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Risk and Return: Diversification and the CAPM

Andrew Detzel, PhD, FRM

FIN 3300

Directions here:

Red Color Value: R=151, G=27, B=30

Learning Objectives

Define systematic risk and describe why investors should only demand higher expected returns to hold it instead of total risk.

Measure systematic risk with Beta

Reading: Ch 6.5, 7

Preliminaries: Properties of covariance

The covariance between two random variables and is defined to be: where is the value of in random state .

Two important properties follow from the definition. If is a constant (not random), and is another random variable, then:

Diversification

Recall: Investors demand positive returns on average to hold risky assets

Total risk of an asset is measured by standard deviation/volatility:

Investors care about the total volatilities of their whole portfolio…

…not just the volatility of individual assets in the portfolio.

Diversification reduces the volatility of portfolio returns relative to the sum of the volatilities of the individual assets in the portfolio:

How should we measure risk of an individual asset?

Should measure how asset incrementally impacts volatility of entire portfolio

Diversification and Market risk

What is the theoretically most diversified possible portfolio?

A: The market portfolio of all assets

Suppose the market portfolio () consists of assets:

How does each stock contribute to the volatility of the whole portfolio?

Each asset contributes to the risk of the portfolio via:

It’s weight in the portfolio

It’s covariance with the whole portfolio return

Diversification and Market risk (cont’d)

Each asset contributes to the risk of the portfolio via:

It’s weight in the portfolio

It’s covariance with the whole portfolio return

Covariance with the market return is Non-diversifiable/systematic risk

Diversification does not eliminate this risk

Hence investors should demand higher expected returns to compensate them for holding higher systematic risk

What does NOT show up in the equation?

Diversifiable risk (aka idiosyncratic risk) = represents the portion of an asset’s return that is not correlated with the market

It gets “averaged” out in the market portfolio across assets

Hence investors care less about it

Measuring systematic risk

What does covariance with the market look like?

The characteristic line is the best-fit line between a security’s excess returns ( and the market’s excess return.

The line represents the portion of returns that is perfectly correlated with market

Its slope is called the beta (), which measures the stock’s systematic risk

Typically proxy for the market return using broad stock market index (e.g. S&P 500)

Measuring systematic risk

The differences between the actual return and the line represents idiosyncratic returns (the volatility of which is diversifiable risk)

The “average out” across stocks

Beta (Formula)

An asset's beta coefficient is a measure of its non-diversifiable or systematic risk.

Beta is measures the asset's non-diversifiable risk per unit of risk of the Market Portfolio

A model of returns 1

The collective behavior of security market participants is characterized by risk aversion

Investments facing high “risk” must be accompanied by high expected return

A simple model:

Investors demand a base level rate of return even for a risk-free investment rf  Rate of return on a risk-free asset (e.g. t-bill rate)

Then, investors demand a risk premium to compensate for the risk of the the investment

Hence:

A model of returns 2

Investors demand a risk premium to compensate for the systematic risk of the the investment

Let be the risk premium per unit of systematic risk measured by :

The market has a beta of 1. Hence:

CAPM:

Example: Suppose the risk-free rate is 2%, the expected return on the market is 9% and you have calculated the beta of your portfolio to be 2. What is the expected return on your portfolio?

CAPM example

Example: Suppose the risk-free rate is 2%, the expected return on the market is 9% and you have calculated the beta of your portfolio to be 2. What is the expected return on your portfolio?

Concept check

You’re considering two investments, Dell and Exxon. If Dell has a  of 1.4 and standard deviation of 10% and Exxon has a  of 0.5 and a standard deviation of 20%, which is riskier?

Why would you prefer to hold the market return that has a beta of 1 instead of a single stock that has a beta of 1?

Does the CAPM work?

CAPM:

Suppose the market return has averaged 10% over recent history, while a mutual fund with a returned an average of 12%. Is this consistent with the CAPM?

What does CAPM predict the fund return should have been?

The Security Market Line

The SML and a Positive-Alpha Stock

How do we test the CAPM?

We test the CAPM (and any other model) using (“alpha”)

= ACTUAL average/expected returns an asset -average/expected returns it SHOULD earn under a model

CAPM

In the example on the previous page, the CAPM alpha is:

We estimate as the intercept from the best fit line of excess returns ( on asset a vs the market’s.

If , we do not reject the CAPM

Figure 7.4 Scatter Diagram/SCL: Google vs. S&P 500, 01/06-12/10

Alpha is a parallel shift

Size and book-to-market anomalies

We’re going to test whether the CAPM explains returns on small-cap value stocks

See “Size BM portfolios” spreadsheet:

6 portfolios formed on firm size (large vs small-cap stocks) and B/M (low, mid, high) from ALL U.S. Stocks

Taken from the website of Kenneth French

B/M is ratio of accounting (“book”) value of a firm’s equity to the market value

High B/M stocks are called value stocks

Low B/M stocks are called growth stocks

Value investing one of the oldest and most common strategies

Do small-cap value stocks earn alpha?

This was one of the first “anomalies” that ”broke” the CAPM.

Testing for 0 alpha

We can estimate alpha as the intercept from regression line.

The Analysis ToolPak add-in in Excel allows us to run a regression with an excess return like small-cap value stocks as the y variable, and the market excess return as the x variable

In this example, we want to know whether the total return on the portfolio of small-cap value stocks is explained by the CAPM.

Is 0 in: ?

Test this by estimating a regression:

Make sure to subtract off the risk-free rate

Testing for 0 alpha

The regression prints t-statistics and p-values for our estimates

The null value is the “default value” that you are testing whether the actual value is different from

For alpha: null value=0

The standard error represents how precisely we can measure the statistic (bigger s.e. means lower precision)

Testing for 0 alpha

The p-value is based on the t-statistic (big t  small p) and says how likely an alpha is to be as big or bigger as the one we observed…

... in our sample, just by chance

…if the true alpha is 0.

If the p-value is less than 5%, we typically conclude that the alpha is statistically significant/different from 0.

If the t > ~2, the p-value will be less than 0.05.

Caveat: Expected values vs estimates

The true alpha is:

Estimated from regression will never be exactly = true alpha

Same with the estimated

There will always be random sampling variation

Do not take your estimates of alpha literally, use methods that are robust to estimation error.

How much would you trust an estimate of alpha using 20 observations vs 200?

Given the sampling variation in this statistic, would you necessarily want to just pick a fund manager that had the highest alpha last quarter?

Regression statistics and beta

By default, statistics software prints out t-statistics with a null value of 0 for any given statistic.

If you want to test the null that for example, you need to make your own t-statistic:

The standard error is standard output from the regression ToolPak

How much should I invest if ?

Investors care about alpha of an individual asset and Sharpe ratio of their whole portfolio.

Suppose you fit a CAPM to your returns on security XYZ:

Relationship between Sharpe ratio and alpha:

Optimal weights: , , where

The portfolio P with these weights has:

Never invest when alpha=0

Concept check

Let denote the optimal portfolio consisting of and .

The is sometimes referred to as the “appraisal ratio” of the asset XYZ. By the equation: the appraisal ratios tells you how much better you can make your portfolio (in terms of Sharpe ratio) by adding asset XYZ.

Trades off benefit of alpha vs cost of diversifiable risk ()

You estimate that mutual fund XYZ has a CAPM alpha of 0

Would you add ANY to your portfolio?

How much should I invest if ?

Assuming I take our estimates of alpha from our regression literally, how much should we invest in small-cap value stocks to maximize Sharpe ratio of our portfolio?

Estimate using the sample average and variance

We get from the “Mean-squared residual” (in Excel, look for (row) column called (Residual) MS.

CAPM failures summary and solutions

The CAPM does not appear to explain the returns on small-cap high-B/M (value) stocks (positive alpha)

Some say this is because value stocks are underpriced (inefficient markets view)

Alternative: Some propose that stock market s alone do not completely capture systematic risk (efficient markets view)

In turn, researchers have proposed multifactor models to capture more common risk factors

Covariances/ with these common factors are still how we measure risk.

More than 75% of CFOs use the CAPM in estimating cost of capital

If the CAPM doesn’t work, why would they do this?