increasing, but the standard deviation of the returns on the portfolio is still less than that of the fixed incomes. This paper assumes that virtually everyone will want to continue increasing the allocation to stocks until the risk is at least equal to that of investing in just the fixed incomes.
The original work of Markowitz measured risk in terms of the standard deviation of the returns on a portfolio. Suppose, however, that the returns on the investments in the portfolio are normally distributed. In this case, the probability distribution of the return on the portfolio is also normally distributed. And it is possible to calculate risk in terms of the lowest returns that are possible with a given small probability.
Many may find this a more helpful way of assessing risk. And this measure of risk is readily calculated when simulations are used to determine the distribution of terminal value. This paper assumes risk is measured by the lowest 5 percent of terminal values.
When there is only one more year the probability distribution of terminal wealth is known under the base case assumptions for the stocks and fixed incomes. This is given the value of the portfolio at the beginning of the year and its allocation. Under these conditions terminal wealth is normally distributed.
To see that this is so, note the base case assumes that the annual real returns on the stocks and the fixed income issues are normally distributed. For the stocks the mean real return is .07 and the standard deviation is .18. For the fixed incomes the mean is .03 and the standard deviation is .07. These values assume normal valuations at the beginning of a year. Justification based on historical real returns is provided in the author’s, The Retrenchment Rule (2012). The effect of variation in these values is considered in the next section of this paper.
In this section it is assumed that the annual real returns for the stocks and the fixed incomes are independently distributed. In the next section, the real return on the stocks for a year is the sum of a fixed proportion of the return on the fixed incomes for that year and an independently and normally distributed variable. The mean and standard deviation of this independent variable are determined so as to keep the annual mean return on the stocks equal to .07 and the standard deviation equal to .18 depending on the value of the fixed proportion.
In any case, given the allocation, the return on the portfolio each year is a fixed weighted sum of independently and normally distributed variables. And such a sum is itself normally distributed. The mean and standard deviation for the portfolio can be calculated based on the allocation and other given parameters.
Suppose for a given allocation that the risk is desired based on the lowest 5 percent of possible real returns on the portfolio. The upper limit on these returns can be calculated using the corresponding percentage point for the cumulative normal distribution, and the mean and standard deviation of the portfolio with the given allocation. Limits can similarly be calculated for the highest 50, 25, and 5 percent of values.
Chart 1 shows how these limits vary with the allocation to stocks for the portfolio when the returns on the stocks and fixed incomes are independently distributed. The lower line shows the upper limit on the lowest 5 percent of real returns on the portfolio. The upper three lines show the lower limits on the highest 50, 25, and 5 percent of real returns. These real returns are easily transformed into terminal real wealth. This is using the value of the portfolio at the beginning of the year and any contribution that is made at the end of the year.
Of particular interest is the variation of the risk, shown by the bottom line, with the allocation to stocks. This risk changes in the same way as just discussed using the standard deviation of the return on the portfolio as the measure of risk. As stocks are added to the portfolio the upper limit on the lowest 5 percent of returns increases. This increase shows that risk at first declines as stocks are added to the portfolio. When the allocation to stocks increases beyond 17 percent, however, risk begins to increase. When the allocation reaches 36 percent the risk is no longer less than when the portfolio is invested just in the fixed income issues
Chart 1. Probability of the real return on a portfolio over a year based on the proportion allocated to stocks at the beginning of that year. The two choices available are stocks and fixed income issues. The real returns attempt to replicate those realized in the past when valuations are normal at the beginning of a year. For the stocks these are the returns on the S&P 500 and for the fixed incomes the returns on intermediate term U. S. government issues. For this chart these returns are independently distributed. The bottom line shows the upper limit on the lowest 5% of returns. The upper three lines show the lower limits on the highest 50%, 25%, and 5% of returns.
Sensitivity
Based on the assumptions for Chart 1, as much as 35 percent can be allocated to stocks with less risk than investing just in the fixed income issues. But how sensitive to these assumptions is this allocation to stocks?One critical assumption is the correlation between the returns on the stocks and the fixed income issues.
Suppose these returns have the high positive correlation that was evident over the period from 1970 to 1989. In this case, only 13 percent of the portfolio can be invested in stocks with less risk than investing just in fixed incomes. On the other hand, suppose that the returns have the negative correlation that was evident over the period from 1990 to 2011. In this case, 44 percent of the portfolio can be invested in stocks with less risk than investing just in the fixed incomes.
These historical correlations indicate that a surge in inflation is likely to cause significant positive correlation between the real returns on stocks and fixed income issues. The extremely high positive correlation over 1970 to 1989 coincided with the unprecedented surge in inflation in the 1970s, and recovery from that inflation in the 1980s. Over the 1990 to 2011 period, on the other hand, general inflation was not a significant problem. There were two major bear markets for stocks over this period, but they were not related to inflation.
It appears that a best guess for the future correlation of the returns on stocks and fixed income issues should average together the correlations observed over the last 50 years. Thus, the effect of the correlation evident over the period from 1960 to 2011 is shown in Chart 2. This correlation reduces from 35 to 28 percent the largest stock allocation that has less risk than investing just in the fixed incomes.
Chart 2. Dashed lines show how Chart 1 changes when the real returns on the stocks and fixed incomes issues are positively correlated. The assumed positive correlation in the chart is that evident in the realized returns over the period from 1960 to 2011. The largest stock allocation with less than the 5% risk of the fixed incomes declines from 35 to 28 percent.
The correlation between the returns was measured by regressing the annual real return for the stocks on the real return for the fixed incomes. Over the 1960 to 2011 period this coefficient was .40. To make the simulations, the return for the stocks for a year was set equal to .40 of the return on the fixed incomes for that year plus an independently and normally distributed variable. The mean and standard deviation of this independent variable were determined so that the mean and standard deviation of the annual return on the stocks remained equal to the base case values of .07 and .18. These values are respectively .058 and .1778.
The regression coefficients over the 1970 to 1989 and the 1990 to 2011 periods were respectively .97 and −.55. Following the same procedure for these cases gives stock allocations respectively of 13 and 44 percent. The stock allocation for the same risk changes by about .02 for a .1 change in the regression coefficient.
Another critical assumption is the spread between the mean returns on the stocks and the fixed incomes. These mean returns may move up and down together over an interest rate cycle. Such concurrent variation, however, has little effect on the stock allocation for the same risk.
A change in the spread between the mean returns on the stocks and fixed incomes, however, does have an effect. A narrowing in the spread by .01 causes a .03 reduction in the stock allocation for the same risk. Anticipating a spread between the mean returns of only .02 instead of the base case .04 reduces the stock allocation for the same risk from 28 to 22 percent.
When the volatilities for the returns on the stocks and the fixed incomes move up and down together there is also little change. If the standard deviation of the return on the stocks is increased to .19 instead of .18, however, the stock allocation for the same risk declines from 28 to 25 percent. And the same decline in the stock allocation occurs if the standard deviation for the returns on the fixed incomes declines from .07 to .065.
Two More Years
Efficient portfolio allocation for a one period investment has been known for over 50 years. And the probability distribution of the return on the portfolio is easily calculated when the returns on the underlying investments are normally distributed. When the returns on such a portfolio are compounded over two or more periods, however, the probability distribution of the terminal returns can no longer be readily calculated.
For two or more periods, however, the terminal returns can readily be simulated on a computer. And with enough iterations for each simulation, sufficient accuracy is achievable with respect to the distribution of returns to make the same assessments as for a one period case. It is possible to determine with sufficient precision how much can be allocated to stocks each period without exceeding the risk of investing in just fixed income issues.
The process for making this assessment uses the well known procedure in financial analysis of backwards recursion. Consider first a two year period and assume that there are not future contributions. First simulate the value that will be obtained after two years when the portfolio each year is invested just in the fixed income issues.
The value obtained at the end of the first year is given by a drawing from a normal distribution with a mean real return of .03 and a standard deviation of .07. This value is reinvested over the second year. Another drawing is made from the same distribution to get the value at the end of the second year, which in this case is the terminal value.
To get a reliable indication of the distribution of terminal values for this paper this process was repeated one million times. And the upper limit on the lowest 5 percent of values was determined. This value provides the desired measure of risk. This same simulation was made a total of 10 times. As will be discussed shortly these repetitions are needed to reliably determine the stock allocations that have the same risk as investing just in the fixed incomes.
The next step is to repeat the simulation of terminal value when a portion of the portfolio at the beginning of each year is allocated to stocks. The real return on the stocks for each year is obtained as discussed in the last section. It is equal to a portion of the drawing for that year for the real return on the fixed incomes plus an independent drawing from another distribution. The latter is a normal distribution with a mean and standard deviation calculated so that the real return on the stocks each year is a drawing from the distribution specified earlier in this paper. This is a distribution with a mean of .07 and a standard deviation of .18.
The stock allocation for the second year of the two year period that has the same risk as investing just in the fixed incomes was obtained in the last section, and is 28 percent. The objective now is to find the stock allocation at the beginning of the first year that has the same 5 percent risk for terminal value over two years as investing just in the fixed incomes. This is using the stock allocation of 28 percent that satisfies this condition for one year as the allocation at the beginning of the second year.
Determining the allocation for the first year of a two year period that satisfies this condition is a trial and error process. Guess a stock allocation for the first year in whole percentage points. Next, run a simulation with a million iterations using this guess as the stock allocation for the first year and 28 percent as the allocation for the second year. Determine the upper limit on the lowest 5 percent of terminal values for these stock allocations.
Suppose this 5 percent value is less than that for any of the 10 simulations when investing just in the fixed incomes. In this case, the initial guess for the stock allocation for the first year is rejected. This allocation is deemed not to have less risk than investing in just the fixed incomes over the two year period. The initial guess is also rejected if the 5 percent value is lower for any one of a total of 10 simulations when the simulation is repeated. If the initial guess is rejected a new lower guess for the stock allocation is tested.
Suppose after 10 trials that a guess is not rejected. This guess is not accepted, however, unless the next higher stock allocation in whole percentage points is rejected by the same process. In view of the latter test, the initial guess is increased for the next simulation if the 5 percent value for the first simulation appears to be too high for the initial guess to be accepted.
Increasing the number of iterations in each simulation reduces the range of the 5 percent values for the simulations. Reducing this range in general reduces the number of simulations required to complete the trial and error process. On the other hand, more iterations, of course, require more time for each simulation.
Using one million iterations for a simulation provides a good balance between the number of simulations and the time for each simulation. The simulations for this paper were made using @Risk 6.2 software and an AMD A10−5745M processor. For a two period accumulation period the run time for a simulation was a little over a minute. The run times increase, of course, with the length of the accumulation period and were over four minutes for 20 periods.
Following the trial and error process just discussed a stock allocation of 36 percent for the first year was easily rejected. It gives a lower 5 percent terminal value than investing just in the fixed incomes. On the other hand, a stock allocation of 35 percent gave a range of the 5 percent terminal values for 10 simulations that was above the range of these values for 10 simulations when investing in just the fixed incomes.
These tests show that the highest stock allocations in whole percentage points that provide less risk than investing in just the fixed incomes over two years are respectively 35 and 28 percent. This is when this risk is viewed from the beginning of either of the two years. The stock allocation for the first year is significantly higher than for the second year.
The stock allocation for the same risk is higher for the first year because of the benefits of hedging over time. The possibility of high returns in one year is hedging against the possibility of low returns in the other year. This hedging reduces risk allowing a larger allocation to stocks to provide the same risk as investing in just fixed incomes. The hedging effect is similar to the hedging for stocks and fixed incomes for a one period investment. This hedging allows an allocation to stocks to have the same risk as investing in just fixed incomes.
The next section shows that adding more periods increases the hedging. This allows larger increases in the allocation to stocks without more risk than investing in just fixed incomes. The benefit of adding more periods, however, gradually diminishes as more periods are added. This diminishing benefit is similar to the diminishing benefit when adding more investments to the portfolio for a one period investment.
Besides the risk, the charts for the one year case also show the improvement in the upper 50, 25, and 5 percent terminal values. The simulations for the two year case also show this improvement. For stock allocations of 35 and 28 percent these values increase respectively by 2.4, 3.2, and 4.4 percent over investing in just the fixed incomes. The improvements become more significant, of course, as the accumulation period increases.
More Than Two More Years
The solution for the two year case indicates how to find the stock allocations for any number of years. For three years, the stock allocations for the last two years are known from the solution for the two year case. And the allocation for the first year of the three years is obtained in a similar way to the first year of the two year case, given the known solution in that case for the last year.
The allocation for the first year of a four year case can then be determined using the known allocations for the last three years. And the allocations can successively be determined in this way for any number of years. Successively carrying out this process for three, four, and five years gives allocations over five years of 49, 45, 41, 35, and 28 percent.
When adding another year or two to the five years the stock allocation at the beginning of the longer period increases by 3 percentage points for each year. To reduce the number of simulations, however, for periods over five years the simulations for longer periods were made in five year segments. And over these five year segments, the stock allocations were assumed in each segment to increase by the same number of percentage points each year. These changes from year to year could vary in terms of half or full percentage points.
With these assumptions, the largest stock allocation at the beginning of 10 years that has no more risk than investing in just fixed incomes is 64 percent. From 64 percent the stock allocations decline by 3 percentage points each year over the following five years. With these declines the allocation becomes 49 percent at the beginning of the last five years as required by the earlier results.
Following the same procedure for the first five years of a 15 year period, the stock allocation at the beginning of the 15 years is 76.5 percent. This allocation declines by 2.5 percentage points each year until it is 64 percent at the beginning of the last 10 years as required by the earlier results. At the beginning of a 20 year period the stock allocation is 84 percent. This allocation declines by 1.5 percentage points a year until it is 76.5 percent at the beginning of the last 15 years as required.
These results illustrate some important general principles with respect to investing over time. The longer the accumulation period, the larger the allocation that can be made to stocks without more risk than investing in just fixed income issues. The size of these increases in stock allocation, however, becomes smaller as the accumulation period becomes longer. This risk refers to the lowest possible 5 percent of values at the end of the accumulation period as viewed from any remaining year in the period.
These principles are explained as the result of hedging over time. Adding an additional year to the accumulation period allows additional hedging of possible low and high returns each year. This increase in diversification allows a larger allocation to stocks without more risk than investing in just fixed income issues. This diversification effect is similar to the hedging that occurs in a single year by combining stocks and fixed income issues in a portfolio.
After more years have been added, however, adding an additional year has a smaller marginal benefit. This declining marginal benefit of hedging over time is also similar to the hedging in a single period investment. After more different investments have been added to the portfolio, there is less marginal benefit in risk reduction by adding an additional different investment.
While these stock allocations do not increase the risk at any time, versus investing in just fixed incomes, they do, of course, significantly increase the wealth accumulated. Compared to investing in just the fixed incomes over 5, 10, 15, and 20 years the median value accumulated increases by 7, 18, 32, and 48 percent respectively. The upper 25 percent of value accumulated increases by 10, 26, 46,and 72 percent. And the upper 5 percent of value increases by 14, 37, 69, and 111 percent.
Future Contributions
The next step is to consider the effect of making future contributions to the portfolio. Future contributions increase the allocations that can be made to stocks without exceeding the risk of investing in just the fixed income issues. The reason is that the future contributions absorb some of the increase in volatility caused by earlier stock allocations.
To facilitate the analysis, it is assumed that the contributions are added at the end of each year, and are constant in real terms. In this case, the effect of the contributions on increasing the allocation to stocks is directly proportional to the size of the annual contribution relative to the current size of the portfolio. This ratio will be referred to as the Contribution Portfolio Ratio (CPR). The proportion of the CPR that adds to the stock allocation increases as the remaining number of years increases in the accumulation period.
This linear relationship substantially facilitates dealing with future contributions in the simulations. The proportion of the CPR for a given accumulation period can be determined with a series of simulations for a single value of the CPR. And this proportion gives the appropriate increase in the stock allocation for this accumulation period whatever the value of the portfolio turns out to be for that period.
These proportions are determined for successively longer periods using a similar backwards recursion process to that used earlier when there are not any contributions. The allocation for a one year period is unaffected by a contribution and remains equal to 28 percent. Next, find the allocation for a 2 year period when there is a future contribution equal in real value to the initial value of the portfolio so that the CPR is 1.0.
The first step is finding the lowest 5 percent of terminal values for the specified contribution when investing just in the fixed incomes. Next, use the same trial and error process as before to find the stock allocation for the first year given the allocation of 28 percent for the second year. For the first year this is the largest stock allocation in whole percentage points that gives 5 percent terminal values that are higher than those obtained when invested just in the fixed incomes.
With an annual contribution equal to the initial value of the portfolio, a stock allocation of 46 percent for the first year is rejected. An allocation of 45 percent, however, gives a range of 5 percent terminal values above the range for investing in just the fixed incomes. Without the contributions, the earlier tests gave a stock allocation for the first year of 35 percent.
Thus, the contributions increase the stock allocation by 10 percentage points. This increase is equal to 0.1 of the ratio of the annual contribution to the current portfolio value, which is 1.0. Running the same test when the contribution is half the initial portfolio reduces the increase in the stock allocation by a half. These tests confirm the linearity of the relationship.
For the first year of a two year period, the stock allocation is 35 percent plus 10 percent of the initial CPR. For a three year period, the stock allocation for the second year is given by this relationship and depends on the value of the CPR at the beginning of the last two years. The stock allocation for the first year of the three year period is based on simulations that take into account an assumed value of the CPR at the beginning of the three years.
The stock allocation for the first year of a three year period is 41 percent plus 22 percent of the CPR at the beginning of the three years. Successively increasing the accumulation period by one year each time, the proportion of the CPR to add can be determined each year in this same way for a period of any number of years. The proportions for accumulation periods from 1 to 10 years are respectively 0, .10, .22, .38, .56, .76, 1.00, 1.26, 1.54, and 1.84.
A stock allocation can be no more than 100 percent in this analysis as borrowing is not allowed. For a CPR of 1.0, the limit on the stock allocation of 100 percent is applicable for any accumulation period of five years or more. In these cases it is not possible to invest enough in stocks initially to provide as much risk as investing just in the fixed income issues. This last statement assumes that the stock allocations will be reduced below the limit of 100 percent when that becomes possible in the future.
The CPR tends to decline over time because the contribution is fixed and the portfolio is increasing. Eventually, the stock allocations determined earlier will necessarily become applicable. For a five year period when the CPR is 1.0, for instance, the stock allocation at the beginning of the next year is sure to be less than 100 percent.
Contributions reduce the effective length of the accumulation period. Thus, it is not surprising that contributions reduce the improvement in terminal value. This is when investing as much as possible in stocks without more risk than investing just in fixed income issues. Consider a 10 year period and assume that the contributions are equal to the initial value of the portfolio so that the CPR is 1.0.. The median value increases by 10.5 percent in this case compared to investing just in the fixed incomes. Without any contributions the increase is 18 percent.
Including contributions tends to increase the variability of the 5 percent terminal values obtained with the simulations. For the contributions, therefore, the number of iterations was increased from one to two million. And the number of simulations for each case to establish rejection was reduced from 10 to 5. For a 10 year period with 2 million iterations the run time for a simulation is about six minutes with the processor given earlier.
Menu for Increasing Rewards and Risks
The focus so far has been on the largest stock allocations over an accumulation period that have less risk each year than investing just in fixed income issues. Presumably, almost everyone will be willing to accept at least this much risk to get terminal values for wealth that are as large as possible. Some, however, will be willing to accept more risk to increase their chances of getting higher values.
When increasing risk, it seems reasonable that risk should be increased uniformly over each of the years of the accumulation period. A uniform increase can be approximately achieved by increasing the allocations obtained earlier by the same number of percentage points each year. Gradually raise these sets of increases. Simulating terminal wealth for each set of these increases generates a menu of increasing rewards and risks.
Such a menu is shown in Chart 3. This chart assumes a 10 year accumulation period and contributions at the end of each year equal in real terms to 10 percent of initial portfolio value. The vertical axis shows terminal wealth in real terms as a percentage of initial portfolio value. The horizontal axis shows the percentage points by which the stock allocation has been increased for each year. This is an increase above the allocation that provides the same risk of terminal wealth for each year as investing just in the fixed income issues. The increases are limited, however, in cases where they would put a stock allocation over 100 percent.
Chart 3. Increasing rewards and risks as the percentage points shown on the horizontal axis increase the stock allocations for each year. The increases are limited, however, when the allocation would become more than 100%. Vertical axis shows terminal wealth in real terms as a percent of initial portfolio value for accumulation over 10 years. This is when a contribution in real terms is made at the end of each year equal to 10% of the initial portfolio. The rewards and risks on the far left are for allocating to stocks each year so that the risk is the same as investing in the fixed incomes. The lower solid line shows the upper limit on the lowest 5% of terminal wealth. The upper three solid lines show the lower limits on the highest 5%, 25%, and 50% of terminal wealth. The dashed lines show these limits when investing in just the fixed income issues.
The total contributed value in terminal wealth is 200. This includes the 100 of initial portfolio value and a total of 100 for the contributions of 10 at the end of each of the 10 years. Note on the far left of the chart that there is a .05 chance that terminal wealth will be slightly less than the total contributed value of 200. This is irrespective of whether investing in just the fixed income issues or using the stock allocations over the 10 years that have the same risk and were obtained in the last section.
When investing in just the fixed incomes the lower limits on the upper 50, 25, aand 5 percent of terminal values are shown by the short dashed lines in the chart. As expected, when using the stock allocations obtained in the last section the lower limits on the upper 50, 25, and 5 percent of terminal values are substantially higher. These values are given by the intersection of the vertical axis with the three solid upper lines.
Moving to the right, the solid lines show the increasing risk and rewards when the stock allocations are increased uniformly over each of the 10 years. There is a small increase in risk shown by the small decline in the lower line. The median values show a comparable small increase to the small increase in risk. The upper 25 and 5 percent of values show larger improvements. Those accumulating wealth will have to decide whether this additional improvement is worth the added risk.
Conclusions
When investing over time hedging occurs as low returns in one year can be offset by high returns in another. This diversification over time allows higher allocations to stocks at the beginning of longer periods. This is without more risk of low values of terminal wealth than occurs when investing in just fixed income issues over the same period.
This paper has used simulated investment returns to make quantitative estimates of how much stock allocations can be increased without increasing such risk. This is as the length of the accumulation period increases. The simulations also show the increased terminal wealth obtained as a result of the higher stock allocations.
Another factor that increases the stock allocations that can be made without increasing risk is the size of future contributions to a portfolio. The larger are future contributions relative to the current value of the portfolio, the larger are the stock allocations that can be made without increasing risk. The reason is that future contributions absorb some of the volatility caused by an increase in earlier stock allocations. Simulations in this paper have estimated the size of this effect and show that it can be very large.
Some investors are willing to accept more risk than investing in just fixed income issues to further improve their prospects. Simulations in this paper also show the increase in terminal wealth obtained by accepting more risk. This is more risk for low values of terminal wealth than when investing in just fixed income issues as viewed from each year of the accumulation period.
Posted May 2014
Addendum*
Addendum** of Sensitivity introduces a set of low return assumptions consistent with the real returns realized on intermediate term U.S. government issues since 2008. These low returns reduce from 3.5% to 2.5% the initial disbursement that provides a sustainable stream of disbursements over retirement under the conditions otherwise generally assumed to illustrate disbursements at this site. Reducing the initial disbursement from 3.5% to 2.5% means that a 40% larger portfolio will be required to get a spending stream of the same size with the low returns. This addendum shows that almost 40% more will also have to be saved over a 20 year period to accumulate a given amount of funds with these low returns. As 40% more savings are required to provide 40% more funds, these effects are compounded. Almost twice as much savings will therefore be required for retirement with the low returns.
Suppose a stream of savings for retirement is assumed to be a stream of contributions for 20 years each of which is equal in real terms to the initial value of the portfolio. The text obtains the stock allocations for the last 10 years of such a 20 year stream using two backwards recursive processes. One of these finds the stock allocation each year when there are not any contributions. The other finds the multiple of the ratio of the contribution to portfolio value (CPR) each year that is added to the stock allocation when there are contributions. These processes are described in detail in the text. The stock allocations for a 20 year stream are obtained by simply continuing these backwards recursive processes for 10 more years. The stock allocations for the low returns are obtained by simply repeating these backwards recursive processes using the low returns. (1)
In the early years for a flat stream of contributions the stock allocations given by the recursive processes can be over 100%. In such cases, the stock allocations are limited to 100% as leverage using the intermediates is not allowed. Future contributions in these cases are absorbing sufficient volatility to make stocks less risky than intermediates due to the much higher expected returns of the stocks. In the future, the value of the portfolio increases relative to the value of the contribution. Eventually, the contributions are no longer large enough relative to the size of the portfolio to absorb enough volatility, and the stock allocations given by the recursive processes decline below 100%.
For the low returns, the stock allocations are smaller than for the higher returns when only a few more years remain for accumulation. These smaller allocations occur because the premium expected return for the stocks over the intermediates is assumed to be the same for the low returns, but the volatility of the return on the stocks declines relatively less making the stocks relatively less valuable. In particular, for the low returns the expected return for the stocks declines from .07 to .04, and for the intermediates by the same amount from .03 to .00. In contrast, the standard deviation for the stocks declines from .18 to .14, whereas for the intermediates the decline is relatively more being from .07 to 04.
On the other hand, the increase in the stock allocations for the low returns is stronger as the time for accumulation increases. Eventually, when the time for accumulation becomes sufficiently long, the stock allocations become higher for the low returns. Note for either the stocks or intermediates that volatility increases relative to expected return for the low returns. Thus, the volatility of the portfolio relative to expected return is higher for the low returns. With volatility higher relative to expected return, the reduction in volatility provided by hedging over time and by future contributions makes the higher expected returns of the stocks more desirable, increasing the allocation to stocks.
Given the stock allocations when saving over 20 years for the low and higher returns the value accumulated in either case can be simulated. The cumulative probabilities of the funds after 20 years as a percent of the total funds contributed are shown in Chart 4. For the higher returns, the median value increases from 125% to 171% of contributed value, or by 37%. To accumulate the same median value for a retirement portfolio, 37% more will have to be saved with the low returns. For the lowest 20th percentile, 25% more will have to be saved, and for the highest 20th percentile, 49% more will have to be saved. Thus, the central tendency is that an increase in savings is required that is almost equal to the 40% increase in funds needed for the same sustainable stream of spending with the low returns. Compounding these two effects, the low returns almost double the savings over a 20 year period required to fund a sustainable stream of spending in retirement.
Note
As discussed in the text, a trial and error process is used to determine for any year the largest stock allocation in whole percentage points that has a 5 percentile terminal value that exceeds the 5 percentile terminal value when the portfolio is entirely invested in intermediates. For the new results, this condition is considered to be satisfied when it is true for each of five simulations with 1 million iterations. As assumed in the text, before the last five years when there are not any contributions, trials are conducted for five year segments with changes in terms of either half or full percentage points. When determining the multiples for the CPR with more than 10 years remaining for accumulation, the multiples were determined for two years at a time assuming the intervening multiple was half way in between. For the higher returns when starting initially with 20 years remaining and using the stock allocations determined by the backwards recursive processes, the stock allocation is limited to 100% for every iteration for the first five years. The allocation is limited to 100% for at least one iteration until after 10 years. For the low returns, the stock allocation is limited to 100% for every iteration for the first seven years, and for at least one iteration until after 11 years.
*Addendum posted December, 2020. Combines and simplifies Addendum* posted August, 2020, and Addendum** posted October, 2020.