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**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
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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
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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
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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
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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
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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
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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
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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
**

Comments on portfolio risk measures

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
**

General comments about risk

σ 35% for an average stock.

Most stocks are positively (though not perfectly) correlated with the market (i.e., ρ between 0 and 1).

Combining stocks in a portfolio generally lowers risk.