# Risk And Rates Of Return, Lecture Notes - Financial Management, Study notes for Financial Management. University of Michigan (MI)

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CHAPTER 5 Risk and Rates of Return

8-1

CHAPTER 8: Risk and Rates of Return Updated: September 20, 2011

All Financial Assets Produce CFs

Risk of Asset Depends on Risk of CFs

Stand-alone Risk of Asset’s CFs

Portfolio Risk of CFs

Diversifiable and Market Risk

Risk & return: CAPM / SML

8-2

Investment returns

The rate of return on an investment can be calculated as follows:

(Amount received – Amount invested) Return = ________________________

Amount invested

For example, if \$1,000 is invested and \$1,100 is returned after one year, the rate of return for this investment is:

(\$1,100 - \$1,000) / \$1,000 = 10%.

8-3

What is investment risk?

 Two types of investment risk

 Stand-alone risk

 Portfolio risk

 Investment risk is related to the probability of earning a low or negative actual return.

 The greater the chance of lower than expected or negative returns, the riskier the investment.

 Risk = Dispersion of Returns around mean, or expected mean: variance or standard deviation

8-4

Probability distributions

 A listing of all possible outcomes, and the probability of each occurrence.

 Can be shown graphically.

Expected Rate of Return

Rate of

Return (%) 100 15 0 -70

Firm X

Firm Y

8-5

Selected Realized Returns, 1926 – 2004

Average Standard

Return Deviation

Small-company stocks 17.5% 33.1%

Large-company stocks 12.4 20.3

L-T corporate bonds 6.2 8.6

L-T government bonds 5.8 9.3

U.S. Treasury bills 3.8 3.1

Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation

Edition) 2005 Yearbook (Chicago: Ibbotson Associates, 2005), p28.

8-6

Investment alternatives

Economy Prob. T-Bill HT Coll USR MKT.

Recession 0.1 5.5% -27.0% 27.0% 6.0% -17.0%

Below avg 0.2 5.5% -7.0% 13.0% -14.0% -3.0%

Average 0.4 5.5% 15.0% 0.0% 3.0% 10.0%

Above avg 0.2 5.5% 30.0% -11.0% 41.0% 25.0%

Boom 0.1 5.5% 45.0% -21.0% 26.0% 38.0%

8-7

Why is the T-bill return independent of the economy? Do T-bills promise a completely risk-free return?

 T-bills will return the promised 5.5%, regardless of the economy.

 No, T-bills do not provide a completely risk-free return, as they are still exposed to inflation. Although, very little unexpected inflation is likely to occur over such a short period of time.

 T-bills are also risky in terms of reinvestment rate risk.

 T-bills are risk-free in the default sense of the word.

8-8

How do the returns of HT and Coll. behave in relation to the market?

 HT – Moves with the economy, and has a positive correlation. This is typical.

 Coll. – Is countercyclical with the economy, and has a negative correlation. This is unusual.

8-9

Calculating the expected return

12.4% (0.1) (45%)

(0.2) (30%) (0.4) (15%)

(0.2) (-7%) (0.1) (-27%) r

P r r

return of rate expected r

HT

^

N

1i ii

^

^







 

8-10

Summary of expected returns Expected return HT 12.4% Market 10.5% USR 9.8% T-bill 5.5% Coll. 1.0%

HT has the highest expected return, and appears to be the best investment alternative, but is it really? Have we failed to account for risk?

8-11

Calculating standard deviation

deviation Standard

2Variance 

i 2

N

1i i P)r(rσ 

 ˆ

8-12

Standard deviation for each investment

15.2%

18.8% 20.0%

13.2% 0.0%

(0.1)5.5) - (5.5

(0.2)5.5) - (5.5 (0.4)5.5) - (5.5

(0.2)5.5) - (5.5 (0.1)5.5) - (5.5

P )r (r

M

USRHT

CollbillsT

2

22

22

billsT

N

1i i

2 ^

i





  

  









2 1

8-13

Comparing standard deviations

USR

Prob. T - bill

HT

0 5.5 9.8 12.4 Rate of Return (%)

8-14

Comments on standard deviation as a measure of risk

 Standard deviation (σi) measures total, or stand-alone, risk.

 The larger σi is, the lower the probability that actual returns will be closer to expected returns.

 Larger σi is associated with a wider probability distribution of returns.

8-15

Comparing risk and return

Security Expected return, r

Risk, σ

T-bills 5.5% 0.0%

HT 12.4% 20.0%

Coll* 1.0% 13.2%

USR* 9.8% 18.8%

Market 10.5% 15.2%

* Seem out of place.

^

8-16

Coefficient of Variation (CV)

A standardized measure of dispersion about the expected value, that shows the risk per unit of return.

r

return Expected

deviation Standard CV

ˆ

 

8-17

Risk rankings, by coefficient of variation

CV T-bill 0.0 HT 1.6 Coll. 13.2 USR 1.9 Market 1.4

 Collections has the highest degree of risk per unit of return.

 HT, despite having the highest standard deviation of returns, has a relatively average CV.

8-18

Illustrating the CV as a measure of relative risk

σA = σB , but A is riskier because of a larger probability of losses. In other words, the same amount of risk (as measured by σ) for smaller returns.

0

A B

Rate of Return (%)

Prob.

8-19

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 a riskless asset, which serves as compensation for investors to hold riskier securities.

8-20

Portfolio construction: Risk and return

 Assume a two-stock portfolio is created with \$50,000 invested in both HT and Collections.

 A portfolio’s expected return is a weighted average of the returns of the portfolio’s component assets.

 Standard deviation is a little more tricky and requires that a new probability distribution for the portfolio returns be devised.

8-21

Calculating portfolio expected return

6.7% (1.0%) 0.5 (12.4%) 0.5 r

rw r

:average weighted a is r

p

^

N

1i

i

^

ip

^

p

^



 

8-22

An alternative method for determining portfolio expected return

Economy Prob. HT Coll Port.

Recession 0.1 -27.0% 27.0% 0.0%

Below avg 0.2 -7.0% 13.0% 3.0%

Average 0.4 15.0% 0.0% 7.5%

Above avg 0.2 30.0% -11.0% 9.5%

Boom 0.1 45.0% -21.0% 12.0%

6.7% (12.0%) 0.10 (9.5%) 0.20

(7.5%) 0.40 (3.0%) 0.20 (0.0%) 0.10 rp ^





8-23

Calculating portfolio standard deviation and CV

0.51 6.7%

3.4% CV

3.4%

6.7) - (12.0 0.10

6.7) - (9.5 0.20

6.7) - (7.5 0.40

6.7) - (3.0 0.20

6.7) - (0.0 0.10

p

2 1

2

2

2

2

2

p



      

      



8-24

 σp = 3.4% is much lower than the σi of either stock (σHT = 20.0%; σColl. = 13.2%).

 σp = 3.4% is lower than the weighted average of HT and Coll.’s σ (16.6%).

 Therefore, the portfolio provides the average return of component stocks, but lower than the average risk.

 Why? Negative correlation between stocks.

8-25