PII: 1057-0810(94)90017-5 FINANCIAL SERVICES REVIEW, 3(2): l-126 Copyright Q 1994 by JAI Press Inc. ISSN: 1057-0810 All rights of reproduction in any form reserved. Asset Allocation, Life Expectancy and Shortfall Kwok Ho Moshe Arye Milevsky Chris Robinson An analytical model provides a solution to the retirement problem of how to allocate investment between risky and risk-free assets. The objective is to minimize the probability that the retiree will be unable to consume at the desired level over his/her expected lifetime. The procedure incorporates mortality tables, real or nominal rates of return, initial wealth, and desired consumption levels. Numerical examples using standard mortality tables, historic rates of return on Canaa’ian equity and treasury bills, and a range of realistic values for wealth and consumption show that equity should play a much bigger role in retirement portfolios than other wn’ters advise. I. INTRODUCTION How does a person who is retired invest his or her wealth to maximize the probability of a secure and sufficient income? This asset allocation decision is very critical, because the person no longer has the opportunity or time to recover from mistakes with increased earnings from work. As an increasing proportion of the population of the developed nations enters retirement years, this personal finance problem is becoming of particular interest. The retiree faces several issues in making the investment decision: 1. How much annual income does one need to provide the desired standard of living? 2. How long does the money have to last? Another way to put this is to ask how the retiree balances lower consumption against running out of money before death. 3. How does the decision incorporate inflation? 4. How should the investment be allocated among the various classes available- shares, bonds, etc.’ Kwok Ho, Moshe Arye Milevsky, and Chris Robinson, Faculty of Administrative Studies, Atkinson College, York University, North York, CANADA M3J lP3. 110 FINANCIAL SERVICES REVIEW, 3(Z) 1994 Basic financial planning answers the first question and we assume the income figure required is known. The third question involves using either real returns and constant dollars or nominal returns and nominal dollars throughout the analysis. The approach we use in this paper works equally well with either one, although for convenience we use real returns and constant dollars in the numerical examples.* This paper provides an analytic solution to questions two and four, under reasonable assumptions. We incorporate standard mortality tables into the decision to arrive at an expected rate of return needed to finance future consumption, with each year’s consumption weighted by the probability of survival, and given the initial wealth available to generate the income. The approach is perfectly generalizable to any mortality schedule or to any individual’s preferred risk schedule. For example, an individual may decide that he or she wants to be sure of consuming until age 90, and weight each year at 100 percent.3 Malkiel(l990) in his chapter on the life cycle guide to investing provides an explicit answer to the allocation question without the same analytic process: As investors age they should start cutting back on the riskier investments and start increasing the proportion of the portfolio committed to bonds. By the age of fifty-five, investors should start thinking about the transition to retirement and moving the portfolio toward income production. . . In retirement, portfolio mainly in a variety of intermediate-term bonds (five to ten years to maturity) and long-term bonds (over ten years to maturity) is recommended. The small proportion of stocks is included to give some income growth to cope with inflation. (pp. 356-7) In the graphs that follow the chapter, he recommends investors in the late sixties and beyond hold 60 percent bonds, 30 percent equity and 10 percent in a money market fund. Investors in their mid-fifties are recommended to have 50 percent in stocks, 45 percent in bonds. We compare a numerical example generated in our model with Malkiel’s advice. The investment allocation in this paper incorporates the required rate of return to minimize the probability of failing to meet that rate of return on average over the weighted lifespan remaining to the person. This implied utility function of minimizing shortfall is somewhat similar to the approach taken by Leibowitz and Kogelman (1991). They “measure risk by the “shortfall probability” relative to a minimum return threshold.” A fund manager can choose any combination of minimum return and probability and allocate the assets between a risk and risk-free asset to attain a desirable position. Their procedure does not endogenize the time horizon of the investor, since fund managers do not necessarily have a specific time constraint. They do observe that for longer time horizons, the proportion invested in equity rises. Many researchers have considered the general question of which investment horizon to use and what effect different horizons have on how we view risk and return. In general, they find that risk declines if assets are held without trading for long periods. Different assets perform better in shorter periods of time so the benefits of changing portfolio composition are considerable if the investor times successfully.4 The conclusion for asset allocation is that you should use more equity for longer horizons. Lloyd and Modani (1983) conclude: In general, the usefulness of time diversification is more evident for portfolios containing common stock. Further, the riskiness of any portfolio position is unclear unless the number of time periods the portfolio will be held is also considered. (p. 11) Asset Allocation, Life Expectancy and Shortfall 111 Butler and Domian (1993) use a simulation to find that equity is almost certain to be superior to bonds for holding periods exceeding 10 years, and is likely to be better for shorter holding periods. Since we are solving the problem for an individual retiree, we incorporate this time dimension explicitly. In addition, we require annual consumption from the portfolio, which does not appear in other researchers’ treatments of this problem. Substitu- tion of standard Canadian mortality tables and reasonable estimates of return and variance for Canadian T-bills and equity provides surprising results. Only at quite high wealth levels or well into retirement do the portfolios contain less than 100 percent equity. Not surpris- ingly, 100 percent equity is optimal for women at an older age than men, since women have a longer expected lifespan to finance. This result highlights the contradiction in the obser- vation that women are generally seen to invest in less risky portfolios than men do. The unrecognized risk for retirees is the risk of living too long. In the rest of the paper we proceed as follows. The next section formulates and solves the retiree’s asset allocation problem. Most of the mathematical details are left to an Appendix. The following section provides the numerical results. We then examine the problem when 100 percent equity is insufficient, and provide an heuristic solution to the question of optimal leverage on personal (margin) account. We discuss the implications of our results for retirement planning. Finally, we conclude with a brief discussion of possible improvements and extensions. II. DEVELOPING A SOLUTION Formulation of the Problem We wish to solve the problem of how a retiree should allocate his/her wealth between a risky and a default risk-free asset. We consider how age, mortality rates (or equivalently, life expectancy of a person at any given age), initial wealth, and the desired level of consumption affect the allocation decision. Assume that, at the point of retirement, the individual of n years of age has wealth of W dollars. Assume that he has no other source of income so that his current and future consumption is entirely financed from this sum and earnings on it. He will invest W in a portfolio of risky and risk-free assets in order to support the level of desired consumption until death. Let C, be this desired annual consumption in nominal dollars5 We do the analysis in before-tax dollars, because the details of tax rules are too difficult to incorporate. Let iP, be the probability that the individual aged n will survive one year to age n + 1. For the first year after retirement, the expected consumption is then ,P, . C,. For the second year after retirement, the expected consumption is *P,, . C,. If the mortality table ends at age T, the expected consumption time path after retirement will be 1,P; C,,,P; C,, . . , T_,,Pn C,,) . The probabilities and the life expectancy for any given age can be found in standard mortality tables. Letting d be one plus the minimum rate of return necessary to support the expected consumption, we have: T-nPn ’ CT-n dT-” ’ Given the wealth, consumption and life expectancies, there is an unique solution for d, which is the level of return required to avoid disaster. That is, d is one plus the minimum FINANCIAL SERVICES REVIEW, 3(2) 1994 rate of return that an individual with initial wealth W must earn to have enough to consume C, per annum, given the average mortality rate. Although we will examine this more formally later, we note that the larger the value of n, the lower the value d for a given Wand C,. In other words, older individuals may earn less in order to maintain their consumption because they have fewer years to live. This is consistent with the observation that older investors usually invest more in ‘safer’ assets, which provide lower rates of return. Our analysis provides an explicit way to determine when they should switch to ‘safer’ assets. If we perform the analysis in real dollars, which is equivalent to assuming that the level of inflation is certain, then C, is a constant, C.6 We can simplify equation (1) for computation purposes to: P P W=ln+2+*. .+ T-npn C d d2 dr_n (2) The solution d is now in real terms. We use constant dollars and real rates of return in our numerical illustrations in a later section for ease of exposition, but the theoretical development is the same. Without loss of generality, we assume that there are two assets: treasury bills (T-bills) and a diversified equity portfolio. The individual allocates W between the two. T-bills are free from default risk, but not from interest-rate risk in the long-run. A security is completely risk-free only if it pays off a known and certain amount of consumption at exactly the date required by the investor. An important point to note is that T-bills are risky, in the sense that they have a standard deviation in either real or nominal returns. A person who holds a T-bill until maturity will get exactly the promised rate of return, but if inflation changes during the period, the return is risky in terms of the consumption it permits. Empirically, we observe that the time series of real T-bill rates has significant variability. We treat the T-bill rate of return as a random variable, and hence even a portfolio invested 100 percent in T-bills has some risk. The investor must redo the calculations and rebalance the portfolio periodically because the required d changes as one ages. In practical terms, annual rebalancing seems reasonable, since mortality tables report one year age differences. An Analytic Solution The individual’s problem is to allocate Wbetween T-bills and shares so as to minimize the probability of failing to earn the minimum gross rate of return d on average over the remaining years of one’s life. We assume: 1. Rates of return on equity and treasury bills, are normally (as opposed to lognor- mally) distributed. This assumption is not crucial for optimal results, however it enables us to secure an analytic solution to our problem. 2. Rates of return on each asset are serially uncorrelated. Thus, we consider a series of decisions in a static framework, without the dynamic consideration of what they will do each year when they come to rebalance their portfolios. 3. Returns on each asset are uncorrelated with the other. This assumption can be relaxed, and a solution is given in the appendix. Asset Allocation, Life Expectancy and Shortfall Let us use the following notation and terminology: 113 a is the proportion of W invested in T-bills pLn is the average annual one plus rate of return on treasury bills, (or any other relatively safe investment .) of, is the variance of the annual rate of return on treasury bills. pe4 is the average annual one plus rate of return on equity, (or any other relatively risky investment.) CY& is the variance of the annual rate of return on equity. Denote by: CLp(a)=Cltr.a+CL,q.(l--O1) (3) ~~(a)=~~~.aZ+a$.(l-CL)* (4) Which represents the mean and variance of the rate of return (which is normally distributed), of the investor’s portfolio, assuming that he has placed a proportion 01, of his wealth, in treasury bills, and a proportion 1 - a in equity. For a given I&, I&, oy,, o& we are looking for an asset allocation proportion a’ that will minimize the probability of earning an annual rate of return that is less than the required rate d. Thus, we are trying to solve the following stochastic optimization problem: s.t. O 0; so the optimal allocation includes equity. The risk-free rate is only free of default risk. Each year the above computation must be done anew, (i.e., the portfolio must be re-balanced once a year) because the individual’s d, one plus the required rate of return, will change as time progresses. To generalize the picture, we calculate a range of results for variations in initial wealth, desired consumption, age, and sex. We combine the mortality rates for females and males at various ages with wealth and consumption in constant dollars to obtain d in real terms. Using the same returns and variances as in the example, we obtain Table 1. The value of d, one plus the required rate of return, are shown in Table 2. Table 1 has a block of Es in the upper left denoting all equity portfolios, which are preferred whenever the required rate of return equals or exceeds the T-bill rate (we will explain shortly). Below them are a few bold-face numbers ranging from 0.171 to 0.849. These are interior optima where the required rate falls between zero and the T- bill rate. Finally, the lower part of the table has values of a ranging from 0.865 to 0.955. These are portfolios where d < 1 (see Table 2). That is, the portfolio need not earn positive returns, but must not lose more than a very small percentage of its value. Regardless of how secure the consumption seems to be, the optimal portfolio includes some equity. The extent to which all equity portfolios dominate is quite surprising at first glance. Equity is always characterized as the riskiest security, even in a portfolio. In fact, the greatest risk for a retiree is outliving the available wealth, and given a relatively long lifespan, high risk/high return investments are necessary to minimize this risk. Thus we see that for a reasonable range of wealth/consumption ratios, an all-equity allocation is preferred into normal retirement years, and is essential for early retirees, even if they have very substantial wealth. Numerically, the upper limit of equation (6) is a = 0.96 for the returns and standard deviations in the example. As a practical matter, an a > 0.9 is essentially all T-bills. We can draw more specific observations from Table 1: 1. The equity requirement is greater for women than for men. We show only five year intervals, and women should invest in all equity until they are about five years older than men with the same wealth-to-consumption ratios. 2. Women with quite low wealth to consumption ratios-seven or less-should invest in all equity as late as 80 years of age. T A B L E 1. O pt im al A ll oc at io n B et w ee n T -B il ls a nd E qu it ie s P A : W om en W ea lth to C on su m pt io n R at io [ 6 7 8 9 10 10 .5 11 11 .5 12 12 .5 13 13 .5 I4 14 .5 1. 5 16 E E E E E E E E E E E E E E E E f q E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E P E E E E E E E E E E E E E E 0. 17 1 0. 71 0 E E E E E E E E E 0. 38 6 0. 69 0 0. 78 0 0. 82 3 0. 84 9 0. 86 6 0. 88 6 E E E E 0. 51 2 0. 76 6 0. 83 3 0. 86 4 0. 88 2 0. 89 4 0. 90 2 0. 90 8 0. 91 3 0. 91 7 0. 92 0 0. 92 4 i k E E 0. 79 5 0. 89 1 0. 91 4 0. 92 0 0. 92 4 0. 92 8 0. 93 0 0. 93 2 0. 93 4 0. 93 5 0. 93 7 0. 93 8 0. 93 9 0. 94 0 0 0. 86 2 0. 91 8 0. 93 2 0. 93 8 0. 94 1 0. 94 2 0. 94 4 0. 94 4 0. 94 5 0. 94 6 0. 94 6 0. 94 7 0. 94 7 0. 94 8 0. 94 8 0. 94 9 0. 94 3 0. 94 7 0. 94 9 0. 95 0 0. 95 1 0. 95 1 0. 95 2 0. 95 2 0. 95 2 0. 95 2 0. 95 3 0. 95 3 0. 95 3 0. 95 3 0. 95 3 0. 95 4 ? B : M en W ea lth fo C on su m pt io n R at io g - 6 7 8 9 10 10 .5 11 11 .5 12 12 .5 I3 13 .5 14 14 .5 15 16 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 0. 29 8 0. 62 3 0. 78 4 E E E E E E E E 0. 25 4 0. 67 0 0. 77 4 0. 82 1 0. 84 8 0. 86 5 0. 87 7 0. 89 3 E E E E 0. 65 1 0. 79 5 0. 84 5 0. 87 1 0. 88 6 0. 89 6 0. 90 4 0. 90 9 0. 91 4 0. 91 7 0. 92 0 0. 92 4 E E 0. 76 0 0. 88 3 0. 91 0 0. 91 6 0. 92 1 0. 92 5 0. 92 8 0. 93 0 0. 93 2 0. 93 4 0. 93 5 0. 93 6 0. 93 7 0. 93 9 0. 74 3 0. 90 3 0. 92 4 0. 93 3 0. 93 7 0. 93 9 0. 94 0 0. 94 1 0. 94 2 0. 94 3 0. 94 4 0. 94 4 0. 94 5 0. 94 5 0. 94 6 0. 94 7 0. 93 3 0. 94 0 0. 94 4 0. 94 6 0. 94 8 0. 94 8 0. 94 9 0. 94 9 0. 94 9 0. 95 0 0. 95 0 0. 95 0 0. 95 1 0. 95 1 0. 95 1 0. 95 1 0. 94 9 0. 95 0 0. 95 2 0. 95 2 0. 95 3 0. 95 3 0. 95 3 0. 95 4 0. 95 4 0. 95 4 0. 95 4 0. 95 4 0. 95 4 0. 95 4 0. 95 4 0. 95 5 A ge 50 55 60 65 70 75 80 85 90 A ge 50 55 60 65 70 75 80 85 90 N ot es : T hi s ta bl e pr es en ts t he f ra ct io n of in ve st m en t ca pi ta l at re tir em en t to b e al lo ca te d to T -b ill s (a lp ha f ro m e qu at io n (6 )) . T he se e st im at es u se th e hi st or ic al r et ur ns in H at ch a nd W hi te ( 19 88 ) a nd th e re qu ir ed r et ur ns ( 6) f ro m T ab le 2 . T he v al ue s of a lp ha i n th e ta bl e m us t b e in te rp re te d ca re fu lly . T he ta bl e sh ow s al ph as f or d if fe re nt r et ir em en t ag es a nd le ve ls o f w ea lth r el at iv e to th e co ns um pt io n in c on st an t do lla rs to b e fu nd ed b y th e w ea lth . F or e xa m pl e, a ra tio o f 10 co ul d be $ 40 ,0 00 d es ir ed c on su m pt io n in r ea l t er m s to b e fu nd ed b y $4 00 ,0 00 in s av in gs . T he ‘ E ’ e nt ri es a re a ll eq ui ty p or tf ol io s. T he n on -E v al ue s ar e op tim al p or tf ol io s co nt ai ni ng b ot h T -b ill s an d eq ui ty . T he v al ue s in b ol d- fa ce a re p or tf ol io s w he re t he v al ue o f d lie s be tw ee n ze ro a nd th e T -b ill r at e, w he re th e al lo ca tio n de ci si on is p ar tic ul ar ly s ig ni fi ca nt . I f t he re qu ir ed r ea l r at e of re tu rn is n on -p os iti ve , th e pr oc ed ur e pr od uc es a llo ca tio ns ra ng in g fr om a bo ut 8 6 pe rc en t (i f th e ra te i s ze ro ) to 9 6 pe rc en t. H er e th e al lo ca tio n de ci si on i s no t to o si gn if ic an t, al th ou gh s om e eq ui ty i s al w ay s de si re d. T he ri sk o f fa ili ng t o ea rn e no ug h is E qu ite l ow , a nd c ha ng in g to a 1 00 p er ce nt T -b ill p or tf ol io w ou ld n ot r ai se t he r is k si gn if ic an tly . V I T A B L E 2 . R eq ui re d R at es o f R et ur n W ei gh te d by S ur vi va l P ro ba bi lit ie s (F or D if fe re nt W ea lth /C on su m pt io n R at io s) A : W om en W ea lth /C on su m pt io n R ad io 9 10 10 .5 I1 Il .5 12 12 .5 13 13 .5 A ze ri 7 8 14 14 .5 15 16 50 1. 15 9 1. 13 4 1. 11 6 1. 10 1 1. 08 8 1. 08 3 1. 07 8 1. 07 4 1. 07 0 1. 06 6 1. 06 2 1. 05 9 1. 05 5 1. 05 2 1. 05 0 1. 04 4 55 1. 15 5 1. 13 0 1. 11 1 1. 09 6 1. 08 3 1. 07 8 1. 07 3 1. 06 8 1. 06 4 1. 06 0 1. 05 6 1. 05 2 1. 04 9 1. 04 6 1. 04 3 1. 03 7 60 1. 14 9 1. 12 3 1. 10 4 1. 08 8 1. 07 5 1. 06 9 1. 06 4 1. 05 9 1. 05 5 1. 05 1 1. 04 7 1. 04 3 1. 04 0 1. 03 6 1. 03 3 1. 02 8 6J 1. 13 9 1. 11 3 1. 09 3 1. 07 7 1 . c6 3 1. 05 7 1. 05 2 1. 04 7 1 . cM 2 1. 03 8 1. 03 4 1. 03 0 1. 02 6 1. 02 3 1. 01 9 1. 01 3 70 1. 12 3 1. 09 7 1. 07 6 1. 05 9 1. 04 5 1. 03 9 1. 03 3 1. 02 8 1. 02 3 1. 01 8 1. 01 4 1. 01 0 1. 00 6 1. 00 3 0. 99 9 0. 99 3 75 1. 09 8 1. 07 1 1. 04 9 1. 03 2 1. 01 7 1. 01 1 1. 00 5 0. 99 9 0. 99 4 0. 99 0 0. 98 5 0. 98 1 0. 97 7 0. 97 3 0. 96 9 0. 96 2 80 1. 06 0 I. 03 1 1. 00 9 0. 99 1 0. 97 6 0. % 9 0. 96 3 0. 95 7 0. 95 2 0. 94 6 0. 94 2 0. 93 7 0. 93 3 0. 92 9 0. 92 5 0. 91 8 85 1. 00 0 0. 97 1 0. 94 8 0. 92 9 0. 91 3 0. 90 6 0. 89 9 0. 89 3 0. 88 7 0. 88 2 0. 87 7 0. 87 2 0. 86 8 0. 86 3 0. 85 9 0. 85 1 90 0. 90 4 0. 87 3 0. 84 9 0. 82 9 0. 81 2 0. 80 4 0. 79 7 0. 79 0 0. 78 4 0. 77 8 0. 77 2 0. 76 7 0. 76 2 0. 75 7 0. 75 3 0. 74 5 B : M en W ea lth /C on su m pt io n R at io A ge 6 7 8 9 10 10 .5 11 11 .5 12 12 .5 13 13 .5 14 14 .5 15 16 5 50 1. 15 3 1. 12 8 1. 10 9 1. 09 3 1. 08 1 1. 07 5 1. 07 0 1. 06 6 1. 06 1 1. 05 7 1. 05 3 1. 05 0 1. 04 7 1. 04 3 1. 04 0 1. 03 5 e 55 1. 14 6 1. 12 0 1. 10 0 1. 08 5 1. 07 2 1. 06 6 I. 06 1 1. 05 6 1. 05 2 l.c .4 7 1. 04 4 1. 04 0 1. 03 6 1. 03 3 1. 03 0 1. 02 4 p 60 1. 13 4 1. 10 8 1. 08 8 1. 07 2 1. 05 9 1. 05 3 1. 04 8 1. 04 3 1. 03 8 1. 03 4 1. 03 0 1. 02 6 1. 02 2 1. 01 9 1. 01 6 1. 01 0 65 1. 11 8 1. 09 1 1. 07 1 1. 05 4 1. 04 1 1. 03 5 1. 02 9 1. 02 4 1. 01 9 1. 01 5 1. 01 0 1. 00 7 1. 00 3 0. 99 9 0. 99 6 0. 99 0 B 70 1. 09 4 1. 06 7 1. 04 6 1. 02 9 1. 01 5 1. 00 9 1. 00 3 0. 99 8 0. 99 3 0. 98 8 0. 98 4 0. 98 0 0. 97 6 0. 97 2 0. 96 9 0. 96 2 5 75 1. 06 0 1. 03 3 1. 01 1 0. 99 4 0. 98 0 0. 97 3 0. 96 7 0. 96 2 0. 95 7 0. 95 2 0. 94 7 0. 94 3 0. 93 9 0. 93 5 0. 93 2 0. 92 5 ij 80 1. 01 2 0. 98 5 0. % 3 0. 94 5 0. 93 1 0. 92 4 0. 91 8 0. 91 2 0. 90 7 0. 90 2 0. 89 7 0. 89 3 0. 88 9 0. 88 5 0. 88 1 0. 87 4 B 85 0. 94 6 0. 91 8 0. 89 6 0. 87 8 0. 86 3 0. 85 6 0. 85 0 0. 84 4 0. 83 9 0. 83 4 0. 82 9 0. 82 4 0. 82 0 0. 81 6 0. 81 2 0. 80 5 90 0. 85 1 0. 82 3 0. 79 9 0. 78 0 0. 76 4 0. 75 7 0. 75 0 0. 74 4 0. 73 8 0. 73 3 0. 72 7 0. 72 2 0. 71 8 0. 71 3 0. 70 9 0. 70 1 E N ot es : T h es e va lu es a re 1 + r at e of r et u rn = d b as ed u po n s ta n da rd C an ad ia n m or ta li ty ta bl es . E qu at io n (2 ) is s ol ve d fo r th e ra te o f in te re st th at eq u at es a c on st an t d ol la r $ va lu e fo r co n su m pt io n , w ei gh te d by p ro ba bi li ty o f li vi n g to t h e en d of e ac h y ea r, w it h t h e cu rr en t i n ve st ab le w ea lt h o f th e in di vi du al . S in ce t h e co n su m pt io n is a $ co n st an t ( se e E qu at io n 2 ). w ea lt h a n d co n su m pt io n ca n b e su m m ar iz ed in a s in gl e ra ti o. F or e xa m pl e, a n in di vi du al w it h S 40 0, O O O to in ve st w h o w is h es to c on su m e “W $4 0$ 00 p a in c on st an t d ol la rs h as a w ea lt h /c on su m pt io n ra ti o of 1 0. T h is y ie ld s th e sa m e d as i f on e h ad $ 60 0, 00 0 to i n ve st a n d w is h ed t o co n su m e $6 0, 00 0 p. a. 8 Asset Allocatim, Life Expectancy and Shortfall 117 3. Virtually all women should invest in all equity at age 65 or earlier. 4. Men with quite low wealth should invest in all equity as late as 75 years of age. 5. Virtually all men should invest in all equity at age 60 or earlier. The specific values of alphaderived from this procedure must be interpreted with some caution, which is why we have shown ‘E’ instead of the specific values. The definition of the problem requires that 0 S a I 1. This is the same as saying that the required d cannot exceed one plus the treasury bill rate. As soon as it does, we would want no treasury bills in the portfolio. The intuition is that you cannot minimize the probability of falling below a rate of return by including in the portfolio any asset which is expected to earn less than that rate of return. Including a high risk, high return asset like equity may yield a greater loss on some occasions, but the probability of earning more than the required minimum is still higher. Given enough years of returns, the long-run return will converge to the expected return. Since so many people are in a position where they need more return to minimize shortfall risk than 100 percent equity will provide, we model borrowing in the next section. IV. OPTIMAL MARGIN POSITION As long as the borrowing rate is less than the return on equity, borrowing to buy more equity provides a higher rate of return than 100 percent equity, but it is also more risky. Persons normally borrow on margin or demand loans, which charge floating rate interest. Therefore, although the equity returns will fluctuate in real terms, the interest expense is essentially fixed in real terms. The investor is faced with the annual (one plus) rate of interest charged on margin loans denoted by r, together with the previously-mentioned l_~,,~,