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Choosing an Investment Portfolio

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Choosing an Investment Portfolio

P.V. Viswanath

Based on Bodie and Merton

- If an investor buys an asset, a, at time 0 for $P0, receives a cashflow at time 1 of $D1 and sells it at time 1, for $P1, the return on the asset is defined as Ra,1 = (P1 + D1 – P0)/P0
- At time 0, the actual value of Ra,1 is unknown.
- Suppose there are three different possible outcomes, corresponding to three different states: states 1, 2 and 3.
- Thus Ra,1 could be equal to 3%, 7% or 10% with probabilities of 0.3, 0.2 and 0.5 for the three different states.
- The mean or expected value of Ra,1 is computed as 0.3(3%) + 0.2(7%) + 0.5(10%) = 7.3%
- The expected value is a kind of average outcome, where the different possible outcomes are weighted by the probabilities.
- E(Ra) is the way we represent the expected return on asset a.

P.V. Viswanath

- Consider another asset, b, which has returns of 1%, 7% and 11.2% in the three different states.
- The expected return of asset b is 0.3(1%) + 0.2(7%) + 0.5(11.20%), which is also 7.3%.
- However, asset b’s returns are said to be more volatile than those of asset a, meaning roughly that the range of possible outcomes is greater, or that more extreme outcomes – both positive and negative – are more likely.

P.V. Viswanath

- The variance of returns is computed as the mean squared deviation from the expected return.
- The squared deviation is a measure of how extreme a return is.
- Taking the mean computes the average value of these deviations.
- The square root of the variance is called the standard deviation or volatility.
- Variance is measured in percent-squared.
- Standard deviation is measured in the same units as returns, viz. percent.

P.V. Viswanath

P.V. Viswanath

- We see that the variance of returns for asset a is 9.21, computed as 0.3(18.49) + 0.2(0.09) + 0.5(7.29) and the standard deviation is 3.035%
- The variance of returns for asset b can be similarly computed and is 19.53; the standard deviation is 4.419%.
- sa is used to represent the standard deviation of returns on asset a, while Var(Ra) or sa2is used to represent the variance of returns on asset a.
- We see that, as expected asset b, which has more extreme outcomes, also has a higher volatility.

P.V. Viswanath

- A portfolio is defined as a set of assets and the proportions of money invested in those assets.
- Thus, if we agree on a set of assets, i = 1,…,n, then a set of weights wi, i=1,..,n represents a portfolio.
- If the assets are all stocks – SBUX, VZ and F, then (0.1, 0.45, 0.45) is a portfolio representing an investment of $100 in SBUX, $450 in VZ and $450 in F; or equivalently, $50 in SBUX, $225 in VZ and $225 in F.
- Even though the dollar returns will vary for the two portfolios, the percentage returns will always be the same; hence the mean return and variance of returns will be the same.

P.V. Viswanath

- Suppose we construct a portfolio, consisting of $30 in asset a and $70 in asset b; i.e. wa = 0.3 and wb = 0.7.
- Then the return on the portfolio in any given state will be waRa + wbRb.
- We can then compute the expected return for this portfolio, as well as its variance of returns.
- A riskfree asset always has the same return independent of the state.
- Hence, its standard deviation of returns is zero.

P.V. Viswanath

- Suppose we combine a risky portfolio, P with a risk-free asset, f.
- Let C denote the complete portfolio, and y the proportion invested in the risky portfolio. Then,
- rC = y rP + (1-y) rf
- E(rC) = y E(rP) + (1-y) rf, and
- sC = y sP
- Suppose E(rP) = 15%, rf = 7%, and sP = 22%, then we can compute the trade-off between expected return and volatility obtained by putting different proportions in the risky portfolio and the risk-free asset.

P.V. Viswanath

P.V. Viswanath

- If we have two assets a and b, the covariance between the returns on the two assets is defined as the weighted average of the product of deviations of each return from its mean return.
- The correlation is the ratio of the covariance to the product of the two standard deviations. The correlation can only take values between -1 and +1.
- We can use the deviations of the returns on assets a and b in our earlier example to compute the covariance coefficient and the correlation coefficient.

P.V. Viswanath

- The covariance can be computed as 0.3(17.49) + 0.2(0.49) + 0.5(19.24) = 13.41
- The correlation coefficient is 13.41/(3.035 x 4.419) = 0.99988

P.V. Viswanath

- The covariance (and correlation coefficient) is a measure of how closely two variables move together.
- In our case, both assets have high returns in state 1 and low returns in state 2; hence they have a high correlation. If there were another asset, c, with low returns in state 1 and high returns in state 2, it would have a negative correlation with asset a.
- An asset, d, with a return of 1%, 7% and 45% in the three states can be shown to have a correlation coefficient of only 0.93 with asset a.
- This correlation is less than the correlation between assets a and b because asset d behaves quite differently from asset a in state 3, even though state 3 is a good state for both assets.
- If we have many different states, we can construct assets with different kinds of correlation coefficients between their returns, corresponding to reality.

P.V. Viswanath

- Suppose we invest positive amounts in two risky assets that are imperfectly positively correlated.
- Then, because of the potential for diversification or offset between the returns of different assets in a portfolio, the standard deviation of returns on a portfolio is less than the weighted average of the standard deviation of returns on the assets making up the portfolio.
- The smaller the correlation coefficient, the greater the possibility of diversification and vice-versa.
- Thus, if we combine assets a and b in a portfolio, there will not be much diversification; however, if we combine assets a and d, there will be more diversification, while there will be even more diversification and potential for variance reduction if we combine assets a and c.

P.V. Viswanath

- Suppose we combine two risky assets, D and E, with portfolio weights wD and wE.
- Then the expected return and the variance of returns on the portfolio are given by

P.V. Viswanath

P.V. Viswanath

- If we have assets D and E, as above, but we have, in addition, a risk-free asset as well.
- Then, we can think of the optimal combination of the three assets as being achieved in two steps.
- First compute the optimal combination, P, of assets D and E. This portfolio is also called the tangency portfolio.
- Next combine this portfolio with the risk-free asset.
- In practice, the optimal risky portfolio will depend upon the risk-free rate, as well, as can be seen from the graph in the next slide.

P.V. Viswanath

P.V. Viswanath

- The proportions of assets D and E in the tangency portfolio are given by the formulas below:

- The particular proportions that the investor would then invest in the tangency portfolio versus the risk-free asset depends on his/her risk aversion. The greater the risk aversion, the more will be invested in the risk-free asset.

P.V. Viswanath

- The proportions of assets D and E in the minimum variance portfolio are given by the formulas below:

- Of course, no rational investor would actually hold this portfolio, unless s/he were extremely risk-averse.

P.V. Viswanath

- The efficient frontier is the set of portfolios of risky assets offering the highest possible expected rate of return for any given standard deviation.
- With two risky assets and one risk-free asset, the efficient frontier is precisely the line starting from the risk-free asset that is tangential to the combination line consisting of combinations of the two risky assets.
- For any standard deviation of returns the highest return is given by some portfolio on the above straight line.

P.V. Viswanath

- Suppose there are many risky assets, but no risk-free asset.
- Then conceptually, we can imagine taking pairs of assets and drawing their combination lines. Any point on one of these combination line is a portfolio of those two particular assets. We could then imagine combination lines where the two risky investments, themselves, are from the “combination” portfolios generated above.
- There would be a very large, potentially infinite number of these combination lines.
- If we consider all of the points on these infinite combination lines, we would have a dense area, bounded on the left by a parabola.

P.V. Viswanath

P.V. Viswanath

- In this case, the optimal combination of risky assets is once again found as the point of tangency between a straight line from the point representing the riskless asset and the efficient frontier of risky assets.
- The straight line connecting the riskless asset and the tangency point representing the optimal combination of risky assets is the best risk-reward trade-off line available – that is, the efficient frontier.

P.V. Viswanath

P.V. Viswanath