Choosing an Investment Portfolio

P.V. Viswanath

Based on Bodie and Merton

P.V. Viswanath 2

Returns and Return Uncertainty

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 3

Return Volatility

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 4

Variance

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 5

Return volatilities of Assets a & b

Probabilities Return (Asset a) Deviation from Mean Return

Squared Deviation

0.3 3 -4.3 18.49

0.2 7 -0.3 0.09

0.5 10 2.7 7.29

Probabilities Return (Asset b) Deviation from Mean Return

Squared Deviation

0.3 1 -6.3 39.69

0.2 7 -0.3 0.09

0.5 11.2 3.9 19.53

P.V. Viswanath 6

Variance and Standard Deviations

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%.

a is used to represent the standard deviation of returns on asset a, while Var(Ra) or a

2 is 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 7

Portfolios

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 8

Portfolio Return

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 9

Combining risky and riskless assets

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 C = y P

Suppose E(rP) = 15%, rf = 7%, and P = 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 10

The Risk-Reward Trade-Off Line

Capital Allocation Line

P

f

p= 22%

E(rp) - rf = 8%

rf = 7%

E(rp)=15%

Figure: Capital Allocation Line

Risk-reward ratio = 8/22

P.V. Viswanath 11

Covariance and Correlation

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 12

Covariance

Probabilities Ra – E(Ra) Rb – E(Rb) Product of deviations

0.3 -4.3 -6.3 27.09

0.2 -0.3 -0.3 0.09

0.5 2.7 3.9 10.53

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 13

Covariance

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 14

Combinations of risky assets

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 15

Combinations of risky assets

E r w E r w E rp D D E E( ) ( ) ( )

p E D E D Ew w w w Cov r rD D E

2 2 2 2 2 2 ( )

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 16

Combinations of risky assets

*

*

*

D

E

Figure: Combination Line of Portfolios constructed from two risky assets

The reward-to-variability ratio at portfolio p is equal to the slope of the tangent to the combination line at p.

Minimum Variance Portfolio

(r)

E(r)

Hypothetical combination line if standard deviations combined linearly

p

P.V. Viswanath 17

Optimal Combinations of risky assets and the risk-free asset

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 18

Optimal Combinations of risky assets and the risk-free asset

*

*

D

E

Figure: Location of the optimal portfolio when there is a risk-free asset

rf

* P

Capital Allocation Line

*

Optimal Portfolio

P.V. Viswanath 19

The Tangency Portfolio

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

),(])()([])([])([

),(])([])([22

2

EDfEfDDfEEfD

EDfEEfDD

rrCovrrErrErrErrE

rrCovrrErrEw

DE ww 1 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 20

The minimum variance portfolio

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

w DCov r r

Cov r rw E w DE D E

D E D Emin min min( )

( , )

( , ); ( ) ( )

2

2 2 21

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

P.V. Viswanath 21

Efficient Frontier with Two Assets

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 22

Efficient Frontier with many assets

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 23

Efficient Frontier with Many Risky Assets

Global Minimum

Variance Portfolio

Efficient Frontier

Individual Assets

MinimumVariance Frontier

Figure: The Minimum-Variance Frontier of Risky Assets

P.V. Viswanath 24

Efficient Frontier with Many Risky Assets and a Risk-free Asset

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 25

Efficient Frontier with Many Risky Assets and a Risk-free Asset

rf

Efficient Frontier

Figure: Optimal Portfolio Selection with Many Risky Assets and a Risk-free Asset

Efficient Portfolio, P, of Risky Assets: Forms the risky part of all investors' optimal portfolios.

Optimal Combination of the risk-free asset and Portfolio P.

E(r)

(r)