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An Asset Allocation Puzzle: Comment
by Isabelle Bajeux-Besnainou, James V. Jordan, Roland Portait
An Asset Allocation Puzzle: Comment
Isabelle Bajeux-Besnainou, James V. Jordan, Roland Portait
The American Economic Review
Updated: August 2nd, 2012
An Asset Allocation Puzzle: Comment
Should the proportion of risky assets in the risky part of an investor's portfolio depend on the investor's risk aversion? According to basic financial theory, in particular the mutual-fund separation theorem with a riskless asset, the answer is no. The theorem states that rational investors should divide their assets between a riskless asset and a risky mutual fund, the com- position of which is the same for all investors. Risk aversion affects only the allocation be- tween the riskless asset and the fund.
However, Niko Canner et al. (1997), CMW hereafter, observed that popular investment ad- vice does not conform to this theory. They reported the stocks, bonds, and cash allocations recommended by four advisors for conservative, moderate, and aggressive investors. As shown in Table 1, which is reproduced from CMW, the advisors recommend a bondlstock ratio that varies directly with risk aversion. For example, Fidelity recommends a bondlstock ra- tio of 1.50 for a "conservative" (more risk- averse) investor, a ratio of 1 .OO for a "moderate" (less risk-averse) investor, and a ratio of 0.46 for an "aggressive" (still less risk-averse) inves- tor. The inconsistency between such advice and the separation theorem is called an asset alloca- tion puzzle by CMW. They attempted to solve the puzzle by relaxing key assumptions in the theory, but finally reached a negative conclu- sion: "Although we cannot rule out the possi- bility that popular advice is consistent with some model of rational behavior, we have so far been unable to find such a model" (p. 181). However, they suggested that consideration of intertemporal trading might help resolve the puzzle.
In the present paper, we provide theoretical
* Bajeux-Besnainou: Department of Finance, School of Business and Public Management, George Washington Uni- versity, 2023 G Street NW, Washington, DC 20052; Jordan: National Economic Research Associates, 1255 23rd Street NW, Washington, DC 20037; Portait: CNAM and ESSEC, Finance Chair CNAM, 2 Rue Conte, Paris, France. This research was supported by a grant from the Institute for Quantitative Investment Research. We thank two anony-
nou us referees for their comments.
support for the popular advice. The two key insights are that the investor's horizon may ex- ceed the maturity of the cash asset and that the investor rebalances the portfolio as time passes. If the investor's horizon exceeds the maturity of cash, which might be a money-market security with maturity of one to six months, then cash is not the riskless asset as is commonly assumed in the basic theory. In a theory allowing portfolio rebalancing, as opposed to a buy-and-hold framework, it is not unreasonable to assume that the investor can synthesize a riskless asset (a zero-coupon bond maturing at the horizon) us- ing a bond fund and cash. Then bonds will be both in the (synthetic) riskless asset and in the risky mutual fund and we show that in this case the theoretical bondlstock ratio varies directly with risk aversion for any hyperbolic absolute risk aversion (HARA) investor.' As an example of the type of results that a specific model can produce, we provide a continuous-time model with closed-form solutions, which produces the- oretical bondlstock ratios similar to the popular advice.
The present paper is organized as follows: in the next section, we analyze the popular advice in terms of the theory of mutual-fund separation of David Cass and Joseph E. Stiglitz (1970). We show that this theory is relevant both in static and dynamic frameworks and use it to analyze the popular advice in complete and incomplete markets. In Section 11, we analyze the popular advice in the context of Robert C. Merton's (1971) continuous-time statement of mutual-fund separation and present an illustrative model in which a CRRA investor makes continuous-time portfolio decisions under inter- est rate and stock price uncertainty. In Section 111, numerical results are compared with the popular advice. Section IV is a conclusion.
' HARA functions include quadratic utility, which is one way of justifying mean-variance preferences, and constant relative risk aversion (CRRA) utility. Both quadratic and CRRA utility were considered in the CMW analysis.
Percent of portfolio Ratio Advisor and of bonds investor tv~e Cash Bonds Stocks to stocks
Fidelity Conservative Moderate Aggressive
Merrill Lynch Conservative Moderate Aggressive
Jane Bryant Quinn Conservative Moderate Aggressive
New York Times
Sources: Jane Bryant Quinn (1991); Larry Mark (1993); Don Underwood and Paul B. Brown (1993); Mary Rowland (1994).
I. Cass-Stiglitz Separation, Dynamic Strategies, and Investor Horizon
After discussing the main features of a frame- work adequate for analyzing the puzzle, we provide an explanation of the popular advice grounded on the Cass-Stiglitz (1970) separation theory.
A. An Adequate Theoretical Framework for Analyzing the Popular Advice
Analyzing the popular advice as represented in Table 1 requires careful attention to four aspects of the investment problem: (1) the in- vestor's horizon, (2) the tradable asset classes included in the popular advice (the "investment basis"), (3) the sources of risk considered, and
(4) the self-financing strategies (funds) ob- tainable from the investment basis. First, specifying the investment horizon is neces- sary for identifying a riskless asset. Although it is common in buy-and-hold analyses to treat cash as the riskless asset, this implies that the investor's horizon is the same as the maturity of a money-market security, perhaps as short as one to six months. Because the popular advice may address investment over several years, we make the less-restrictive assumption that the investor's horizon is T, which may be equal to or greater than the maturity of cash. Second, because the popular advice involves stocks, bonds, and cash, we define the investment basis as consisting of a stock fund (S), a bond fund (B,) with dura- tion K, and a money-market fund designated as cash (C) with maturity E. Here, a fund's duration is defined as in John Cox et al. (1979), namely that a fund with duration K has the same risk (and return) from interest rate changes as does a zero-coupon bond with maturity K. Third, a theoretical framework adequate to address the three-asset allocation problem should involve, at least, interest rate risk and stock market risk. Fourth, the popular advisors probably would assume that portfo- lios will be rebalanced over long horizons and therefore the analysis should not preclude rebalancing.
Therefore, the most parsimonious framework adapted to an analysis of the popular advice involves three assets, interest rate risk, and stock market risk and allows rebalancing over the investment period. We claim that portfolio theory based on such a framework is compatible with the popular advice.
B. A JustiJication of the Popular Advice Based on Cuss-Stiglitz Separation Theory
Cass and Stiglitz (1970) derived the condi- tions for two-fund separation. In complete mar- kets, two-fund separation holds for a broad class of utility functions, including HARA functions. In incomplete markets with a riskless asset, two-fund separation holds only for HARA func- tions (without a riskless asset, it holds only for quadratic and CRRA preference^).^ Although Cass-Stiglitz two-fund separation is usually ap- plied in a static framework without consider- ation of portfolio rebalancing, it is also valid for analyzing dynamic strategies. With continuous rebalancing, each admissible self-financing strategy can be viewed as a contingent claim; these contingent claims, infinite in number, can be considered as the traded securities in a static framework. The Cass-Stiglitz separation results
Two-fund separation was also shown by Fischer Black (1972) for the mean-variance case.
can then be applied in this infinite-claim, static frame~ork.~
When two-fund separation is used to analyze the puzzle, the cases T = E and T > E must be distinguished. It is clear that the puzzle exists when T = E. Cash is then the riskless asset (at least when inflation uncertainty is ignored) and two-fund separation holds for the broad class of utility functions in complete markets or for the HARA class in incomplete markets. The two funds are cash and a risky fund consisting of the bond fund and the stock fund. Within the risky fund, the weights are the same regardless of the investor's risk aversion. Theory and the popular advice are indeed inconsistent in this case.
For T > E, we consider first the case of complete markets. The assumption of complete markets would be unrealistic if rebalancing were not allowed. Moreover, to have complete markets with an investment basis containing only three assets, no more than two sources of risk, such as interest rate and stock market risk, can be considered. If we ignore inflation uncer- tainty so that interest rate risk means real rate risk, then, for T > E, the riskless asset is a zero-coupon (nominal) bond (B,) that matures at the investor's horizon. Because this bond does not belong to the investment basis, the
More precisely, consider a standard dynamic continuous- time problem (P) of pure portfolio optimization (without consumption) between 0 and T, the control variables being the portfolio weights x (the corresponding strategies are constrained to be self-financing)
(PI Max E(U[X(T)I); X(0) = X,,
where X, is the initial endowment and X(7') is terminal wealth. Stanley R. Pliska (1986), Ioannis Karatzas et al. (1987), and Cox and Chi-fu Huang (1989) showed that (P) is equivalent to a two-step optimization. Step 1, solve (P')
Max E(U[X(T)]) s.t. EQ
EQ is the risk-neutral expectation (taken at time 0) and C(T) is the value at T of $1 invested in cash at time 0. (P') yields the optimal terminal wealth X"(7') but not the strategy (asset weights) that attains it. Step 2, find the admissible strategy ~"(t) that produces the optimal X*(T). From the equivalence between (P) and (P'), it is clear that the dynamic optimization (P), which involves continuous re- balancing of a finite number of assets, is equivalent to the static problem (P'), which involves an infinite number of contingent claims X(T) from which the (static) choice must be made.
Synthetic Riskless Risky Fund Fund
( replicated with
Note: C = cash, B, = constant K-duration bond fund, B, = bond with initial maturity (T) equal to investor horizon, S = stock fund.
investor must synthesize it in a replicating port- folio that is a dynamic combination of C and B,.~ The risky fund is a dynamic combination of C, B,, and S. This case is illustrated in Figure 1.
In this situation, the bondstock ratio varies directly with risk aversion. For example, a less risk-averse investor will hold less of the syn- thetic riskless fund and therefore less of B,.* Because the weights of B, and S within the risky fund do not change, the total bondstock (B,IS) ratio will decrease. Therefore, when T > E, markets are complete and the invest- ment basis is restricted to consist of the three assets C, BK, and S, two-fund separation theory supports the popular adviceS6
We assume implicitly that B, can be replicated using only C and B,. Therefore, not only is the market complete but the bond market itself can be spanned with only two securities (C and B,). Synthesizing B, requires a positive weight on B, (positive on C if K > T, negative if K < T). The replication of a zero-coupon bond maturing at T by a dynamic combination of fixed-income securities of different durations (to obtain at each date t a portfolio duration of T -t) is known among bond portfolio managers as "pas- sive immunization." See, for example, H. Gifford Fong (1990) and Frank J. Fabozzi (1996).
It should be noted that there is a restriction to the Cass-Stiglitz two-fund separation. When considering dif- ferent HARA investors characterized by a marginal utility of wealth u'(W) = (A + BW)-", where A, B, and c are utility parameters and for which risk aversion is BcI(A + BW), two-fund separation holds only if parameter c is held constant (see, for example, Jonathan E. Ingersoll, Jr., 1987
If the riskless bond B, is included in the investment basis, then the riskless fund consists of B, only and the risky fund consists of C, B,, and S. Within the risky fund, weights are the same regardless of risk aversion, therefore
What if markets are incomplete when T > E? This could be the case if rebalancing were excluded or if additional sources of risk such as inflation uncertainty were included in the framework. In this case, the risk-free asset cannot be synthesized and the theory yields ambiguous results. It can be shown numeri- cally, for instance, that the optimal bond/ stock ratio for a buy-and-hold investor may vary either directly or inversely with risk aversion, depending on the choice of market parameters. Moreover, the results are extremely sensitive to the assumed values of the parameters.7 Because the theory in the incomplete-markets case does not yield a def- inite prediction about the relationship between risk aversion and the bondlstock ratio, it is not clear that a puzzle exists.'
No ambiguity exists, however, if the complete- markets analysis is acceptable. The main assumptions that support the complete-markets analysis are that rebalancing is allowed, that the bond with a maturity matching the investor's horizon can be adequately synthesized (with a dynamic combination of cash and the bond fund), and that inflation uncertainty can be ig- nored. When dynamic strategies are considered, the ability to create contingent claims through rebalancing suggests that the assumption of dy- namic completeness may reasonably approxi- mate actual markets (see Darrell Duffie and Huang, 1985). Dynamic strategies are not merely a theoretical construct. They are required in many portfolio applications, such as
the ratio B,IS is the same for all investors. However, the ratio of all bonds (B, and B,) to stock increases with risk aversion. Therefore, with this broader definition of the in- vestment basis, it might be claimed that there is no incon- sistency with theory. However, we cannot be sure what the popular advice would be if the investor horizon and the corresponding zero-coupon bond were part of the advice. We can only say that such advice remains unobserved.
See able A1 in Appendix A on the American Economic Review web site.
There is a case of incomplete markets in which no puzzle exists. As long as riskless asset B, can be synthe- sized by dynamic combinations of C and B,, the analysis provided in the complete-markets case is valid. There could be any number of sources of risk affecting the stock market that are not hedgeable by the stock fund in the investment basis. Although the market would then be incomplete, as long as these sources of risk did not prevent the synthesis of B, using C and B,, the bondlstock ratio would increase with risk aversion.
asset allocation funds, portfolio insurance, and portfolio immunization. The neglect of inflation uncertainty is perhaps more troublesome, espe- cially when long horizons are considered, but some empirical evidence (George Pennacchi, 1991) indicates that in the United States in- flation volatility is smaller than real interest rate volatility.9 For all of these reasons, the complete-markets analysis appears to be reasonable.
11. Continuous-Time Analysis
In this section, an additional theoretical jus- tification of the popular advice, based on Mer- ton fund separation, and an illustrative model are provided.
A. A Just$cation of the Popular Advice
Based on Merton Separation Theory
Within an explicit dynamic framework, the multiple fund separation characterized by Mer- ton (1971) provides an alternative theoretical explanation of the asset allocation puzzle. This approach also allows an analysis of the popular advice in a framework involving intermediate ~onsum~tion.'~
In a continuous-time economy with n state variables, all optimal portfolios can be constructed from n + 2 funds. One fund is the instantaneously riskless portfolio. Another fund is the growth-optimal or logarithmic port- folio, the portfolio that maximizes the expected logarithm of wealth. The n remaining funds are portfolios with returns perfectly correlated with the state variables. These last n funds allow hedging against adverse shifts in the investment
Set. We consider one-state-variable model (n = 1) that implies complete markets when the investment basis
three assets. in ~ ~ r t ~ ~ ,
the instan- taneous interest rate is taken to be the single state variable driving the investment opportu- nity set.
In this context, the intertemporal consumption- portfolio choice problem of an individual inves- tor may be written
we thank an anonymous referee for bringing this re- search to our attention.
lo Merton's analysis requires no restrictions to the way risk-aversion parameters may vary, as in the Cass-Stiglitz approach (see footnote 5).
Riskless Growth-Optimal Hedging Fund Fund Fund
Note: C = cash, B, = constant K-duration bond fund, S = stock fund.
Max I?[\-u(c(t)) dt + U(X(T))
where u and U are "well-behaved" utility func- tions, c(t) is the consumption stream, r(t) is the instantaneous risk-free rate, x(t) is the vector of weights on risky assets S and B,, dR(t) is the vector of their instantaneous rates of return, 1is the unit vector, X(t) is the wealth process, and X, is the initial wealth. Merton (1973) showed that the solution x*(t) of this consumption- portfolio problem can be written, for any indi- vidual, as a dynamic combination of three mutual funds. These thee separating mutual funds are the instantaneous risk-free security that we define as C, the growth-optimal portfo- lio containing C, B,, and S, and the hedging portfolio that contains only B, because the bond return is perfectly correlated with the state variable." Figure 2 illustrates this case.
The puspose of the hedging portfolio is to hedge against adverse shifts in the investment opportunity set, which in this model are trig- gered by decreases in the interest rate (the ex-
'' The prices of all of the zero-coupon bonds are locally perfectly correlated with the interest rate (the only state variable).
pected returns of all of the securities decrease when the interest rate decreases, thereby wors- ening investment opportunities).12 Because a decrease in the interest rate causes an increase in the value of B,, the weight of B, within the hedging portfolio should be positive. With in- creasing risk aversion, the weight on the hedg- ing portfolio should increase (increasing B,) and the weight on the growth-optimal poi-tfolio should decrease (decreasing both B, and S). Therefore, the bondlstock ratio will increase. This analysis supports the conclusions derived from Cass-Stiglitz separation.
In addition, it is known that the growth-optimal portfolio is highly aggressive and thus levered (negative weight in cash).13 Increasing risk aversion implies decreasing weight in the growth-optimal portfolio and, consequently, in- creasing weight in cash. This relationship be- tween risk aversion and cash weight also conforms to the popular advice.
B. An Asset Allocation Model Supporting the Popular Advice
The following one-state-variable model yields closed-form solutions that will be used in Sec- tion I11 to generate numerical results to compare with the popular advice.14 Markets are assumed to be arbitrage-free, frictionless, and continu- ously open between dates 0 and T. The invest- ment basis consists of thee assets: an instantaneously riskless money-market fund with the price at t given by C(t), a stock index fund with price S(t), and a bond fund paying $1 at maturity with price B,(t) that maintains a constant duration equal to K through continuous rebalancing during the period 0 to T.
The investment opportunity set is a function of one state variable, the instantaneous siskless inter- est rate r that follows an Ornstein-Uhlenbeck pro- cess with constant parameters given by
l2 A decrease in the interest rate decreases the expected return on all securities provided that the risk premiums are constant or, if dependent on the interest rate, that they do not increase enough to offset the effect of the rate decrease.
l3 The growth-optimal portfolio maximizes the expected continuously compounded growth rate of portfolio value subject to no constraint on risk.
The model is similar to the one presented in the last section of Bajeux-Besnainou and Portait (1998).
where a,, b,, and a, are positive constants and dz, is a standard Brownian motion. The market price of interest rate risk is assumed to be con- stant; thus, this is an Oldrich A. Vasicek (1977)type market.
The stochastic processes for the securities are given by
where 0, is the risk premium of the bond fund, which is a constant because the market price of risk is assumed to be constant and the duration of the fund is maintained as a constant, 0, is the risk premium of the stock, also assumed to be constant, a,, a,, and a, are positive constants, and dz is a standardized Brownian motion or- thogonal to dz,. This model is a simple exten- sion of Black-Scholes and a variant of the Merton one-state-variable model. It allows for stochastic fluctuations of interest rates that are negatively correlated with bond and stock re- turns and that induce parallel movements of the expected returns of all of the risky assets.15 The framework implies that the financial markets are dynamically complete.
Portfolio weights are given by xi(t),i = C, S,and K. The corresponding portfolio value is X(t). Our main interest in this article is the ratio x,(O)lx,(O). The choice theoretic prob-
l5 An implication of the Vasicek model is that
The Ornstein-Uhlenbeck process [equation (I)] is writ- ten with a negative sign on the volatility parameter. Because all volatilities in the model are positive [in particular, we assume that u,in equation (3) is positive and a, in equation
(4) is positive according to the previous formula], this negative sign induces the negative correlation between in- terest rates and stock and bond returns that is typically observed.
lem of an investor with constant relative risk aversion can be stated as the following opti- mization program:
The solution can be written16
The constants hs and h, are the weights of the stock index and the bond index, respec- tively, in the growth-optimal portfolio. The val- ues of the parameters h,, h,, and a,_, can be calculated and are given by
with h and A, ilnplicitly given by
0, = a,h, and 0, = alh + a,h,
l6 See Appendix B on the Aiilericciil Ecor?orilic Reviekt web site.
The optimal weights for the CRRA investor are deterministic functions of time and are inde~en- dent of the level of the interest rate.17
The optimal bondlstock ratio at any point in time is
This ratio is increasing in investor risk aversion and thus is consistent with the popular advice. This result holds at any time t and for any value of the parameters. Moreover, the weight in cash conforms to other aspects of the popular advice. Indeed, x,(t) in equation (5) is decreasing with a,-, and therefore increasing with t as long as risk-aversion parameter y is greater than 1. Therefore, in conformity with another aspect of the conventional wisdom, the investor will in- crease the weight in cash as time passes and the horizon gets closer. It can also be shown that the weight in cash increases with risk aversion for all reasonable values of the parameters, in con- formity with the popular advice (see Section 111).
The consistency between this model and the popular advice was predictable because the un- derlying framework implies that financial markets are complete and the theoretical justi- fication, based on Merton's separation theory, applies. The analysis has shown that a continu- ous-time portfolio model, grounded on the sim- plest framework involving stock market risk and interest rate risk, supports theoretically dif- ferent aspects of the popular advice. It is also useful to see whether the numerical weights produced by this model resemble the weights in the popular advice and to explain any differ- ences that arise.
111. Comparing Model Weights to the Popular Advice
Table 2 contains the results for CRRA inves- tors with horizons ranging from three months to 10 years. The bond fund constant duration is K = 10 years. The assumed instantaneous in- terest rate is r, = 4 percent. The parameters of the Vasicek model, based on K. C. Chan et al.
l7 These features are specific to CRRA utility functions.
TABLE 2-ASSET ALLOCATIONSFOR CRRA INVESTORS: BASE CASE
Percent of portfolio Ratio
Notes: The bond fund constant duration is K = 10 years. The current instantaneous interest rate is r, = 4 percent. The parameters of the interest rate stochastic process (Chan et al., 1992) are a, = 18 percent, b, = 4 percent, and a, = 2 percent. The other assumptions are 8, = 1.5 percent, 8, = 6 percent, a, = 19 percent, and a, = 6 percent.
(1992), are a, = 18 percent, b, = 4 percent, and a, = 2 percent.18 The risk premia are assumed to be 6 percent for the stock index and
1.5 percent for a 10-year bond fund; the assumed volatilities are a, = 19 percent and a, = 6 percent. Therefore the stock volatility, as =
(a: + is 20 percent and the correlation between stock returns and changes in the short rate, a,/as, is -0.30.
l8 Although there are more recent parameter estimates for the Vasicek model, there does not appear to be a con- sensus in the literature. For example, Robert R. Bliss and David C. Smith (1998) reported a lower rate of mean reversion and lower volatility when the period of the Fed- eral Reserve Board policy change (October 1979-Septem- ber 1982) is excluded. The parameter values are a, = 9.29 percent, b, = 10.65 percent, and a, = 1.4 percent. The model results with these parameters are very similar to the results reported in Table 2.
The bondstock ratios decrease with risk aversion. As expected, only as the horizon de- creases toward the maturity of cash, which in this model is infinitesimal, does the ratio ap- proach a constant. For instance, for a one-year horizon, as risk aversion decreases from y = 8 to y = 2, the ratio decreases from 1.60 to 0.75. In Table 1, the average ratios for the four advi- sors decrease from 1.20 to 0.25 as risk aversion decreases.
As noted in the theoretical analysis, the model results also correspond to the popular advice in that the weight in cash decreases as risk aversion decreases. For shorter horizons and higher risk aversion, the weight in cash remains positive. For longer horizons and lower risk aversion, the investor may lever the port- folio by borrowing cash.19 This deviation from the popular advice is not surprising for two reasons. First, it is not abnormal that a model containing no short-sale constraints yields le- vered positions (negative weight in cash) for aggressive investors. Indeed, for long horizons, the T-maturity bond is the riskless asset and cash is a risky asset (with a negative risk pre- mium). For less risk-averse investors, the diver- sification obtained through cash is not sufficient reason to hold such an asset and a short position in cash is optimal. It is likely that the popular advice, which is intended to apply to all inves- tors regardless of wealth, sophistication, and investment experience, does not consider the possibility of levered portfolios. A model with short-sale constraints could more easily produce agreement with this aspect of the popular ad- vice. Second, a model that would integrate in- flation uncertainty would probably yield higher cash weights, in particular for long horizons. Indeed, rolling over a money-market instrument yields a real return that is less sensitive to un- expected changes in inflation than is a long-term bond.20
Table 2 also shows, for a given investor (val- ue of y), how portfolio weights are modified as time passes. All investors with relative risk aversion exceeding 1 will increase determinis- tically the weight in cash as the horizon gets
l9 In Table 4, we show a choice of reasonable assump- tions that produces closer agreement with the popular ad- vice, including no leverage.
20 We thank an anonymous referee for reminding us of the role of cash as an inflation hedge.
closer. This feature of the optimal portfolio strategy partially conforms to another form of popular advice cited in CMW: as an investor's horizon gets closer, the investor should switch from risky assets to cash. Because of the CRRA assumption, Table 2 shows a constant weight in stocks at all horizons, holding risk aversion constant. This is a generalization of Merton's (1973) result of a constant stock weight for CRRA investors when interest rates are constant and the stock price follows geometric Brownian motion.21 Because of the constant stock weight, in our model the investor reallocates money from bonds to cash as the horizon draws closer.
Table 3 provides some indications of the sen- sitivity of the results to different parameters for an investor with a five-year horizon.
Increasing interest rate volatility causes the investor to shift money from the 10-year bond fund into cash and stock and the bondstock ratio decreases. This result can be explained by the fact that the instantaneous bond fund return becomes more risky for the same risk premium (8, = 1.5 percent). At this higher interest rate volatility, the cash weight is positive for all but the most aggressive (here, y = 2) investor.
Decreasing the speed of mean reversion (a,) is similar to increasing interest rate volatility because rates are more likely to diverge from the long-run mean. The weights and the bond stock ratio move in the same direction, as in the case of an increase in interest rate volatility. In both cases, leverage is avoided except for the most aggressive investor (y = 2).
Changing the long-run mean (b,) has no ef- fect. This is linked to the assumption of CRRA utility, which yields portfolio weights that are mainly a function of the risk premia and are independent of the interest rate level.
Because the popular advisors do not specify the duration of the bond fund, the effect of changing the duration of the fund is also worth noting. When the bond fund duration equals the investor horizon, the cash weights are all nega- tive because the investor borrows cash (which is risky) mainly to invest in the now-riskless bond fund that yields a higher expected return. In this case, by virtue of the two-fund separation
Constant stock weights would not be obtained with the more general HARA utility functions. The results are avail- able from the authors.
TABLE3-NEW ASSET ALLOCATIONS
WHEN PARAMETERS AND ASSUMPTIONS FROM THE BASE CASE (TABLE
DIFFER 2) FOR HORIZON((T) OF 5 YEARS
Percent of portfolio Ratio Investor type and of bonds investor horizon Cash Bonds Stocks to stocks
Incrense Volatility of Interest Rntes (a,) from 2 to 4
Decrease Speed of Men11 Reversion (a,) from 18 to 10 Percent
8 24 58 18 3.14 5 16 55 29 1.87 2 -17 43 73 0.59
Cllnnge in b,
8 10 73 17 4.18 5 -2 74 28 2.65 2 -47 78 70 1.12
Decrense Bond F~md Duration (K)from 10 to 5 Years
8 -20 102 17 5.87 5 -32 104 28 3.72 2 -79 109 70 1.57
Increase Bond Fund D~rrntion (K)frorn 10 to 15 Years
Y 8 17 65 17 3.74 5 6 66 28 2.37 2 -39 70 70 1.OO
principle, the ratio of weights on the two risky assets (cash and stocks) is independent of the investor's risk aversion (this ratio is approxi- mately -1.1 in Table 3). When the bond fund duration is extended to 15 years, the weights in cash become positive, except for the most aggressive investor.
The previous cases conform to the popular advice except for very aggressive investors with long horizons, for whom cash weights are neg- ative. Even if we presume that the popular ad- vice is not directed toward such investors, we might wonder if the model can produce all positive cash weights without requiring unrea- sonable values of parameters. Table 4 provides an affirmative answer for an investor with a five-year horizon. All of the cash weights are
TABLE4-CASE OF ALL POSITIVE ASSET ALLOCATIONS Percent of portfolio Ratio
Investor type and
Notes: The bond fund constant duration is K = 10 years. The current instantaneous interest rate is r, = 4 percent. The parameters of the stochastic processes are a, = 10 percent, b,. = 4 percent, a, = 7 percent, 8, = 1.0 percent, 8, = 8 percent, a, = 19 percent, and a, = 6 percent.
positive, ranging from 2 to 22 percent, which is within the range of 0 to 50 percent in Table
1. With few exceptions, the other weights and the bondlstock ratios either are within the ranges of Table 1 or not far outside. For exam- ple, the "conservative" bondlstock ratio is 1.58 compared with the extreme of 1.50 in Table
1. Only four of the seven parameter values were adjusted to achieve these results: the speed of mean reversion was cut approximately in half but remains at 10 percent, interest rate volatility was increased by five percentage points, the risk premium on 10-year bonds was reduced from
1.5 to 1 percent, and the risk premium on stocks was increased from 6 to 8 percent.22 Changing all seven parameters would likely allow similar results with smaller changes in each parameter, but further analysis along these lines seems superfluous. The sensitivity analysis shows that this model can adequately account for the es- sential aspects of the popular advice.
Portfolio theory unambiguously supports the popular advice about allocations to stocks, bonds, and cash under two conditions: (I) com- plete markets and (2) the investor's horizon exceeding the maturity of cash. Under these two conditions, there is no asset allocation puzzle concerning the changing ratio of the weights of
"A lower speed of mean reversion is consistent with the results of Yacine Ait-Sahalia (1996), who found that interest rates are a random walk within a range of 4 to 17 percent. A risk premium for the stock of 8 percent corresponds to the long-tenn average in the United States.