Risk and Rates of Return

   Stand-alone risk Portfolio risk Risk & return: CAPM / SML

5-1

Investment returns

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

5-2

What is investment risk?

   Two types of investment risk  Stand-alone risk  Portfolio risk Stand-alone risk: The risk an investor would face if he or she held only one asset.

Portfolio risk: The riskiness of assets held in portfolios.

5-3

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 8%, regardless of the economy.

No, T-bills do not provide a 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 risk-free in the default sense of the word.

5-4

Company

IBM Expected Rate of return  The rate of return expected to be realized from an investment.

Expected Rate of Return

-22% -2 20 35 50

Probability

10% 20 40 20 10

5-5

Return: Calculating the expected return for each alternative ^ k  expected rate of return k ^  i n   1 k i P i ^ k IBM  (-22%) (0.1)  (-2%) (0.2)  (20%) (0.4)  (35%) (0.2)  (50%) (0.1)  17.4%

5-6

Summary of expected returns for all alternatives

IBM Market USR T-bill Shell Exp return 17.4% 15.0% 13.8% 8.0% 1.7% IBM has the highest expected return, and appears to be the best investment alternative, but is it really? Have we failed to account for risk?

5-7

Risk: Calculating the standard deviation for each alternative   Standard deviation   Variance   2   i n   1 ( k i  kˆ ) 2 P i

5-8

Standard deviation calculation

  i n   1 (k i  k ^ ) 2 P i  IBM     (-22.0

 17.4) (20.0

2 17.4)   (50.0

17.4) 2 2 (0.1)  (0.4) (0.1) (-2.0

 (35.0

17.4) 2 17.4) (0.2) 2 (0.2)     1 2   IBM  20.04%  T bills  0.0%  USR Shell  13.4%  13.8%  M  15.3%

5-9

Comments on standard deviation as a measure of risk

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

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

Difficult to compare standard deviations, because return has not been accounted for.

5-10

Comparing risk and return

Security T-bills IBM Shell USR Market Expected return 8.0% 17.4% 1.7% 13.8% 15.0% Risk, σ 0.0% 20.04% 13.4% 13.8% 15.3% 5-11

Coefficient of Variation (CV)

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

 Very useful in comparing the risk of assets that have different expected returns.

CV  Std dev Mean   ^ k

5-12

Risk rankings, by coefficient of variation

T-bill IBM Shell USR Market CV 0.000

1.152

7.882

1.000

1.020

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

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

5-13

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.

5-14

Portfolio construction: Risk and return

Assume a two-stock portfolio is created with $50,000 invested in both IBM and Shell.

 Expected return of a portfolio is a weighted average of each of the component assets of the portfolio.

5-15

Calculating portfolio expected return ^ k p  i n   1 w i ^ k i ^ k p  0.5

(17.4%)  0.5

(1.7%)  9.6%

5-16

Calculating portfolio standard deviation

Forecasted return Portfolio Return Year IBM Shell Calculation

2004 8% 2005 10 2006 12 2007 14 2008 16 16% 14 12 10 8 (.50*8%) + (.50*16%) (.50*10%) + (.50*14%) (.50*12%) + (.50*12%) (.50*14%) + (.50*10%) (.50*16%) + (.50*8%)

Expected Portfolio Return

12% 12% 12% 12% 12%

5-17

Calculating portfolio standard deviation (cont.)

 Expected value of portfolio return, 2004-2008 K P = 12% + 12% + 12% + 12% + 12% 5 = 12%

5-18

Calculating portfolio standard deviation (cont.)

 P  i n   1 (k i  k ) 2 /n 1  P  (12% 12%) 2  (12% 12%) 2  (12% 12%) 2  (12% 12%) 2  (12% 12%) 2 /( 5  1 )  0 %

5-19

Alternative Formula for Calculating portfolio standard deviation  p  W1 2  1 2  W2 2  2 2  2W1W2  1  2

r

12 W1  Proportion of Asset 1 W2  Proportion of Asset 2  1  Standard Deviation of Asset 1  1  Standard Deviation of Asset 2

r

12  Correlatio n Coefficien t between th e return of assets 1 and 2

5-20

Returns distribution for two perfectly negatively correlated stocks (ρ = -1.0)

25 15 0 Stock W 25 15 0 Stock M 25 15 0 Portfolio WM -10 -10 -10 5-21

Returns distribution for two perfectly positively correlated stocks (ρ = 1.0)

25 15 0 -10 Stock M 25 15 0 -10 Stock M’ 25 15 0 Portfolio MM’ -10 5-22

Illustrating diversification effects of a stock portfolio 

p (%) 35 Company-Specific Risk Stand-Alone Risk,



p 20 Market Risk 0 10 20 30 40 2,000+ # Stocks in Portfolio 5-23

Breaking down sources of risk

Stand-alone risk = Market risk + Firm-specific risk   Market risk – portion of a security’s stand-alone risk that cannot be eliminated through diversification. Measured by beta. (e.g. War, Inflation, High Interest Rates) Firm-specific risk – portion of a security’s stand-alone risk that can be eliminated through proper diversification.

5-24

Capital Asset Pricing Model (CAPM)

  Model based upon concept that a stock’s required rate of return is equal to the risk-free rate of return plus a risk premium that reflects the riskiness of the stock after diversification.

CAPM : Ke= Rf + β(Rm – Rf) Rf = Risk free rate of return Rm = Market Return β = Beta Coefficient Ke = Required Return

5-25

Beta

  Measures a stock’s market risk, and shows a stock’s volatility relative to the market.

Indicates how risky a stock is if the stock is held in a well-diversified portfolio.

5-26

Comments on beta

     

If beta = 1.0, the security is just as risky as the average stock.

If beta > 1.0, the security is riskier than average.

If beta < 1.0, the security is less risky than average.

Most stocks have betas in the range of 0.5 to 1.5.

The beta coefficient for the market = 1 Betas May be positive or negative. But, positive is the norm.

5-27

The Security Market Line (SML): Calculating required rates of return SML: k i = k RF + (k M – k RF ) β i    Assume k RF = 8%, k M = 15% and β i =1.3

The market (or equity) risk premium is k i RP M = k M – k RF = 15% – 8% = 7%.

= 8.0% + (15.0% - 8.0%)(1.30) = 8.0% + (7.0%)(1.30) = 8.0% + 9.1% = 17.10%

5-28

What is the market risk premium?

  Additional return over the risk-free rate needed to compensate investors for assuming an average amount of risk.

Its size depends on the perceived risk of the stock market and investors’ degree of risk aversion.

5-29

An example: Equally-weighted two-stock portfolio   Create a portfolio with 50% invested in HT and 50% invested in Collections.

The beta of a portfolio is the weighted average of each of the stock’s betas.

β P β P β P = w 1 β 1 + w 2 β 2 = 0.5 (1.30) + 0.5 (-0.87) = 0.215

5-30

Factor that shifts the SML

18 15 11 8

 What if investors raise inflation expectations by 3%, what would happen to the SML?

k i (%)

D

I = 3% SML 2 SML 1 0 0.5

1.0

1.5

Risk, β i 5-31

Factors that change the SML

18 15 11 8

 What if investors’ risk aversion increased, causing the market risk premium to increase by 3%, what would happen to the SML?

k i (%)

D

RP M = 3% SML 2 SML 1 0 0.5

1.0

1.5 Risk, β i 5-32