Notes Portfolio
Copyright c 2005 by Karl Sigman
1
Portfolio mean and variance
Here we study the performance of a one-period investment X0 > 0 (dollars) shared among several different assets. Our criterion for measuring performance will be the mean and variance of its rate of return; the variance being viewed as measuring the risk involved. Among other things we will see that the variance of an investment can be reduced simply by diversifying, that is, by sharing the X0 among more than one asset, and this is so even if the assets are uncorrelated. At one extreme, we shall find that it is even possible, under strong enough correlation between assets, to reduce the variance to 0, thus obtaining a risk-free investment from risky assets. We will also study the Markowitz optimization problem and its solution, a problem of minimizing the variance of a portfolio for a given fixed desired expected rate of return.
1.1
Basic model
You plan to invest a (deterministic) total of X0 > 0 at time t = 0 in a portfolio of n ≥ 2 distinct assets, and the payoff X1 comes one period of time later (at time t = 1 for simplicity). Apriori you do not know how to distribute the amount X0 among the n assets, your objective being to distribute X0 in such a way as to give you the best performance. If X0i is the amount to be invested in asset i, i ∈ {1, 2, . . . , n}, then X0 = X01 + X02 + · · · + X0n . The portfolio chosen is described by the vector (X01 , X02 , . . . , X0n ) and its payoff is given by X1 = X11 +X12 +· · ·+X1n , where X1i is the (random) payoff from investing X0i in asset i, that is, the cash flow you receive at time t = 1. Ri , called the total return, is the payoff per dollar invested in asset i1 , Ri = X1i . X0i
We define the rate of return as the corresponding rate ri = R i − 1 = and it holds then that X1i = (1 + ri )X0i . But note that unlike fixed-income securities, here the rate ri is a random variable since X0i is assumed so. The expected rate of return (also called the mean
1
Portfolio mean and variance
Here we study the performance of a one-period investment X0 > 0 (dollars) shared among several different assets. Our criterion for measuring performance will be the mean and variance of its rate of return; the variance being viewed as measuring the risk involved. Among other things we will see that the variance of an investment can be reduced simply by diversifying, that is, by sharing the X0 among more than one asset, and this is so even if the assets are uncorrelated. At one extreme, we shall find that it is even possible, under strong enough correlation between assets, to reduce the variance to 0, thus obtaining a risk-free investment from risky assets. We will also study the Markowitz optimization problem and its solution, a problem of minimizing the variance of a portfolio for a given fixed desired expected rate of return.
1.1
Basic model
You plan to invest a (deterministic) total of X0 > 0 at time t = 0 in a portfolio of n ≥ 2 distinct assets, and the payoff X1 comes one period of time later (at time t = 1 for simplicity). Apriori you do not know how to distribute the amount X0 among the n assets, your objective being to distribute X0 in such a way as to give you the best performance. If X0i is the amount to be invested in asset i, i ∈ {1, 2, . . . , n}, then X0 = X01 + X02 + · · · + X0n . The portfolio chosen is described by the vector (X01 , X02 , . . . , X0n ) and its payoff is given by X1 = X11 +X12 +· · ·+X1n , where X1i is the (random) payoff from investing X0i in asset i, that is, the cash flow you receive at time t = 1. Ri , called the total return, is the payoff per dollar invested in asset i1 , Ri = X1i . X0i
We define the rate of return as the corresponding rate ri = R i − 1 = and it holds then that X1i = (1 + ri )X0i . But note that unlike fixed-income securities, here the rate ri is a random variable since X0i is assumed so. The expected rate of return (also called the mean