Risk, Return, and Discount Rates Capital Market History and The Risk/Return Relation.

Risk, Return, and Discount RatesRisk, Return, and Discount Rates

Capital Market History

and

The Risk/Return Relation

How Are Risk and Expected How Are Risk and Expected Return Related?Return Related?

There are two main reasons to be concerned with this question. (1) When conducting discounted cash flow analysis, how should we adjust discount rates to allow for risk in the future cash flow stream?

(2) When saving/investing, what is the tradeoff between taking risks and our expected future wealth?

Here we concentrate on the first of these questions. We will however develop the answer to the first question by asking the second. They are opposite sides of the same coin.

Discounting Risky Cash FlowsDiscounting Risky Cash Flows

How must the discount rate change in our standard NPV calculation if the cash flows are not riskless?– This is more easily answered from the “other side.”

What must be the expected return on a risky asset so you are happy to own it rather than a riskless asset?

– Risk averse investors say that to hold a risky asset they require a higher expected return than they require for holding a riskless asset. E(rrisky) = rf + .

» Note now that we have to start to talk about expected returns since risk is being explicitly examined.

– Since a “comparable alternative” to a risky project is owning a risky security the discount rate must also rise.

How does the expected return on a risky asset relate to the risk of that asset? It is easy to see that expected return should increase with the amount risk.

ExpectedReturn

RiskBut, how should risk be measured?At what rate does the line slope up? At what level does it start?Is it even a line?Lets look at some simple but important historical evidence.

The Future Value of an Investment The Future Value of an Investment of $1 in 1926of $1 in 1926

0.1

10

1000

1930 1940 1950 1960 1970 1980 1990 2000

Common StocksLong T-BondsT-Bills

$40.22

$15.64

63.845,2$)1()1()1(1$ 199919271926 rrr

Source: © Stocks, Bonds, Bills, and Inflation 2000 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved.

Rates of Return 1926-1999Rates of Return 1926-1999

-60

-40

-20

0

20

40

60

26 30 35 40 45 50 55 60 65 70 75 80 85 90 95

Common Stocks

Long T-Bonds

T-Bills


Risk, More FormallyRisk, More Formally

Many people think intuitively about risk as the possibility of an outcome that is worse than what one expected.

Useful but limited, if things can be worse they can also be better than expected. Risk is the possibility of an outcome being different than expected. – For those who hold more than one asset, is it the risk of

each asset they care about, or the risk of their whole portfolio?

The standard construct for thinking rigorously about risk:– The “probability distribution.”– A list of all possible outcomes and their probabilities.

Example: Two Probability Distributions Example: Two Probability Distributions on Tomorrow's Share Price.on Tomorrow's Share Price.

Which implies more risk?

00.10.20.30.40.50.6

10 12 13 14 16

0

0.1

0.2

0.3

0.4

10 12 13 14 16

In very simple cases, we specify the probability distributions completely.

More often, we use parameters of the historical distribution to summarize important information. The The expected valueexpected value, which is the center or , which is the center or

meanmean of the distribution. Thus when we think of the distribution. Thus when we think of returns as random this is expected return.of returns as random this is expected return.

The The variancevariance or or standard deviationstandard deviation, which , which measure the dispersion of possible outcomes measure the dispersion of possible outcomes around the mean. We noted this captures risk.around the mean. We noted this captures risk.

Risk and Probability DistributionsRisk and Probability Distributions

Historical Returns, 1926-1999 Historical Returns, 1926-1999


– 90% + 90%0%

Average Standard Series Annual Return Deviation Distribution

Large Company Stocks 13.0% 20.3%

Small Company Stocks 17.7 33.9

Long-Term Corporate Bonds 6.1 8.7

Long-Term Government Bonds 5.6 9.2

U.S. Treasury Bills 3.8 3.2

Inflation 3.2 4.5

The Risk-Return TradeoffThe Risk-Return Tradeoff

2%

4%

6%

8%

10%

12%

14%

16%

18%

0% 5% 10% 15% 20% 25% 30% 35%

Annual Return Standard Deviation

Ann

ual R

etur

n A

vera

ge

T-Bonds

T-Bills

Large-Company Stocks

Small-Company Stocks

Risk PremiumRisk Premium

We wrote the expected return on an asset as E(rrisky) = rf + .– We referred to as the risk premium, the return you get

for bearing risk. Defining the “market” portfolio to have one unit of

risk, the premium per unit risk is the expectation of the return on this portfolio less the riskless rate.– Measured this way, using the historical average, the risk

premium is about 9% per year (see chart).– Most believe this to be overstated, even vastly so.

SummarySummary

An old saying on Wall Street is “You can either sleep well or eat well.”

More than being simply colorful the idea is to constantly remind people that greater reward (expected reward) comes only with exposure to greater risk.

The “other side” of the question then suggests that for projects with risky cash flows the appropriate discount rate must be higher than that for a project with risk free cash flows.

The Capital Asset Pricing ModelThe Capital Asset Pricing Model

The Risk Return Relation Formalized

SummarySummary E(r) = rf + . = amount of risk premium per unit risk. Defined the “market portfolio” as having one unit of risk;

Var(rm) = 1 “unit of risk.” Risk premium per unit of risk is then E(rm) – rf. Diversification implies that an asset’s contribution to the risk of

a large portfolio is measured not by Var(ri) or STD (ri) but Cov(ri, rm).

Standardize Cov(ri, rm) by Var(rm) (to get it as the number of “units”) and get i as the measure of risk of any asset i.

Then: E(ri) = rf + i (E(rm) – rf) the famous SML.

Risk and ReturnRisk and Return When we consider only large portfolios risk and

return can be measured, as we discussed, using expected return and standard deviation of return.

When we want to consider individual assets, risk and return are more complex.

There are two types of risk for individual assets:– Diversifiable/nonsystematic/idiosyncratic risk– Nondiversifiable/systematic/market risk

Think of an asset’s risk as being the sum of two parts: risk = M + D

Nonsystematic/diversifiable risksNonsystematic/diversifiable risks

Examples– Firm discovers a gold mine beneath its property– Lawsuits– Technological innovations– Labor strikes

The key is that these events are random and unrelated across firms. For the assets in a portfolio, some surprises are positive, some are negative. If your portfolio is made up of a large number of assets, across assets, the surprises offset each other .

Systematic/Nondiversifiable riskSystematic/Nondiversifiable risk

We know that the returns on different assets are positively correlated with each other on average. This suggests that economy-wide influences affect all assets.

Examples:– Business Cycle– Inflation Shocks– Productivity Shocks– Interest Rate Changes– Major Technological Change

These are economic events that affect all assets in a similar way. The risk associated with these events does not disappear in well diversified portfolios as they tend to be in the same direction for all/most firms.

DiversificationDiversification

One of the most important lessons in all of finance has to do with the power of diversification.

Part of the risk of an asset can be diversified away without loss of expected return. A great benefit.

This also means that no compensation needs to be provided to investors for exposing their portfolios to this type of risk.

Which in turn implies that the risk/return relation is a systematic risk/return relation.

Diversification ExampleDiversification Example

Suppose a large green ogre has approached you and demanded that you enter into a bet with him.

The terms are that you must wager $10,000 and it must be decided by the flip of a coin, where heads he wins and tails you win.

What is your expected payoff and what is your risk?

Example…Example…

The expected payoff from such a bet is of course $0 if the coin is fair.

We can calculate the standard deviation of this “position” as $10,000, reflecting the wide swings in value across the two outcomes (winning and losing).

Can you suggest another approach that stays within the rules?


If instead of wagering the whole $10,000 on one coin flip think about wagering $1 on each of 10,000 coin flips.

The expected payoff on this version is still $0 so you haven’t changed the expectation.

The standard deviation of this version, however, is $100.

Why? If we bet a penny on each of 1,000,000 coin flips?

The risk, measured by standard deviation, is $10.


The example works so well at reducing risk because the coin flips are “independent.”

If the coins were somehow perfectly correlated we would be right back in the first situation.

With 10,000 flips, for “flip correlations” between zero (independence) and perfect correlation the measure of risk lies between $100 and $10,000.

This is one way to see that the way an “asset” contributes to the risk of a large “portfolio” is determined by its correlation or covariance with the other assets in the portfolio.

Portfolio RiskPortfolio Risk

Above, we characterized the risk of an asset as: M + D.

What we have just seen is that the D part (the part that is independent across assets) goes away in large portfolios at no cost (no reduction in expected return).

We have also argued that M (the risk that is left after diversification) is measured by the covariance between the assets in the portfolio.

Once we standardize covariance to put it in terms of our “one unit of risk” we wind up with the very famous beta coefficient (β) for stocks.

BetaBeta

Beta is just as we said “standardized covariance” βi = Cov(ri, rm)/Var(rm).

Alternatively we can think of it as a sensitivity measure.– The beta of the “market portfolio” is 1; one unit of risk.– When the market is unexpectedly up 5%, an asset with

a beta of ½ will, on average, have a return 2.5% above its expected return.

– What about an asset with a beta of 2.0? Most importantly beta measures the systematic

risk (its contribution to portfolio risk) of an asset.

Betas and PortfoliosBetas and Portfolios

The beta of a portfolio is the weighted average of the component assets’ betas.

Example: You have 30% of your money in Asset X, which has X = 1.4 and 70% of your money in Asset Y, which has Y = 0.8.

Your portfolio beta is:

P = .30(1.4) + .70(0.8) = 0.98. Why do we care about this feature of betas?

– It shows directly that an asset’s beta measures the contribution that asset makes to the systematic risk (β) of a portfolio!

Expected Returns and PortfoliosExpected Returns and Portfolios

Just as with betas, the expected return on a portfolio is a weighted average of the expected returns on the individual assets in the portfolio.

Example: You have 30% of your money in Asset X, which has E(rX) = 16.2% and 70% of your money in Asset Y, which has E(rY) = 11.4.

Your portfolio expected return is:

E(rP) = .30(16.2) + .70(11.4) = 12.84.

CAPM IntuitionCAPM Intuition

E[ri] = rF (risk free rate) + θ (Risk Premium)

= Appropriate Discount Rate Risk free assets earn the risk-free rate (think of

this as a rental rate on capital). If the asset is risky, we need to add a risk

premium. The risk premium is simply the number of units of

systematic risk times the price per unit of systematic risk.

The CAPM Intuition FormalizedThe CAPM Intuition Formalized

]][E[r)Var(r

)r,Cov(r]E[r FM

M

MiFi rr

]][E[r]E[r FMiFi rr

• The expression above is referred to as the “Security Market Line” (SML).

Number of units of systematic risk () Market Risk Premium

or the price per unit risk

or,

ProblemsProblems

The current risk free rate is 4% and the expected risk premium on the market portfolio is 7%. – An asset has a beta of 1.2. What is the expected return

on this asset? Interpret the number 1.2.

– An asset has a beta of 0.6. What is the expected return on this asset?

– If we invest ½ of our money in the first asset and ½ of our money in the second, what is our portfolio beta and what is its expected return?

ProblemProblem

The current risk free rate is 4% and the expected risk premium on the market portfolio is 7%.– You work for a software company and have been asked

to estimate the appropriate discount rate for a proposed investment project.

– Your company’s stock has a beta of 1.3.

– The project is a proposal to begin cigarette production.

– RJR Reynolds has a beta of 0.22.

What is the appropriate discount rate and why?

Risk, Return, and Discount Rates Capital Market History and The Risk/Return Relation.

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