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portfolio_selection.pdf

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Rev. Aug. 29, 2017

PORTFOLIO SELECTION

AND THE CAPITAL ASSET PRICING MODEL

What portfolio would you recommend to a 28-year-old who has just been promoted

to a management position, and what portfolio would you recommend to a 60-yearold who has just retired?

The perennial question in the world of personal finance seems to be “What should I buy?”

A far better question would be “What portfolio should I hold?” The research exercise described in

this note provides an opportunity to explore the reasoning behind this distinction. More

importantly, it will become apparent that the answers to the portfolio question provide critical

insights into how we measure risk and determine appropriate rates of return for a given level of

risk. In particular, the analysis described here will develop familiarity with the capital asset pricing

model (CAPM), a model of appropriate returns based on the relation between the returns on an

individual asset and the returns on a broad market portfolio.

The analysis described here is organized around the question stated at the top of this page.

To focus our analysis, we will ignore issues related to the amount of wealth these individuals might

have, the tax situation each may face, and specific expenditure plans they might have. Simply

assume that both have relatively large amounts to invest, but not so large that they are indifferent

to the returns they might earn. Both would, of course, prefer higher returns to lower returns. The

newly promoted manager, however, would be able to accept a higher degree of risk than the retiree.

The analysis will proceed as follows. The accompanying spreadsheet (UVA-F-1604X)

presents historic returns of three stocks as well as the returns on the S&P 500 index. You will be

guided through the analysis of these data. Each analysis will be accompanied by a series of

questions. The first set of steps examines various portfolios of investments; the second set explores

the implications of the portfolio results for individual stocks.

This technical note was prepared by Associate Professor Marc Lipson. Copyright 2009 by the University of Virginia

Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail to

[email protected] No part of this publication may be reproduced, stored in a retrieval system,

used in a spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording,

or otherwise—without the permission of the Darden School Foundation.

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Portfolio Selection

We will begin this analysis by considering just single investments and then proceed to add

additional alternatives, including portfolios combining individual investments.1 The analysis will

make use of basic statistical descriptions of returns and employ a very simple regression analysis.

Step 1: Calculate the realized returns for a hypothetical portfolio invested equally in the three

stocks. You may simply average the three returns.2 We will refer to this as the equally weighted

portfolio. Calculate the mean (average) realized return and standard deviation of the realized

returns for each stock, the S&P 500 index, and the equally weighted portfolio.

Compare and contrast the mean and standard deviations of these five possible

investments. What are the noticeable differences?

When choosing between the three individual stock investments and ignoring

combinations of these investments, which would you suggest to our hypothetical

investors? Why?

How does your answer change if you add the index as an allowable investment?

Step 2: Calculate the correlation between each of the three stocks and between the S&P 500

and the equally weighted portfolio. Assume that the average return, standard deviation and

correlation of each of these assets from 1990 to 2009 are a good estimate of these same

measures for the future. Using the template provided in the second tab, construct a graph that

shows the expected future relation between the returns (vertical axis) and standard deviations

(horizontal axis) of a pair of the stock investments (your choice which pair).3

1

A portfolio can consist of a single investment and need not have more than one component.

Averaging the returns implicitly assumes that each month, you are rebalancing the portfolio so that an equal

value is invested in each stock.

3

The template produces the graph by first generating a set of observations and then plotting those observations.

This is done as follows. A table was created that has one column with the weight assigned to one stock (using

increments of 1%); a second column with the weight assigned to the second stock (one minus the weight of the first

stock); a third column with the anticipated standard deviation of this two-asset portfolio based on the weights, standard

deviations, and correlation. The formula for the mean would be rp w1 r1 w2 r2 where rp is the portfolio return, r1

2

is the mean historic return of stock number one, r2 is the mean historic return of stock number two, and w refers to

the proportional weight invested in each subscripted security. For example, if 30% of a portfolio is invested in security

number one, then w1 = 0.30 and, since all weights must sum to 1, w2 would equal 0.70. For the standard deviation, the

formula would be p sqrt (w1212 w 22 std22 2w1w

2

1 2 corr1,2 ) where σ denotes the standard deviation of historic

returns for the subscripted security, corr denotes the correlation between the subscripted securities, and the p subscript

indicates the portfolio you are examining. The resulting expected returns are plotted against the standard deviations.

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Is there an intuitive explanation for the shape of the curve you have generated?

How does the shape of the curve change as you alter the correlation (which is bounded

by −1 and 1, inclusive)?

What do the correlations between the individual stocks indicate? What insight is

provided by the correlation between the equally weighted portfolio and the S&P 500

index?

Step 3: Using the template provided, construct a graph that shows the expected returns and

standard deviations of all possible combinations of the three stocks. This graph was constructed

in a fashion analogous to the two-asset graph, but expanded to three stocks.4 This graph

describes the complete investment space offered by the three stock investments.

Restricting yourself to combinations of the three stocks, what portfolio of stock

investments would you recommend for our two hypothetical investors?

Locate the S&P 500 index on this graph. Explain the location of this index relative to

the set of possible three-stock combinations.

Step 4: We can expand the set of financial assets by considering a risk-free bond (U.S.

government security). We will assume that the return on the bond is 0.45% a month, which is

about 5.5% per year.

What would be the diversification benefits from adding a bond?

This risk-free asset can be combined with the market portfolio to yield another set of possible

investments. This line is referred to as the capital market line. Add the capital market line to

your graph from Step 3. The can be done by adding points to the graph that represent linear

combinations of the bond and the market.

4

Explain the shape of the capital market line.

Now consider all the possible investments you have identified (the three individual

stocks, all combinations of the three stocks, the index, and the security market line).

What would you recommend to our two hypothetical investors? Be very specific in

your recommendation.

The formula for the mean would be

rp w1 r1 w2 r2 w3 r3 and for the standard deviation for three stocks

2

would be p sqrt (w 21 21 w 22 22 w 2

3 3 2w w1 21

2 corr1,2

2w1w 3

1 3 corr1,3 2w 2w

3

2

3 corr 2,3 )

. Given

the number of possible combinations, the graph uses increments of 5% in the weights and starts by generating all 5%

incremented combinations of weights on the three stock investments.

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Individual Asset Returns

It should be clear from the previous steps that portfolios of stocks offer significant

advantages over individual stocks. In fact, a case could be made that an extremely broad portfolio

(a market index) is the optimal portfolio of risky investments, and this portfolio would be combined

with an appropriate amount of riskless bonds to achieve a desired optimal overall investment

portfolio.

The optimality of holding portfolios has important implications for individual stocks.

When considering the riskiness of an individual stock, investors will ignore the total risk of the

stock (since much of it will be diversified away) and consider only that level of risk that cannot be

diversified away. This can be described as the risk an individual stock adds to a particular portfolio.

The remaining steps develop a measure of this risk and link that measure to returns.

Step 5: To measure the risk a stock adds to a given portfolio, one simply regresses the returns

of the individual stock on the returns of the portfolio. The regression coefficient on the portfolio

reflects exactly that risk. For example, if we regress an individual stock’s returns on a broad

market index, the resulting coefficient on the market is called a market beta (or, more

commonly, simply the beta). The beta reflects the variation in individual stock returns that

cannot be diversified away. One simple form for the regression is the following:5

ri, t alpha beta rm, t t

(1)

In the regression model above, returns are observed for each time period t for an individual

stock i and market return m. The regression acknowledges individual error terms each period

of ε since stock returns will deviate from this relation due to firm-specific events. Please

calculate the alpha and beta for each of the three stocks relative to the S&P 500 index. This

can be done using the regression tool from the Data Analysis Toolkit in Excel or using the

following simple formulas:

beta

corri,m i

m

alpha ri beta rm

(2)

(3)

The regression analysis will provide additional statistical information on the relationship, but

the above equations are sufficient.

5

Market beta regressions are often run using returns in excess of a risk-free rate and are often also adjusted for

various statistical problems; however, this simple form typically provides very similar results.

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Based on the betas, which firm is the most risky? Least risky? How does your answer

compare with the answer you provided based on standard deviations? Which one is

appropriate for our hypothetical investors? Explain why.

What is the economic meaning of the alpha?

Step 6: Since the beta provides a measure of risk, one should be able to link expected return to

this risk measure. By expected return, we mean the return that an individual investor would

expect to earn from holding this stock. This can be calculated quite easily. First, one recognizes

that a riskless bond has a beta of zero and that the market portfolio has a beta of one. Since all

the diversification benefits associated with portfolios have already been accounted for, the

graph of the relationship between beta and returns for portfolios combining the risk-free bond

and the market portfolio is linear. All individual stocks must lie on this same line. Thus, the

equation for returns for every security is the equation of the line that connects the market return

and the bond return in a graph of returns (vertical axis) against beta (horizontal axis). 6 That

equation and the theories that justify it are referred to as the CAPM). The equation is

ri rrf (rm rrf ) rrf (MRP)

(4)

where ri is the expected return for an individual stock, rrf is the expected risk-free rate of return,

β is the beta, rm is the expected return on the market, and (in the simplified version) MRP is

the market risk premium (the difference between the expected return on the market and the

expected risk-free rate of return).

Given the calculated beta for each stock, calculate the expected monthly return for each stock

assuming that the monthly risk-free rate is 0.45% (about 5.5% annually) and the monthly risk

premium is 0.50% (about 6% annually). Calculate the annual expected returns implied by these

monthly returns. Calculate the realized annual returns from the mean returns in Step 1.

How do the realized returns compare with the expected returns? Assuming the CAPM

describes the appropriate expected returns for these stocks, describe how prices might

respond if, at some point, the expected returns on the three stocks differed from what

was predicted by the CAPM.

Summary

These steps illustrate the essential link between risk and returns as it would be viewed in a

world where individuals can (and therefore do) hold portfolios of investments. We have used past

data to generate a description of the world and considered our choices assuming the past behavior

would indicate future (expected) behavior. There are a number of reasons one needs to be cautious

6

It is crucial to recognize that this graph is plotting returns relative to beta while the earlier graphs plot returns

relative to standard deviations.

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about this assumption, and in many cases the results for past data would not be a reasonable

estimate for the future. Applying the ideas developed here should be based on best estimates of

future expected return behavior.

…

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