AMS Portfolio Theory and Capital Markets

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1 AMS Portfolio Theory and Capital Markets I Class 09 - Optimal Portfolio Growth Robert J. Frey Research Professor Stony Brook University, Applied Mathematics and Statistics frey@ams.sunysb.edu This lesson focuses on optimal portfolio growth over time. Optimal is meant in the sense of maximizing the expected utility of terminal wealth. The material is drawn from Chapter 15 of Luenberger s Investment Scince. April 7, Investments in Stationary, Independent Return (SIR) Processes The return R on an asset with price S over a period from t to t+d is SHt + DL R = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅ The rate of return r is stated in terms of the relative change in value SHt + DL - r = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ï R = 1 + r (1) () Example - Equity Markets The stock market is usually modeled as process whose returns are stationary with independent increments. The Efficient Market Hypothesis assumes that all known information is already priced into the stock; therfore, subsequent price changes must be the result of new information and, hence, represents indpendent events. It certainly not true that stock market returns are stationary, but over short- to mid-term horizons, the assumption of stationarity can be a reasonable approximation. Example - Constant Coefficient Geometric Brownian Motion Clearly, the rate of return r on constant coefficient geometric Brownian motion is a stationary process with independent increments d rhtl = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ d = m d t + s d WHtL (3). Log Utility and SIR Processes Let X t represent wealth at time t. If that wealth is invested in a SIR process, then, with R t representing the r.v. of return: X t = R t X t-1 (4)

2 ams-q01-lec-09-p.nb If X 0 is the initial wealth, then applying (4) recursively gives i n X n = R n R n-1 R n- R R 1 X 0 = X 0 j k t=i y R t z { (5) Under log utility we have U@X n D = log@x n D = log@x 0 D + log@r t D n t=1 (6) The individual returns are i.i.d. and we can, therefore, assert that E@U@X n DD = E@log@X n DD = log@x 0 D + n E@log@RDD (7) For the special case of log utility, if we wish to maximize the expected utility of our terminal wealty X n, then that is equivalent to maximizing the expected log return. If we rearrange the terms in (7) and then taking the antilog of both sides E@log@X n DD - log@x 0 D = n E@log@RDD (8) EC X n ÅÅÅÅÅÅÅÅ X 0 G = e n E@log@RDD (9) Thus, the investment grows exponentially with n at a rate of e E@log@RDD. We use the term growth rate for this quantity less one g = EC X n ÅÅÅÅÅÅ Å X 0 G - 1 = e n E@log@RDD - 1 (10) Example - Kelly Betting Suppose you have a chance to place a bet b representing some proportion of your total wealth that with probabiliy p will double your bet and with probability 1 p will return nothing. We can use Mathematica to solve for the value of b which maximizes the single period expected log return. Solve@ b Hp Log@1 + bd + H1 - pl Log@1 - bdl == 0, bd {{b Ø -1 + p}} Thus, an optimal policy is to bet ( p 1) times your wealth at each round. 3. Volatility Pumping Consider two (admitted rather odd investments). One pays off with a return of h or 1/h, each with probability 1/. Another remains fixed in value. ote that both investments have zero drift in the sense that each has an expected growth rate of 0. A log optimal policy is Ä Solve ÇÅ i l j 1 É ÅÅÅÅ k HLog@l h + H1 - lld + Log@l ê h + H1 - lldly z == 0, l { ÖÑ ;;l -> 1 ÅÅÅÅ?? That is, to split our wealth each period equally between the two investments. Consider the case in which h = 1/. the expected log return is

3 ams-q01-lec-09-p.nb 3 E@log@RDD = ÅÅÅÅÅ 1 Hlog@1.5D + log@0.75dl = (11) We have e = 1.069, so the growth rate is about 7%. Thus, even though each asset individually has a growth rate of zero a suitably constructed portfolio of such instruments actually has a positive growth rate. 4. Optimal Continuous Time Growth A multivariate geometric Brownian motion is described by the following equations d r i HtL = d S ihtl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = m i d t + s i d W i HtL (1) In addition, cross products of d WHtL terms are scaled by their covariances Cov@d W i HtL d W j HtLD = s i,j d t (13) If we form a portfolio of instrumets with weights x = {x 1, x, x 3,, x } d VHtL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ VHtL = x i d S ihtl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = x i Hm i d t + s i d W i HtLL (14) The variance of the stochastic term is E i j k x i s i d W i HtL y z { = x i x j s i, j d t j=1 (15) So V(t) is log normal with EClogC ÅÅÅÅÅÅÅÅÅÅÅÅÅ VHtL i VH0L GG = j k x i m i - ÅÅÅÅÅ 1 y x i x j s i,j z t j=1 { (16) Example - Understanding Volatility Pumping Consider a collection of independent instruments with the same drift and volatility d r i HtL = d S ihtl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = m d t + s d W i HtL (17) The expected log return for each stock is m s /; however, if build an equally weighted portfolio of such instruments, then VHtL EClogC ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ VHt - 1L GG = m - ÅÅÅÅÅÅÅÅÅÅÅ s (18) Growth is reduced by volatility, so by diversifying the portfolio, we realize a lower volatiliy and, hence, a higher rate of growth. 5. Portfolio Efficiency We realize the optimal porfolio by maximizing its expect log return

4 4 ams-q01-lec-09-p.nb lo max m x o n x i m i - ÅÅÅÅÅ 1 x i x j s i, j j=1 ƒ o 1 T x = 1} o ~ (19) 6. ot-quite-continuous Time In continuous time we are dealing with an infinitesimal rebalancing period of dt. In practice, operational and cost considerations force us to rebalance on something more like a quarterly or semi-annual period. Consider a Taylor series expansion for the log return about r = 0 U@RD = log@rd = log@1 + rd = r - r ÅÅÅÅÅÅÅ + OAr3 E (0) For r near zero and roughly symmetric about zero it is straightforward to show that the expected impact of O[r 3 ] is small. Thus, we can write the following approximate expression for the expected log return E@U@RDD = E@log@1 + E@rD - E@r D ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ (1) However, we also know that Var@rD = EAr E - E@rD () Substituting one expression into the other yields Var@rD + E@rD E@U@RDD = E@log@1 + E@rD - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ (3) If we assume that we follow the continuous time log optimal policy, then we have an error based on E@rD caused by the less frequent rebalancing. Depending upon the circumstances we can choose to ignore this effect or adjust the standard continuous time formulation by adjusting the covariance matrix. ote that (m T r) = r T (m m T ) r, so we use the following adjusted covariance matrix in our portfolio optimizations Q = Q + Im m T M ï Hq L i, j = s i,j + m i m j (4) Example - the S & P 500 The annual mean and standard deviation of log return for the S&P 500 is about 1% and 0%, respectively. Consider a quarterly rebalancing frequency. The quarterly mean and variance as 0.5 (1%) = 3% and è!!!!!!!!! 0.5 H0 %L = 10 %. We now need to estimate the mean and variance of the ordinary return. The continuous distribution package in Mathematica provides the necessary tools << Statistics`ContinuousDistributions` We can now compute the mean and standard deviation of r, the ordinary return. Mean@LogormalDistribution@0.03, 0.1DD - 1 è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Variance@LogormalDistribution@0.03, 0.1DD Var@rD + E@rD U@RD = E@log@1 + E@rD - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = If we ignore the E@rD and use the unmodified continuous time formulation, then (5)

5 ams-q01-lec-09-p.nb 5 U@RD = E@log@1 + E@rD - Var@rD ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ This results in a small, %, difference in expected utility. = ÅÅÅÅÅÅÅÅÅÅÅÅ 0.01 = (6) We still have the O[r 3 ] term hanging around. We can again rely on Mathematica to get an idea of the scale of this term. ExpectedValueAH# - 1L 3 &, LogormalDistribution@0.03, 0.1DE The actual third-order term in the Taylor series is divided by 3 so this effect is also quite small. Thus, the continuous time log optimal results still provides reasonable guidance in most realistic investment decisions. If we are concerned about the approximation, then there is a simple adjustment that can be made to account for most of the discrepancy.