Maximize the Sharpe Ratio and Minimize a VaR 1

Transcription

1 Maximize the Share Ratio and Minimize a VaR 1 Robert B. Durand 2 Hedieh Jafarour 3,4 Claudia Klüelberg 5 Ross Maller 6 Aril 28, 2008 Abstract In addition to its role as the otimal ex ante combination of risky assets for a risk-averse investor, ossessing the highest otential return-for-risk tradeoff, the tangency or Maximum Share Ratio ortfolio in the Markowitz (1952, 1991) rocedure lays an imortant role in asset management, as it minimizes the robability that a future ortfolio return falls below the risk-free or reference rate. his is a kind of Value at Risk (VaR) roerty of the ortfolio. In this aer we demonstrate the way this VaR, and related quantities, vary along the efficient frontier, emhasizing the secial role layed by the tangency ortfolio. he results are illustrated with an analysis of the market crash of October 1987, as an eisode of extreme negative market movements, where the tangency ortfolio erforms best (loses least!) among a variety of ortfolios. 1 his work is artially suorted by ARC Grant DP University of Western Australia, M250, Nedlands WA 6009, Australia: Robert.Durand@uwa.edu.au. 3 Centre for Financial Mathematics, MSI, and School of Finance & Alied Statistics, Australian National University, Canberra AC 0200, Australia, and Islamic Azad University of Shiraz: Hedieh.Jafarour@anu.edu.au. 4 Corresonding author. 5 Center for Mathematical Sciences, Munich University of echnology, D Garching, Germany: cklu@ma.tum.de, htt:// 6 Centre for Financial Mathematics, MSI, and School of Finance & Alied Statistics, Australian National University, Canberra AC 0200, Australia: Ross.Maller@anu.edu.au. 1

2 1. Introduction: the Maximum Share Ratio and "angency" Portfolios Given a universe of d 2 risky assets having raw return vector µ and excess mean return vector µ = µ ri (relative to a reference rate r ), and returns covariance matrix Σ, form a ortfolio by taking an allocation, that is, a linear combination with coefficients given by a vector, x, say, of the assets. 1 he "reference rate" could be the revailing risk-free rate, if there is one, or some other benchmark rate, such as the exected market return for the eriod, etc. Let R be the ex ost excess return achieved from this ortfolio, after the ortfolio has been in lace for a secified, fixed, time eriod. Suose R has exectation µ and variance 2 σ. hus 2 E( R ) = µ =xµ and Var( R) = σ =x Σx. (1.1) Assume that, for all such ortfolios, the standardized ex ost excess returns R µ σ are identically distributed, having the same distribution as a random variable Z, say, where E( Z ) = 0 and Var( Z ) = 1. Define the oulation Maximum Share Ratio as SR : = max ix = 1 x µ x Σx. (1.2) his quantity has long been used in ortfolio theory and ractice (Share 1963), either in an ex ante fashion, where it can be used to decide on an otimal allocation giving an otimal returnrisk tradeoff, or ex ost, as a ortfolio erformance evaluation tool. It lays a significant role in 1 Here i denotes a d-vector, each of whose elements is one, and a rime will denote a vector or matrix transose. 2

3 both discrete and continuous time finance, and is an object of interest in research right u to the resent day (see, e.g., Christensen and Platen 2007). he maximization in (1.2) is over all ortfolios satisfying the total allocation constraint ix =1, that a unit amount of resources is invested. here is no requirement that the comonents of the vector x be nonnegative, so short selling of assets is allowed. he ratio in (1.2) is maximized taking its sign into account, as advocated, e.g. by Share (1994); we are interested in maximizing the actual (risk-adjusted) return that is, a measure sensitive to losses, as well as to gains. 2 In this aer, we consider (in Section 2) an otimality roerty of the Maximum Share Ratio ortfolio, that is, the ortfolio achieving the maximum value in Eq. (1.2), which it ossesses with regard to Value at Risk. he ideas are illustrated with a textbook examle in Section 3. We then go on in the fourth section to develo some ideas regarding realized returns on efficient ortfolios, which are illustrated with the same textbook data, and in the fifth section, we examine the erformance of a sectrum of ortfolios calculated from monthly data on US stocks rior to the October 1987 stock market crash, showing how the tangent ortfolio, and various other selected ortfolios, erformed rior to, and on, the day of the crash. o conclude this section we mention some further facts we will need, concerning the connection between the maximum Share ratio and what we will call the "tangency" ortfolio. he quantity SR in Eq. (1.2) is the maximum Share ratio achievable from the d assets. In textbooks, and in alications, the corresonding ortfolio is often found or illustrated by drawing a tangent line in the ( σ, µ lane from the oint (0, r) (where r is the risk-free or ) 2 In some studies, the quantity in Eq. (1.2) is squared before the maximization is done. While this simlifies the algebra, it unrealistically ignores ossible exected losses on the ortfolio. 3

4 reference rate) to the efficient frontier constructed from µ and Σ. he coordinates of this oint, ( σ, µ ), say, give the location of the maximum Share ratio ortfolio in the ( σ, µ ) lane, and the sloe of the tangent line gives the maximum Share ratio available for any ortfolio constructed from this universe of assets. he corresonding allocation vector calculated from Equation (41) in Merton (1972). x can be Merton (1972) showed further, however, that this rocedure can be misleading or in error, since a tangency oint roducing a maximum Share ratio need not in fact exist. He gave a necessary and sufficient condition for this to be the case (heorem II in Merton (1972)). Of course a maximum value of the Share ratio still exists (and is finite), but it has to be found by other means; see, e.g., the method outlined in Maller & urkington (2002). he robability calculation in (2.1) below uses only the existence of the maximum Share ratio ortfolio, however calculated; it does not require the existence of a tangent oint to the efficient frontier. Nevertheless, we shall continue to refer to the ortfolio with maximum Share ratio as the tangent ortfolio whether or not such exists. For all the data considered in this aer, it turns out that the tangent ortfolio does in fact exist, so no confusion should result from this. 2. A Value at Risk Proerty of the Maximum Share Ratio Portfolio Let x be the allocation vector corresonding to the ortfolio obtained as a result of the maximization in Eq. (1.2). As discussed in the revious section, we will refer to this as the tangency ortfolio. Recall that the excess mean return vector µ equals µ = µ ri, where µ is the mean raw return vector, and r is the reference rate. We can write r SR µ = = µ σ σ, 4

5 where µ = x µ, µ = x µ and σ = x Σx. he maximum Share Ratio ortfolio ossesses a certain otimality roerty with resect to VaR, as the following simle calculation shows. For an arbitrary ortfolio with allocation, we have x SR xµ = x Σx µ σ. Letting R be the excess return on the tangency ortfolio, we can calculate R µ µ P( R 0) = P σ σ µ = P Z σ P ( Z SR) R µ µ = P σ σ = P 0. ( R ) (2.1) It follows that ( R ) ( R ) min P 0 P 0 x and the minimum is achieved for the tangency ortfolio. hus, an allocation of assets according to the tangency ortfolio has the lowest robability of the investor receiving a return below the reference rate; in other words, it has the smallest VaR relative to this rate. As ortfolios move away from the maximum Share Ratio allocation, this robability increases. We can illustrate the magnitude of this increase by lotting the robability for ortfolios on the efficient frontier, that is, those having exected return and standard deviation ( µ, σ ), against σ, thus obtaining a reresentation of the way this VaR changes along the efficient frontier. We have to assume a distribution for Z, the standardized return, and for this, 5

6 we will consider a standard normal, as well as a t-distribution with 4 degrees of freedom. hese reresent extreme distributions between which returns distributions are likely to lie. While the normal distribution is often assumed for returns, esecially over longer eriods, it has been long recognized that returns distributions in reality are more heavy tailed and letokurtic than the normal distribution (Fama, 1965; Embrechts et al. 1997; Platen and Sidorowicz 2007); therefore, we utilize a t-distribution with small degrees of freedom to simulate this feature of the data. We illustrate the concets using some data from Ruert s text book (2004,.150). here are d = 3 assets for which the (raw) mean vector and covariance matrix are µ 0.08 = and Σ = he efficient frontier for this examle is shown on.155 of that book. In Figure 2.1 we lot the function P( R 0) = P( Z µ / σ ) for ortfolios on the efficient frontier, as a function of the ortfolio risk, σ. As exected, the curves have a minimum at the tangent oint, and the curve for the t-distribution is higher than for the normal; the robability of a return below the risk-free rate is much higher for the heavier-tailed t-distribution. A Value at Risk is usually thought of as a quantile below which a return falls with a secified (low) robability; thus, we should also consider P( R q), for values of q not equal to zero. It is not the case in general that this quantity is minimized for the Maximum Share Ratio allocation, but by observation this seems to remain aroximately true for q not too far from zero (recall that we are otimizing excess returns, relative to a benchmark). In Figure 2.2, the robabilities of efficient ortfolio returns lower than q are shown for various values of q. For examle, from the lowest curve in Figure 2.2 (left lot) can be read that the robability of a 6

7 future excess return less than for the tangent oint ortfolio is aroximately hus the tangency ortfolio is exected to return more than 0.02 below the reference rate at most 44.1% of the time. For the t 4 distribution in the right lot, on the other hand, such a loss haens aroximately with robability Although it is not necessarily the case that the minima of the curves in Figure 2.2 should occur at the tangent oint (exected for the cases q = 0 ), in fact this haens for this data (and also for the data analyzed in Section 4). Figure 2.1: Ruert Data, Normally and t 4 Distributed Returns he robability of receiving a negative return, as a function of the standard deviation of the efficient ortfolio. he curve labeled N deicts normally distributed returns and the curve labeled deicts t 4 distributed returns. he curves start at the standard deviation of the minimum variance ortfolio on the left, and show the osition of the tangent oint ortfolio (indicated by dot oints). 7

8 Figure 2.2: Ruert Data, Values at Risk for Normally and t 4 Distributed Returns he robability of receiving an excess return lower than q, where q is secified by the numbers at the right hand ends of the curves, as a function of the standard deviation of the efficient ortfolio. he left hand diagram deicts normally distributed returns; the right hand diagram t 4 distributed returns. he curves start at the standard deviation of the minimum variance ortfolio on the left, and show the osition of the tangent oint ortfolio (indicated by dot oints). 3. Efficient Portfolio Returns o investigate the erformances of ortfolios on the efficient frontier, we need some facts concerning them. hese are derived from Merton (1972). In our notation, the quantities on.1853 of his aer are: A 1 = i Σ µ, B, 1 = µ Σ µ C 1 = i Σ i, D BC A 0 2 = >. (Recall that µ denotes the raw returns and µ = µ ri are the excess returns on the d assets.) We assume that a tangent oint exists, so the quantity 1 PC = i Σ µ = A rc is ositive (Merton (1972),. 1863). he coordinates in the ( σ, µ ) lane of the minimum variance and tangent oint ortfolios are given by and 2 1 A σ m =, µ m =, C C 8

9 1 2 µ Σ µ σ = 1 2 ( i Σ µ ), µ = r +. 1 µ Σ µ 1 i Σ µ he corresonding ortfolio allocations are 1 m = Σ i x and C x 1 Σ µ = 1 i Σ µ. (Note: we have µ, not µ, in σ, µ and x.) he equation of the efficient frontier in ( σ, µ ) sace is 2 ( σ 1) A+ D C F( σ ) =. (3.1) C (We use the F notation for frontier, rather than Merton's E notation, which we reserve for exectation.) he ortfolio allocation corresonding to a ortfolio with coordinates ( σ, µ ) on the efficient frontier is given by the vector x = ( Σ ) F( σ ) C µ AΣ i + BΣ i AΣ µ D (3.2) (Merton (1972),.1856 and.1845). It is easily checked by differentiation that the curve F( σ ) r σ (3.3) in ( σ, µ ) sace has a maximum at the oint σ which satisfies σ 2 = 2 B 2rA + r C ( A rc) 2 ; (3.4) this of course is the variance of the tangent oint ortfolio. (Note that the denominator in (3.4) 2 is ( PC ) > 0.) he function in (3.3) increases for σ < σ and decreases for σ > σ. Figure 3.1 shows the curve for the Ruert data, taking r = 0.02 as on.155 of Ruert (2004). 9

10 Figure 3.1: Share Ratios for Efficient Portfolios from the Ruert Data Plot of the Share ratio (Eq. (3.3)) for ortfolios on the efficient frontier, against their standard deviation, Ruert textbook data. he tangent oint ortfolio is indicated by a dot. Now suose we have a new observation vector, R, on the returns of the d assets. We can think in terms of the efficient ortfolio with mean F( σ ) and standard deviation σ being ut in lace at a certain time, then evaluated using the future return some algebra, we can write R. Using (3.2), and after ( 1 1 )( ( ) ( ) + ) Rx r F( σ ) r C A B A = + R µ Σ µ Σ i σ σ σd. (3.5) Here Rx reresents the return on the efficient ortfolio corresonding to the returns R on the d assets, and the quantity on the left of (3.5) is the standardized excess return, i.e., the ex ost Share ratio for the ortfolio. On the right of (3.5) is the oulation Share ratio for the ortfolio lus a random term corresonding to the new return, R. If R is drawn from the same oulation as that from which the efficient ortfolio was constructed, so that E( R ) = µ and 10

11 Var( R ) = Σ, it is clear that the exectation of the random term in (3.5) is zero, and its variance is one (as can also be checked after some algebra). Figure 3.2 shows a lot of Eq. (3.5) for 13 returns generated randomly as observations on N( µ, Σ), using Ruert's values of µ and Σ. (Ruert does not suly the original returns for which his µ and Σ were calculated, so we simulated the observations.) We took 13 returns so as to corresond with the 1987 crash data in the next section. It is clear from Figure 3.2 that Eq. (3.5), as a function of σ, need not resemble Eq. (3.3), as shown lotted in Figure 3.1. For this data, the random comonent in Eq. (3.5), which has a standard deviation of one, overwhelms its exectation, which for this data eaks at about 0.13 (cf. Figure 3.1). Figure 3.2: Standardized Returns on Efficient Portfolios for Ruert Data Ex ost Share ratios for returns on ortfolios on the efficient frontier, corresonding to a new return, against their standard deviations, Ruert textbook data. he tangent oint ortfolio is indicated by a dot. 11

12 While the simulated future return curves sometimes eak close to the tangency oint, at other times the maximum occurs for much higher risk ortfolios, and sometimes the curves are even convex. For such data (and the data in the next section has similar features), unfortunately, investing in the tangency ortfolio roduces very little benefit for individual future returns. Only when averaged over a relatively large number of returns will curves calculated from Eq. (3.5) begin to resemble those from Eq. (3.3). 4. Fama-French Data In this section we analyze a more realistic examle. For oulation µ and Σ, we take values estimated from monthly data on US stocks for twenty-five value-weighted size and book-to-market ortfolios (Fama and French, 1993) downloaded from Ken French s website ( htt://mba.tuck.dartmouth.edu/ages/faculty/ken.french/data_library.html). his classic set of data has been used in many definitive studies of ortfolio and other analyses; see for examle Jagannathan and Ma (2003). We refer to these ortfolios as the Fama-French ortfolios. Choosing an aroriate selection of assets from the real world to demonstrate the VaR minimizing roerties of the Share ratio is roblematic given the large number of assets from which an investor may choose. he Fama-French ortfolios are reresentative of asset classes that cature factors that aear to be imortant to investors; it is reasonable that an investor might use these as reresentative asset classes from which to derive an otimal return to risk trade-off. Figure 4.1 shows the efficient frontier calculated from this data set. 12

13 Figure 4.1: Efficient Frontier constructed from 25 Fama-French ortfolios. he efficient frontier estimated from monthly returns on the 25 Fama-French ortfolios over the eriod October 1982-Setember 1987, with the tangent line, and the maximum Share ratio oint lotted as a dot. Figure 4.2 shows the robability of a negative excess return (return less than the riskfree rate, r = ) for the normal and t4-distributions. he minimum robability at the tangent oint is clearly aarent, but the minimum is not so well defined as it was for the textbook data (in Figure 2.1). his is a reflection of the fact that the tangent oint is not well defined in Figure 4.1; although PC > 0 for this data, PC is close to zero, and the efficient frontier is ractically a straight line to the right of the tangent oint. Consequently, the Share ratio is ractically the same for all ortfolios with higher risk than the tangent oint ortfolio. he robability of a negative return is of course much higher for the more extreme t4-distributed returns. 13

14 Figure 4.2: Fama-French Monthly Data, t 4 and Normally Distributed Returns he robability of receiving a negative return, as a function of the standard deviation of the efficient ortfolio. he curve labeled N deicts normally distributed returns and the curve labeled deicts t 4 distributed returns. Parameters estimated from 60 monthly returns on 25 Fama-French ortfolios over the eriod October 1982-Setember he curves start at the standard deviation of the minimum variance ortfolio on the left, and show the osition of the tangent oint ortfolio (indicated by dot oints). Figure 4.3 shows the family of curves obtained by lotting P( R q) against σ, again for ortfolios on the efficient frontier, for various values of q. he same values of µ and Σ are taken as for Figure 4.1, and Z is N(0,1). For examle, from the lowest curve in Figure 4.3 can be read that the robability of a future excess return less than for the tangent oint ortfolio is aroximately hus the tangency ortfolio is exected to return less than at most 5.5% of the time, for this data, if returns are normally distributed. In other words, the minimum VaR in this data, corresonding to a 5.5% quantile, if a normal distribution is as- 14

15 sumed for returns, is a return below Since the Fama-French data is in ercentage terms, this means a otential loss of 0.15%, on a monthly basis, or about 1.8%.a. Figure 4.3: Fama-French Monthly Data, Normally Distributed Returns he robability of receiving an excess return lower than q, where q is secified by the numbers at the right hand ends of the curves, as a function of the standard deviation of the efficient ortfolio, assuming normally distributed returns. Parameters are estimated from 60 monthly returns on 25 Fama-French ortfolios over the eriod October 1982-Setember he curves start at the standard deviation of the minimum variance ortfolio on the left, and show the osition of the tangent oint ortfolio (indicated by dot oints). Figure 4.4 shows similar information as Figure 4.3, but with Z having a t-distribution with 4 degrees of freedom, rather than a normal distribution. he same values of µ and Σ are taken as for Figure 4.2 and Figure 4.3. With this more extreme distribution for returns, robabilities of large negative returns are much higher than for the normal distribution, and values at risk are corresondingly much higher too. he VaR corresonding to a 5% quantile is about -0.70, thus, a loss of 0.70%, on a monthly basis, or about 8.4%.a., substantially higher then 15

16 for a normal distribution, but still woefully inadequate for describing the losses on efficient ortfolios on Black Monday, October , as we shall see in the next section. Note that the curves in Figure 4.3 and Figure 4.4 are much shallower than in Figure 2.2, again reflecting the difficulty in locating the tangent oint accurately in this data. Figure 4.4: Fama-French Monthly Data, t 4 Distributed Returns he robability of receiving a return lower than q, where q is secified by the numbers at the right hand ends of the curves, as a function of the standard deviation of the efficient ortfolio, assuming t 4 distributed returns. Parameters estimated from 60 monthly returns on 25 Fama- French ortfolios over the eriod October 1982-Setember he curves start at the standard deviation of the minimum variance ortfolio on the left, and show the osition of the tangent oint ortfolio (indicated by dot oints) Crash Performance In this section we make some evaluations of the erformance of a sectrum of ortfolios using real data sets for illustration. Minimizing VaR matters most when rices are falling; 16

17 therefore, we examine the erformance of tangent ortfolios on Black Monday, October 19, 1987, when the S&P 500 index fell by 20.5%. An advantage of using the Black Monday crash is that this is erhas the only case where the reality of an observed event is more extreme than something that might reasonably have been simulated! As in Section 4, we take monthly data on the Fama-French ortfolios over a eriod of 60 months rior to the 1987 crash, and use it to set u a sectrum of ortfolios, each of whose erformance (excess return, risk-adjusted) is then evaluated a day after the crash. he sectrum of ortfolios consists of those on the efficient frontier, including the minimum variance and tangency ortfolios, together with selected ortfolios such as the equally weighted ortfolio. o evaluate the erformances of the ortfolios, we consider the case of an investor whose strategy involves utting in lace one of the above-mentioned ortfolios; for examle she may maximize the ex ante Share ratio of her ortfolio. Of course with the benefit of hindsight, our investor would have shorted the entire market, but to kee the analysis realistic we assume she reviews and rebalances her ortfolio regularly, without foresight, and that rebalancing takes some nonzero time. Introducing such a friction is not unreasonable, though it may mean that we are erring on the side of conservativism. 3 hus, to summarize, the investor makes her allocations at the beginning of October 1987, using a ortfolio constructed from the Fama-French ortfolios, and the information on their returns over the receding 60 months (that is, using the monthly data from October We also confine the illustration to the direct use of the historic data, again, erring on the side of conservativism; for examle, using a adjustment, such as that analyzed in Jorion (1996) or Ledoit and Wolf (2004), might result in better estimates to the otimization and, consequentially, imroved outcomes for the tangent ortfolio 17

18 to Setember, 1987 used in the analysis in the receding section). As benchmark reference rate for the calculation of the tangency ortfolio we take the exected risk-free rate at the end of October, he crash takes lace on October 19, 1987, and we evaluate the return on each ortfolio at the close of business on that day. able 5.1 shows the absolute returns on some selected ortfolios (not all of which are efficient), on that day. Portfolio Return on Black Monday angent Minimum variance Equal weighted ortfolio of the 25 Fama-French ortfolios S&P 500 index (equal-weighted) S&P 500 index able 5.1: Black Monday Returns Returns (in ercentages) on Black Monday for selected ortfolios constructed from 60 monthly returns on 25 Fama-French ortfolios over the eriod October 1982-Setember he returns in able 5.1 are far below anything that could be exected from Figure 4.3 and Figure 4.4. Even under a t-distribution with four degrees of freedom, the robability of receiving a return below is only Nonetheless, our investor would undoubtedly have been very leased with the relatively large return (smaller loss) she achieved! 4 he risk-free rate at the end of October 1987 was also downloaded from Kenneth French's data library. Using the end-of-october risk-free rate assumes that our analyst had erfect exectations about this asect of the return calculation. 18

19 Figure 5.1 shows a lot of the exected Share ratio, that is, the function in Eq. (3.3), for this data. he maximum occurs at the tangent oint, as it should, and is reasonably well defined. Figure 5.1: Share Ratios for Efficient Portfolios from the Fama-French Data A lot of the Share ratio for ortfolios on the efficient frontier, against their standard deviations. he tangent oint ortfolio is indicated by a dot. Figure 5.2 shows the risk-adjusted excess returns from efficient ortfolios calculated from the Fama-French data, lotted against the standard deviations of the ortfolios, for the first 14 trading days in October Forewarned by Figure 3.2 and the analysis of the Ruert data, we exect high variation around the oulation Share ratio of he curves in the to art of Figure 5.2 cover the first 13 trading days in October. In this eriod some of the curves tend to show a maximum near the tangent oint ortfolio, but the curvature is very slight and the maxima are not at all well-defined. his feature disaears as October goes on, and the curves become monotone increasing. he lowest curve is for October 19. he return on 19

20 the tangent ortfolio is not the highest on this day. here is a monotonic increase in riskadjusted excess returns as we increase the risk of the ortfolio along the efficient frontier, from the minimum ossible risk, on. he least losses would have been given by taking ositions at extreme risks. Of course the crash eriod reresents an extreme situation whereas our theory assumes that returns have a constant distribution. his was almost certainly not the case on October 19, We discuss the imlications further in the next section. Figure 5.2: Returns for efficient ortfolios from 25 Fama-French ortfolios, first 14 trading days in October he risk-adjusted returns on efficient ortfolios calculated from monthly returns on the 25 Fama-French ortfolios over the eriod October 1982-Setember 1987, evaluated on the first 14 trading days in October 1987, lotted against the standard deviation of the ortfolios. he number at the right hand end of a curve indicates the date on which the ortfolio was evaluated. he lowest curve corresonds to Black Monday, October 19, he minimum variance ortfolio is at the left end of each curve and the tangent ortfolio is indicated by a dot oint. Excess returns are adjusted for risk by dividing by the standard deviation of the original ortfolio. 20

21 6. Discussion and Conclusion Maximizing the return to risk trade-off through investing in the tangency ortfolio is very well-known and understood by educated investors. Rational investors, esecially investors whose trustees focus only on returns, will want to guard at all cost against the ossibility that their ortfolio will earn less than the risk-free, or reference, rate. Our work demonstrates the way that maximizing the exected Share ratio through selecting the tangency ortfolio minimizes the chances, not only of a return lower than the reference rate, but of even lower returns as well, across the range of efficient ortfolios. hese VaR minimizing roerties of the tangency ortfolio have not, to our knowledge, been imlemented in a ractical situation, and, as a result, the very desirable consequences of imlementing a simle black-box aroach to ortfolio selection have not been thoroughly exlored. he ex ante allocation of assets to a ortfolio is always based on imerfect foresight. In using the crash of 1987 to illustrate the VaR minimizing roerties of the tangency ortfolio, we have chosen a articularly extreme examle where investors cared deserately about the downside risk of their ortfolios. By constructing a ortfolio based on five-years of monthly data before the crash, we have ut ourselves in the situation of an investor following a reasonably realistic tangency-ortfolio strategy. he in-samle estimates of exected loss resented in Figure 4.2 and Figure 4.3 demonstrate that, should the distributions have remained stable, our investor's exected losses would have been minimized by holding the tangency ortfolio. However, in the extreme circumstances of October 1987, the assumtion that ast returns distributions would remain the same was almost certainly violated. But after the damage had been done, it turns out that the tangency ortfolio erformed best (lost least!) out of the range of benchmarks resented in able 5.1. Perhas this kind of damage control is the most we can 21

22 hoe for with this kind of event. If any bonuses were awarded at the end of 1987, our tangentortfolio investor would have deserved hers. Acknowledgements We are grateful to John Gould for some technical assistance. References Christensen, M.M., and Platen, E., (2007) Share Ratio Maximization and Exected Utility when Asset Prices Have Jums", Int. J. heoret. Al. Finance, 10, , Embrechts, P., Klüelberg, C., and Mikosch,. Modelling Extremal Events. For Insurance and Finance. Sringer-Verlag, Berlin, Fama, E.F., (1965), he Behavior of Stock-Market Prices, Journal of Business 38, Fama, E., and K. French, (1993), Common risk factors in the returns on stocks and bonds, Journal of Financial Economics 33, Jagannathan, R. and Ma,. (2003) Risk reduction in large ortfolios: why imosing the wrong constraints hels. J. Finance 58, Jorion, P., (1986) Bayes-Stein estimation for ortfolio analysis, Journal of Financial and Quantitative Analysis 21, Ledoit, O. and Wolf, M. (2004) Honey, I Shrunk the Samle Covariance Matrix, Journal of Porfolio Management 30(4), Maller, R.A. and urkington, D.A. (2002) New light on the ortfolio allocation roblem, Math. Meth. Oer. Res., 56, Markowitz, H. (1952) Portfolio Selection, J. Finance, 7, Markowitz, H. (1991) Portfolio Selection: Efficient Diversification of Investment, Blackwell, Cambridge, Mass. 22

23 Merton, R.C. (1972) An analytic derivation of the efficient ortfolio frontier, J. Fin. Quant. Anal., 7, Platen, E. and Sidorowicz, R. (2007) Emirical evidence on Student-t log-returns of diversified world indices. J. Statist. heory and Practice, 2, Ruert, D. (2006) Statistics and Finance. An Introduction. Sringer, New York. Share, W.F. (1963) A simlified model for ortfolio analysis, Management Science, Share, W.F. (1994) he Share Ratio, J. Portfolio Management (Fall),