Decision Theory gives us a conceptual framework for formalizing optimal choice. This framework is almost indispensible if we want to solve for an optimal choice, and it is also useful for thinking about a choice problem even if we are going to use a combination of intuition and informal analysis to make the final decision.
The essential pieces of a choice problem are the choice variables, the objective function, and the constraints. We may also have parameters, which are inputs to the choice problem that can be varied.
Choose c1, c2, and c3 to maximize U(c1,c2,c3) subject to p1 c1 + p2 c2 + p3 c3 = W, c1 >= 0, c2 >= 0, and c3 >= 0.
The choice variables are c1, c2, and c3, which are expenditures on three classes of consumption goods. The objective function is U(c1,c2,c3), which represents the consumer's preferences for different consumption patterns. There are four constraints, the budget constraint and three positivity constraints. Some parameters of the problem include the prices p1, p2, and p3, and wealth W. There may be additional parameters describing the features of the utility function. The parameters are not chosen as part of the decision; we leave them flexible to allow us to study, for example, the sensitivity of the consumption of good one to its price.
Choose q1, q2, and q3 to maximize E[u(w1)] subject to q1 + q2 + q3 = 1 and w1 = w0(q1(1+r1) + q2(1+r2) + q3(1+r3)).
The choice variables are q1, q2, and q3, which are proportional investments in three different securities. The objective function is E[u(w1)], which represents the investor's preferences over different random payoffs. The first constraint is the budget constraint that proportions in the securities sum to one, and the second constraint defines the payoff w1 given the amounts invested and the random returns r1, r2, and r3 on the three securities. To complete this specification, we would have to specify the particular utility function u and the joint probability distribution of the returns.
Suppose there are two assets, a riskless asset 1 with return 10% and a risky asset 2 with equally probable returns -10% and +50%. Assuming that the utility function is u(w1) = w1-.004w1^2 and the initial wealth is 100, we can substitute in the constraints (e.g. q1 = 100%-q2) and use elementary algebra to reduce the previous choice problem to the following.
Choose q2 to
2
maximize 61.6 + 1.2 q2 - 4.0 q2
The solution is q2 = .15, i.e. 15% of wealth should be in the risky asset and 85% in the riskless asset. (One way of proving this is by writing the objective function as 61.69 - 4.0 (q2 - 0.15)^2.) This is a very conservative strategy; a less conservative solution would arise if we replaced .004 with a smaller number, representing a smaller aversion to risk-taking.
E[u(w1)] = 0.5u(w1u)+0.5u(w1d)
= 0.5(u(100(1.1q1+1.5q2))+u(100(1.1q1+.9q2)))
= 0.5(u(100(1.1(1-q2)+1.5q2))+u(100(1.1(1-q2)+.9q2)))
= 0.5(u(110+40q2)+u(110-20q2))
= 0.5((110+40q2)-.004(110+40q2)^2
+(110-20q2)-.004(110-20q2)^2)
= 0.5(110+40q_2-48.4-35.2q_2-6.4q_2^2
+110-20q_2-48.4+17.6q_2-1.6q_2^2)
= 61.6+1.2q_2-4q_2^2
= 61.69-4(.15-q_2)^2
<=61.69
(since the square cannot be negative), which is the value when q2=0
For an individual's portfolio choice, the choice problem in the previous slides is a good start. In most institutional settings, there are at least two levels of management. At the highest level, the plan sponsor (the representative of the beneficiaries) must choose proportions of the portfolio to be allocated to different asset classes and more specifically how to allocate funds within each asset class to different managers (who may or may not be in-house). We call this selection of broad asset classes asset allocation. The problem of the specific manager (who may manage a portfolio of equities or government bonds or convertible bonds), we refer to as subportfolio management.
Traditionally, academic finance has not looked at subportfolio problems, which are significantly different from asset allocation problems. Subportfolio managers are typically judged relative a benchmark appropriate to the asset class and are constrained directly and indirectly in how far they can deviate from the benchmark. The traditional tools for asset allocation can be modified in a natural way to study subportfolio management.
Mean-variance theory is an important model of investments based on decision theory. It is the simplest model of investments that is sufficiently rich to be directly useful in applied problems. Mean-variance theory was developed in the 50's and 60's by Markowitz, Tobin, Sharpe, and Lintner, among others. Ironically, it is still called Modern Portfolio Theory (MPT) by some people. While no longer the most modern model, mean-variance theory continues to be the main workhorse on which analytical portfolio management is based. The equilibrium version of mean-variance theory is called the Capital Asset Pricing Model (CAPM).
The nicest feature of mean-variance theory is its simplicity. By assuming that preferences depend only on the mean and variance of payoffs and not on other features, we obtain a number of robust results.
The assumptions of no taxes, transaction costs, or information asymmetry are sometimes known collectively as the assumption of ``perfect capital markets.''
For these statistics, there are ``population'' versions (which is what you would expect or what you would see in a hypothetical large sample comprised of the entire population of possible events) and ``sample'' versions telling what actually happened. For example, if we have a coin we believe to be fair, the population probability of ``heads'' is 1/2. If we have seen 1000 flips of this coin, 508 heads and 492 tails, the sample probability of ``heads'' is 508/1000 = 0.508. We will define various statistics in terms of their sample versions, but it is important to keep in mind the difference between sample and population values. In many contexts, the sample version is a good estimate of the population version. However, due to the amount of volatility in security returns, sample means may be very bad estimates of the population means. As a result, using the sample version into a model expecting the population version can produce bizarre prescriptions. In particular, it it may instruct you to put extreme amounts of money into securities and sectors that performed better than expected in the past.
The mean return, a measure of a typical value, is the usual arithmetic average:
1 n
mean = - Sum R
n i=1 i
The variance, a measure of volatility or dispersion, is the average squared deviation from the mean:
1 n 2
variance = --- Sum (R - mean)
n-1 i=1 i
The covariance, a measure of co-movement, is the average product of deviations from the mean:
xy 1 n x x y y
covariance = --- Sum (R - mean )(R - mean )
n-1 i=1 i i
Covariances are important in portfolio theory because they tell us whether risks cancel or compound when assets are combined in portfolios.
Consider two assets, 1 and 2, whose returns have variances S1^2 and S2^2, respectively, and whose returns have covariance S12. Then the variance of a portfolio with weight W1 in the first asset and weight W2 in the second asset (with residual weight 1-W1-W2 in the riskless asset) has variance
2 2 2 2 W1 S1 + 2 W1 W2 S12 + W2 S2.
It can be shown that - S1 S2 <= S12 <= S1 S2 (or equivalently the correlation coefficient S12/(S1 S2) must always lie between -1 and 1).
For a portfolio with many assets, there are many cross terms like the middle one here. If there are n assets, there are n variance terms and n(n-1)/2 cross terms. In a portfolio with a typical universe of assets, estimating all the covariances needed for the cross terms is an important practical question, since the number of covariances may exceed the number of data points.
For shares of stock, we can think of the return being equal to a mean return plus a random part from market-level noise plus a random part idiosyncratic to the firm. Mathematically, this means that
R = mean + beta z + e i i i m i
where meani and betai are constants, and the market noise zm and the idiosyncratic noise terms ei are all uncorrelated (and therefore have zero covariances). This assumption that idiosyncratic risk is uncorrelated across assets is not strictly true (and multifactor models seem to fit better), but this assumption will give us the right intuition.
The variance of a single asset is given by
2
beta var(z ) + var(e ).
i m i
For a typical large stock, we might have betai = .8, var(zm)=.04, var(ei)=.09, and therefore the asset's return variance is .1156. In this example, the market standard deviation is 20% and the asset's standard deviation is sqrt(.1156)=34%. For a typical small stock, we might have betai = 1.5, var(zm)=.04, and var(ei)=.16, and therefore the asset's return variance is .25. In this example, the market standard deviation is 20% and the asset's standard deviation is 50%.
As an example, assume that we put equal amounts of money into n assets. Then the resulting portfolio has variance
1 2 1 1 (- Sum beta ) var(z ) + -(- Sum var(e )). n i i m n n i i
As the portfolio gets larger, the idiosyncratic risk term becomes less and less important, and we can approximate the portfolio's risk by the first term.
Following on the previous slide, suppose we form portfolios with assets all having the same beta = 1 and isiosyncratic standard deviation .3. Suppose further that the market standard deviation is .2. Then using the formula on the previous slide, we find that the standard deviation of a portfolio is computed as
2 1 2
sqrt(.2 + -(.3) ).
n
Here are some values of this function:
| number of assets |
portfolio standard deviation |
| 1 | 36.06% |
| 2 | 29.15% |
| 5 | 24.08% |
| 10 | 22.14% |
| 20 | 21.10% |
| 50 | 20.45% |
| 100 | 20.22% |
| 1000 | 20.02% |
Although not all assets have the same idiosyncratic variance, this still gives an accurate qualitative picture of the value of diversification. Note that diversification does not affect the mean return, which will be the mean of the individual asset mean returns.
We will not go through the algebra of optimal portfolio choice. The main message I take from the algebra is that we tilt into each uncorrelated source of risk in proportion to the expected return we get and in inverse proportion to variance and risk aversion. This basic rule tells how to take consistent exposures to different risks.
As a first simple problem, suppose our optimal mix when choosing only between riskless asset and the market portfolio is 50% in each, where the riskless rate is 5%, the mean return on the market is 15%, and the standard deviation of the market is 20%. Suppose then that we have a stock ``Island'' that has a beta of zero (this means all of its risk is uncorrelated with the market), a mean return of 10%, and a standard deviation of 50%. How much of our portfolio should we invest in each of Island, the market, and the riskless asset?
We know that the investment in uncorrelated sources of risk should be in proportion to their mean excess return over variance. In the case of the market, the mean excess return over variance is (15%-5%)/.04=2.5 and this justifies a commitment of 50% of the portfolio. In the case of Island stock, we have mean excess return over variance of (10%-5%)/.25=0.2. Since this is proportional to the commitment, we should commit 50%(0.2/2.5)=4% of the portfolio to Island stock. After committing half to the market portfolio and 4% to Island stock, we have left 100%-50%-4%=46% to invest in the riskless asset.
Our example of Island Corporation stock was made particularly simple because of the beta of zero which meant that no manipulation was required to treat uncorrelated sources of risk. In general, we should regroup the assets into portfolios that provide ``pure plays'' in uncorrelated sources of risk.
Assume the market portfolio and base problem are as in the previous example, i.e., that our optimal mix of riskless asset and the market portfolio is 50% in each, where the riskless rate is 5%, the mean return on the market is 15%, and the standard deviation of the market is 20%. Suppose then that we have a stock ``Hitek'' with a beta of 1.5, a mean return of 30%, and an idiosyncratic standard deviation of 50%. What is our optimal holding of Hitek, the market portfolio, and the riskless asset?
In the problem, we are not given our level of risk aversion, but we will take that to be implicit in our optimal mix between the market and the riskless asset. In that case, the mean excess return over variance 10%/.04=2.5 induced us to invest half our wealth. To use this in the new problem, we need to focus on uncorrelated risk, that is a net position with the market risk removed. In this case, consider an investment that is long 100% share of Hitek, short 150% in the market, and long 150% in the riskless asset. This new investment has a beta of 0 (it is uncorrelated with the market as we wish), a mean return of 15%, and a standard deviation of 50%. The mean excess return of this portfolio is 30% - (1.5)15% + (1.5)5% - 5% = 10%. Therefore, the mean excess return over variance of this is 10%/.25=.4, and we should invest a proportion of wealth (.4/2.5)50% = 8% in this strategy. This implies overall holdings of 50% - (1.5)8% = 38% in the market portfolio, 8% in Hitek stock, and 100% - 38% - 8% = 54% in the riskless asset.
Suppose our optimal mix of riskless asset and the market portfolio is 50% in each, where the riskless rate is 5%, the mean return on the market is 15%, and the standard deviation of the market is 20%. Suppose then that we favor a stock ``Bloochip'' with a beta of 1.0, a mean return of 20%, and an idiosyncratic standard deviation of 25%. What is our optimal holding of Bloochip, the market portfolio, and the riskless asset?