Chapter 6

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Chapter 6. Risk and Rates of Return. Chapter 6 Objectives. Inflation and rates of return How to measure risk (variance, standard deviation, beta) How to reduce risk (diversification) How to price risk (security market line, CAPM). Historical Risk and Return.

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Chapter 6

Risk and Rates of Return

Chapter 6 Objectives
• Inflation and rates of return
• How to measure risk

(variance, standard deviation, beta)

• How to reduce risk

(diversification)

• How to pricerisk

(security market line, CAPM)

Historical Risk and Return
• Annual From 1926 to 1999

Avg. ReturnStd Dev.

Small Stocks 17.6% 33.6%

Large Co. Stocks 13.3% 20.1%

L-T Corp Bonds 5.9% 8.7%

L-T Govt. Bonds 5.5% 9.3%

T-Bills 3.8% 3.2%

Inflation 3.2% 4.6%

Why are these rates different?
• 90-day Treasury Bill 1.7%
• 90-day Commercial Paper 1.8%
• 2-year US Treasury Note 1.9%
• 10-year US Treasury Note 3.8%
• 10-year AAA Corporate Bond 5.0%
• 10-year BBB Corporate Bond 6.1%

Conceptually:

Real

risk-free

Interest

Rate

k*

Nominal

risk-free

Interest

Rate

krf

Inflation-

risk

IRP

=

+

Interest Rates

Conceptually:

Real

risk-free

Interest

Rate

k*

Nominal

risk-free

Interest

Rate

krf

Inflation-

risk

IRP

=

+

Mathematically:

(1 + krf) = (1 + k*) (1 + IRP)

This is known as the “Fisher Effect”

Interest Rates
Proof of Fisher Effect Equation
• You have a \$100 to buy items costing a \$1 each. You can buy 100 items now.
• Instead of spending the \$100 now, you decide to invest the money at 7% (nominal risk-free rate) for a year giving you \$107 at the end of the year so you can hopefully buy more than 100 items at the end of the year.
Proof of Fisher Effect Equation
• At the end of the year, the items now cost \$1.04 each (4% inflation).
• You can buy \$107/\$1.04 = 102.88 of these items at the end of the year.
• This represents a 2.88% increase in your real purchasing power (real interest rate).
• We used (1+krf) = (1+k*)(1+IRP)
• (1.07) = (1+k*)(1.04):
• 1+k* = 1.07/1.04 = 1.0288; k* = .0288 = 2.88%

yield

to

maturity

time to maturity (years)

Term Structure of Interest Rates
• The pattern of rates of return for debt securities that differ only in the length of time to maturity.

yield

to

maturity

time to maturity (years)

Term Structure of Interest Rates
• The yield curve may be downward sloping or “inverted” if rates are expected to fall.

Required

rate of

return

Risk-free

rate of

return

=

Since Treasuries are essentially free of default risk, the rate of return on a Treasury security is considered the “risk-free” rate of return.

Required

rate of

return

Risk-free

rate of

return

Risk

= +

How large of a risk premium should we require to buy a corporate security?

Returns
• Expected Return - the return that an investor expects to earn on an asset, given its price, growth potential, etc.
• Required Return - the return that an investor requires on an asset given itsriskand market interest rates.
Holding Period (Actual) Returns
• The realized return over a period of time (HPR).
• HPR=(Ending Price - Beginning Price + Distributions Received)/Beginning Price
• Example: What is your HPR if you buy a stock for \$20, receive \$1 in dividends, and then sell it for \$25.
• HPR = (\$25-\$20+\$1)/\$20 = 0.3 = 30%
Calculation of Expected Returns
• Expected Rate of Return (Expected Value) given a probability distribution of possible returns(ki): E(k) or k

_ n

E(k)=k = ki P(ki)

i=1

• Realized or Average Return on Historical Data:

- n

k = 1/n k i

i=1

Expected Return and Standard Deviation Example
• MAD E(r) = .25(80%) + .60(30%) + .15(-30%) = 33.5%
• CON E(r) = .25(5%) + .60(10%) + .15(15%) = 9.5%
Definition of Risk
• Risk is an uncertain outcome or chance of an adverse outcome.
• Concerned with the riskiness of cash flows from financial assets.
• Namely, the chance that actual cash flows will be different from forecasted cash flows.
• Standard Deviation can measure this type of risk.
• For a stock, we can examine the standard deviation of the stock’s returns.

n

i=1

s

S

Standard Deviation

= (ki - k)2 P(ki)

Expected Return and Standard Deviation Example
• MAD E(r) = .25(80%) + .60(30%) + .15(-30%) = 33.5%
• CON E(r) = .25(5%) + .60(10%) + .15(15%) = 9.5%

s

= (ki - k)2 P(ki)

n

i=1

S

( 80% - 33.5%)2 (.25) = 540.56

(30% - 33.5%)2 (.6) = 7.35

(-30% - 33.5%)2 (.15) = 604.84 Variance = 1152.75%

Stand. dev. = 1152.75 = 34.0%

Expected Return and Standard Deviation Example
• MAD E(r) = .25(80%) + .60(30%) + .15(-30%) = 33.5%
• CON E(r) = .25(5%) + .60(10%) + .15(15%) = 9.5%

s

= (ki - k)2 P(ki)

n

i=1

S

Contrary Co.

(5% - 9.5%)2 (.25) = 5.06

(10% - 9.5%)2 (.6) = 0.15

(15% - 9.5%)2 (.15) = 4.54

Variance = 9.75%

Stand. dev. = 9.75 = 3.1%

Which stock would you prefer?

How would you decide?

Return

Risk

It depends on your tolerance for risk!

Remember, there’s a tradeoff between risk and return.

Coefficient of Variation
• A relative measure of risk. Whereas, s is an absolute measure of risk.
• Relates risk to expected return.
• CV = s/E(k)
• MAD’s CV = 34%/33.5% = 1.01
• CON’s CV = 3.1%/9.5% = 0.33
• CONtrary is the less risky of the two investments. Would choose CON if risk averse.
Portfolios
• Expected Portfolio Return is weighted average of the expected returns of the individual stocks = Σwjkj.
• However, portfolio risk (standard deviation) is NOT the weighted average of the standard deviations of the individual stocks.
• Combining several securities in a portfolio can actually reduce overall risk.
• How does this work?

Suppose we have stock A and stock B. The returns on these stocks do not tend to move together over time (they are not perfectly correlated).

kA

rate

of

return

kB

time

kA

kp

rate

of

return

kB

time

Diversification
• Investing in more than one security to reduce risk.
• If two stocks are perfectly positively correlated, diversification has no effect on risk.
• If two stocks are perfectly negatively correlated, the portfolio is perfectly diversified.
Some risk can be diversified away and some cannot.
• Market risk (systematic risk) is nondiversifiable. This type of risk cannot be diversified away.
• Company-unique risk (unsystematic risk) is diversifiable. This type of risk can be reduced through diversification.

portfolio

risk

Market risk

number of stocks

portfolio

risk

company-

unique

risk

Market risk

number of stocks

Note:
• The market compensates investors for accepting risk - but only for market risk. Company-unique risk can and should be diversified away.
• So - we need to be able to measure market risk. We use beta as a measure of market risk.
The Concept of Beta
• Beta(b) measures how the return of an individual asset (or even a portfolio) varies with the market portfolio.
• b= 1.0 : same risk as the market
• b< 1.0 : less risky than the market
• b> 1.0 : more risky than the market
• Beta is the slope of the regression line (y = a + bx) between a stock’s return(y) and the market return(x) over time, b from simple linear regression.
• bi = Covariancei,m/Mkt. Var. =rimsism/sm2
Relating Market Risk and Required Return: the CAPM
• Here’s the word story: a stock’s required rate of return = risk-free rate + the stock’s risk premium.
• The main assumption is investors hold well diversified portfolios = only concerned with market risk.
• A stock’s risk premium = measure of systematic risk X market risk premium.
CAPM Equation
• krp= market risk premium = km - krf
• stock risk premium = bj(krp)
• kj = krf + bj(km - krf )

= krf + bj (krp)

Example: What is Yahoo’s required return if its b = 1.75, the current 3-mo. T-bill rate is 1.7%, and the historical market risk premium of 9.5% is demanded?

Yahoo k = 1.7% + 1.75(9.5%) = 18.3%

Portfolio Beta and CAPM
• The b for a portfolio of stocks is the weighted average of the individual stock bs.
• bp = Swjbj
• Example: The risk-free rate is 6%, the market return is 16%. What is the required return for a portfolio consisting of 40% AOL with b = 1.7, 30% Exxon with b = 0.85, and 30% Fox Corp. with b = 1.15.
• Bp = .4(1.7)+.3(0.85)+.3(1.15) = 1.28
• kp = 6% + 1.28(16% - 6%) = 18.8%
More CAPM/SML Fun!
• According to the CAPM and SML equation with k = 6% + b(16% - 6%)
• How would a change in inflation affect required returns? (Say inflation increases 2% points)
• How would a change in risk aversion (market risk premium) affect required returns? (Say market risk premium decreases 2% points.)
Limitations of CAPM/SML
• Don’t really know what the market portfolio is, which makes it hard to estimate market expected or required return.
• Beta estimates can be unstable and might not reflect the future.
• Maturity debate over proper risk-free estimate.
• Most investors focus on more than systematic risk.