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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%

Real

risk-free

Interest

Rate

k*

Nominal

risk-free

Interest

Rate

krf

Inflation-

risk

premium

IRP

=

+

Interest RatesReal

risk-free

Interest

Rate

k*

Nominal

risk-free

Interest

Rate

krf

Inflation-

risk

premium

IRP

=

+

Mathematically:

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

This is known as the “Fisher Effect”

Interest RatesProof 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%

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.

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.

rate of

return

Risk-free

rate of

return

=

For a Treasury security, what is the required 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.

rate of

return

Risk-free

rate of

return

Risk

premium

= +

For a corporate stock or bond, what is the required rate of return?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.

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%

= (ki - k)2 P(ki)

n

i=1

S

MAD, Inc.

( 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%

= (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?

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

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.

risk

Market risk

number of stocks

As you add stocks to your portfolio, company-unique risk is reduced.

risk

company-

unique

risk

Market risk

number of stocks

As you add stocks to your portfolio, company-unique risk is reduced.

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.

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