5-1

CHAPTER 5Risk and Rates of Return

5-2

5.1 Rates of Return

Holding Period Return: Rates of Return over a given period

Suppose the price of a share is currently $100, and your time horizon is one year. You expect the cash dividend during the year to be $4. Your best guess about the price of the share is to be $110 after one year from now. So,

priceBeginning

dividendCashpriceBeginningpriceEndingHPR

%14100$

4$100$110$

HPR

5-3

Measuring Investment Return over Multiple Period

YearAnnual Holding Period Return 

2000 0.2

2001 0.05

2002 -0.05

2003 0.15

2004 0.3

Arithmetic Mean (20+5-5+15+30)/5 =0.13

Geometric Mean (1.2*1.05*.95*1.15*1.3)1/5-1 =0.1

5-4

5.2 Risk and Return

Risk is the concept of fluctuations. This fluctuations can be (i) a deviation of the actual return from the expected return, or (ii) a deviation of average return from the year to year return. Higher the fluctuations, higher is the risk.

Measures of risk are: i. Standard Deviation ii. Coefficient of Variance iii. Beta

5-5

Calculation of Risk-Return

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ondistributiyprobabilitforRRPRisk

dataseriestimeForn

RRRisk

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RRREeturnRExpected

ii

i

iii

5-6

Calculation of Risk-Return (Historical Data)

Year Return (%) Dev. (Ri-E(R)) Dev. Square2000 20 7 49

2001 5 -8 64

2002 -5 -18 324

2003 15 2 4

2004 30 17 289

Mean Return= 13% Sum of Dev sq= 730

Stand.Deviation2   730/(5-1)= 182.5

Stand.Deviation=    Square root (182.5)= 13.5%

5-7

Calculation of Risk-Return: Scenario Analysis (Probability Distribution)

State of

EconomyProbability

(Pi)Return (Ri)

(%)Exp. Value

(Pi*Ri)Deviation (Ri-E(R))

DeviationSquare Dev sq* Pi

Boom 0.25 25 6.25(25-13.25)

=11.75(11.25)2

=138.0625(138*.25)=34.52

Normal 0.5 14 7(14-13.25)

=0.75(0.75)2

=0.5625(.56*.5)=0.28

Recession 0.25 0 0(0-13.25)=-13.25

(-13.25)2

=175.5625(175*.25)=43.89

13.25% 78.69

Return =13.25% Risk=8.9% Stand Dev 8.87

m

iiiRPRE

1

)(

m

iiii RERp

1

22 })]({[(

5-8

Figure:5.3:Normal DistributionA normal distribution looks like a bell-shaped curve.

Probability

99.74%

– 3 – 13.36%

– 2 – 4.5%

– 1 4.4%

013.25%

+ 1 22.12%

+ 2 31%

+ 3 39.86%

Mean=13.25%

Standard Deviation=8.87%

68.26%

95.44%

5-9

5.3 THE HISTORICAL RECORDBills, Bonds, and Stocks:1926-2006

5-10

Figure 5.1 Frequency Distributions of Holding Period Returns

5-11

Probability distributions

With the same average return more standard deviation means more risk. Shown graphically. Note that as risk increases height goes down and width increases.

Expected Rate of Return

Rate ofReturn (%)100150-70

Firm X

Firm Y

%40

%10

y

x

5-12

Effectiveness of Diversification of Portfolio

Climate Probability Return of Umbrella

ReturnIce-cream

PortfolioReturn

Rainy 0.25 25 -5 10

Moderate 0.5 14 10 12

Dry 0.25 0 15 7.5

E(R) = 

13.3% 7.5% 10.4%

Risk 

8.9% 7.5% 1.9%

ii PR *

ii PRR 2)(

5-13

Portfolio Effects on Risk and Return

0

2

4

6

8

10

12

14

0 2 4 6 8 10

Risk

Ret

urn

Series1

5-14

Optimum portfolio and CML: Given the feasible set highest possible utility function gives us O.P. and the tangency is CML

.R

etu

rn

Risk (σ)

CML

Feasible set

Lending

Borrowing

U1

O.P

5-15

Investor attitude towards risk

Risk aversion – assumes investors dislike risk and require higher rates of return to encourage them to hold riskier securities.

Risk premium – the difference between the return on a risky asset and less risky asset, which serves as compensation for investors to hold riskier securities.

5-16

Risk Premiums and Risk Aversion

If T-Bill denotes the risk-free rate, rf, and variance, , denotes volatility of returns then:

The risk premium of a portfolio is:

2P

2P

2

To quantify the degree of risk aversion with parameter A:

If the risk premium of a portfolio is 8%, and the standard deviation is 20%, then risk aversion of the investor is: A=.08/(.5x.22)=4.

Compare this with risk premium of 10%, and A=.1/(.5x.22)=5

5-17

The Sharpe (Reward-to-Volatility) Measure

5-18

Coefficient of Variation (CV)

A standardized measure of dispersion about the expected value, that shows the risk per unit of return. When, both return and risk increase then coefficient of variance (CV) should be used.

^

k

Mean

dev Std CV

5-19

Use of coefficient of variance

Example: We have 2 alternatives to invest. Security A has a mean return of 10% and a standard deviation of 6%, and security B has a mean return of 13% with a standard deviation of 8%. Which investment is better.

%5.61100*%13

%8

%60100*%10

%6

B

A

CV

CV

So, security A is better as the Coefficient of variance of A is less than the that of B.

5-20

5.4 INFLATION AND REAL RATES OF RETURN

5-21

Real vs. Nominal Rates Fisher effect: ApproximationLet, nominal rate=R

Real rate=rInflation rate (CPI)=i

nominal rate = real rate + inflation rate: R ≈ r + i or r = R - i

Example r = 3%, i = 6%R = 9% = 3% + 6% or 3% = 9% - 6%

Fisher Effect:

2.83% = (9%-6%) / (1.06)

5-22

5.5 ASSET ALLOCATION ACROSS RISKY AND RISK-FREE PORTFOLIOS

5-23

Risk tolerance: Slope of Indifference curves Return

ConservativeNormal

Aggressive

Risk (σ)

15%

10%

7%6%

5-24

Optimum portfolio with different risk tolerance

ReturnConservative

Normal

Aggressive

Risk (σ)

CML

Efficient Frontier

σm

Rm

5-25

Risk Aversion and Allocation

Greater levels of risk aversion lead to larger proportions of the risk free rate

Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets

Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

5-26

Possible to split investment funds between safe and risky assets

Risk free asset: proxy; T-bills Risky asset: stock (or a portfolio)

Issues Examine risk/ return tradeoff Demonstrate how different degrees of risk aversion

will affect allocations between risky and risk free assets

Allocating Capital

5-27

The Risky Asset:Text Example (Page 149)

Total portfolio value = $300,000Risk-free value = 90,000 Risky (Vanguard and Fidelity) = 210,000Vanguard (V) = 54% =$113,400 Fidelity (F) = 46% = $96,600Total =$210,000y=210,000/300,000 =0.7 (portfolio in risky assets)1-y=90,000/300,000 =0.3 (proportion of Risk-free investment)

Vanguard 113,400/300,000 = 0.378

Fidelity 96,600/300,000 = 0.322

Portfolio P 210,000/300,000 = 0.700

Risk-Free Assets F 90,000/300,000 = 0.300

Portfolio C 300,000/300,000 = 1.000

5-28

Change in risk exposure (p.150)

Suppose, the investor decides to decrease the risk exposure from 0.7 to 0.56.

Now, the risky portfolio would be=0.56*300,000=$168,000 This requires a sale of (210,000-168,000)=$42,000 of risky

holdings, and use the sale proceeds to purchase risk-free asset

How would the investment of risky asset change? Sale of Vanguard=42,000*.54=$22,680;

New Vanguard holding=113,400-22,680=$90,720 Sale of Fidelity=42,000*.46=$19,320

New Fidelity holding=96,600-19,320=$77,280 Total Vanguard & Fidelity=90,720+77,280=$168,000 y=168,000/300,000=.56

5-29

Capital Allocation Line (CAL)(Test of linearity)

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5-30

Figure 5.5 Investment Opportunity Set with a Risk-Free Investment

5-31

CAL and CMLre

turn

P

rf

CML CAL 1

CAL 2

CAL3

CAL4

Efficient Frontier

Rm

m

CAL1 is dominant over CAL2, CAL2 is dominant over CAL3.

CML is dominant over CAL1.

5-32

5.6 PASSIVE STRATEGIES AND THE CAPITAL MARKET LINE

5-33

Table 5.5 Average Rates of Return, Standard Deviation and Reward to Variability

5-34

Use of historical data to predict CML

Use of old data is popular Old data may not be representative for future Weight of old data keeps changing Correction of old data for future is largely

subjective, but customary.

5-35

Costs and Benefits of Passive Investing

Active strategy entails costs Free-rider benefit Involves investment in two passive

portfolios Short-term T-bills Fund of common stocks that follows a

broad market index. Diversification can be based on asset allocation like industry classification, large and small firms, local and foreign firms