risk and rates of return

1

Risk and Rates of Risk and Rates of ReturnReturn

Chapter 6Chapter 6

4Interest RateInterest Rate

Interest rate represents the cost of moneyInterest rate represents the cost of moneyIt is the opportunity cost of money:It is the opportunity cost of money:

It shows the return lost from not investing in a comparable risk investment.

It is expected to compensate the investor for the time, inflation, and risk.

5Interest Rates

Conceptually:

6Interest Rates

Conceptually:

Nominalrisk-freeInterest

Rate

krf

7Interest Rates

Conceptually:


Rate

krf

=

8Interest Rates

Conceptually:


Rate

krf

=

Realrisk-freeInterest

Rate

k*

9Interest Rates

Conceptually:


Rate

krf

=


Rate

k*

+

10Interest Rates

Conceptually:


Rate

krf

=


Rate

k*

+

Inflation-risk

premium

IRP

11

Conceptually:


Rate

krf

=


Rate

k*

+

Inflation-risk

premium

IRP

Mathematically:

Interest Rates

12

Conceptually:


Rate

krf

=


Rate

k*

+

Inflation-risk

premium

IRP

Mathematically:

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

Interest Rates

13

Conceptually:


Rate

krf

=


Rate

k*

+

Inflation-risk

premium

IRP

Mathematically:

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

This is known as the “Fisher Effect”

Interest Rates

14

Suppose the real rate is 3%, and the nominal Suppose the real rate is 3%, and the nominal rate is 8%. What is the inflation rate premium?rate is 8%. What is the inflation rate premium?

(1 + k(1 + krfrf) = (1 + k*) (1 + IRP)) = (1 + k*) (1 + IRP)(1.08) = (1.03) (1 + IRP)(1.08) = (1.03) (1 + IRP)(1 + IRP) = (1.0485),(1 + IRP) = (1.0485), so so

IRP = 4.85%IRP = 4.85%

Interest Rates

15Term Structure of Interest Rates

The pattern of rates of return for debt The pattern of rates of return for debt securities that differ only in the length securities that differ only in the length of time to maturity.of time to maturity.



yieldto

maturity

time to maturity (years)



yieldto

maturity



yieldto

maturity


The yield curve may be downward sloping or The yield curve may be downward sloping or “inverted” if rates are expected to fall.“inverted” if rates are expected to fall.

20For a Treasury security, what is the required rate of return?


RequiredRequired

rate of rate of

returnreturn==


Since Treasuries are essentially Since Treasuries are essentially free of default free of default riskrisk, the rate of return on a Treasury security , the rate of return on a Treasury security is considered the is considered the ““risk-freerisk-free”” rate of return. rate of return.

RequiredRequired

rate of rate of

returnreturn==

Risk-freeRisk-free

rate of rate of

returnreturn

23

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

24


RequiredRequired

rate of rate of

returnreturn==

25


RequiredRequired

rate of rate of

returnreturn==

Risk-freeRisk-free

rate of rate of

returnreturn

26


How large of a How large of a risk premiumrisk premium should we require should we require to buy a corporate security? to buy a corporate security?

RequiredRequired

rate of rate of

returnreturn== + +

Risk-freeRisk-free

rate of rate of

returnreturn

RiskRisk

premiumpremium

27Returns

Expected ReturnExpected Return - the return that an - the return that an investor expects to earn on an asset, investor expects to earn on an asset, given its price, growth potential, etc.given its price, growth potential, etc.

Required ReturnRequired Return - the return that an - the return that an investor requires on an asset given investor requires on an asset given itsits riskrisk and market interest rates.and market interest rates.

28Risk and Rates of ReturnRisk and Rates of Return

Two Components of returnTwo Components of returnPeriodic cash flowsPeriodic cash flows


Two Components of returnTwo Components of returnPeriodic cash flowsPeriodic cash flowsPrice Change (capital gains)Price Change (capital gains)


Holding Period return Holding Period return



PPtt + D + Dtt

= ---------- - 1= ---------- - 1 PPt-1t-1



PPtt + D + Dtt

= ---------- - 1= ---------- - 1 PPt-1t-1

(P(Ptt - P - Pt-1t-1) + D) + Dtt

= ---------------- = ---------------- PPt-1t-1


Expected ReturnExpected ReturnExpected return is based on expected cash flows (not

accounting profits)

Return can be expressed as Cash Flows or Percentage Return

Return can be expressed as Cash Flows or Percentage Return



accounting profits)In an uncertain world future cash flows are not known

with certainty



accounting profits)In uncertain world future cash flows are not known with

certaintyTo calculate expected return, compute the weighted

average of all possible returns





average of possible returnsCalculating Expected Return:

k k P ki ii

N

( )

1





average of possible returnsCalculating Expected Return:

k k P ki ii

N

( )

1

whereki = Return state iP(ki) = Probability of ki occurringN = Number of possible states


Expected Return CalculationExpected Return Calculation

ExampleExampleYou are evaluating ElCat Corporation’s common stock. You estimate the following returns given different states of the economy

State of Economy Probability Return

Economic Downturn .10 –5%Zero Growth .20 5%Moderate Growth .40 10%High Growth .30 20%






= –0.5%x

k k Pi ii

N

(k )

1






= –0.5%= 1%

xx

k k Pi ii

N

(k )

1






= –0.5%= 1%= 4%

xxx

k k Pi ii

N

(k )

1






= –0.5%= 1%= 4%= 6%

xxxx

k k Pi ii

N

(k )

1






= –0.5%= 1%= 4%= 6%

k = 10.5%

xxxx

k k Pi ii

N

(k )

1






= –0.5%= 1%= 4%= 6%

k = 10.5%

Expected (or average) rate of return on stock is 10.5%

Expected (or average) rate of return on stock is 10.5%

xxxx

k k Pi ii

N

(k )

1


RiskRiskRisk is the uncertainty of future outcomes



ExampleExampleYou evaluate two investments: ElCat Corporation’s common stock and a one year Gov't Bond paying 6%. The return on the Gov't Bond does not depend on the state of the economy--you are guaranteed a 6% return.




100%

Return

Probability of Return

T-BillT-Bill

6%




100%

Return


T-BillT-Bill

6%

10%Return


ElCat CorpElCat Corp

5%

20%30%40%

–5% 10% 20%




100%

Return


T-BillT-Bill

6%

10%Return


ElCat CorpElCat Corp

5%

20%30%40%

–5% 10% 20%

There is risk in Owning ElCat stock, no risk in owning the Treasury Bill

There is risk in Owning ElCat stock, no risk in owning the Treasury Bill


Measuring RiskMeasuring RiskStandard Deviation () measure the dispersion of

returns.



returns.

(k ) (k )i ii

N

k P2

1



returns.

(k ) (k )i ii

N

k P2

1ExampleExampleCompute the standard deviation on ElCat common stock. the mean (k) was previously computed as 10.5%



returns.

(k ) (k )i ii

N

k P2






returns.

(k ) (k )i ii

N

k P2




x ( – 10.5%)2 = 24.025%2



returns.

(k ) (k )i ii

N

k P2




xx

((

– 10.5%)2 = 24.025%2

– 10.5%)2 = 6.05%2



returns.

(k ) (k )i ii

N

k P2




xxx

(((

– 10.5%)2 = 24.025%2

– 10.5%)2 = 6.05%2

– 10.5%)2 = 0.10%2



returns.

(k ) (k )i ii

N

k P2




xxxx

((((

– 10.5%)2 = 24.025%2– -- 10.5%)2 = 6.05%2

– 10.5%)2 = 0.10%2

– 10.5%)2 = 27.075%2



returns.

(k ) (k )i ii

N

k P2




xxxx

((((

– 10.5%)2 = 24.025%2

– 10.5%)2 = 6.05%2

– 10.5%)2 = 0.10%2

– 10.5%)2 = 27.075%2

2 = 57.25%2



returns.

(k ) (k )i ii

N

k P2




xxxx

((((

– 10.5%)2 = 24.025%2

– 10.5%)2 = 6.05%2

– 10.5%)2 = 0.10%2

– 10.5%)2 = 27.075%2

2 = 57.25%2

= 57.25%2



returns.

(k ) (k )i ii

N

k P2




xxxx

((((

– 10.5%)2 = 24.025%2

– 10.5%)2 = 6.05%2

– 10.5%)2 = 0.10%2

– 10.5%)2 = 27.075%2

2 = 57.25%2

= 57.25%2

= 7.57%



returns.

(k ) (k )i ii

N

k P2




xxxx

((((

– 10.5%)2 = 24.025%2

– 10.5%)2 = 6.05%2

– 10.5%)2 = 0.10%2

– 10.5%)2 = 27.075%2

2 = 57.25%2

= 57.25%2

= 7.57%Higher standard deviation, higher riskHigher standard deviation, higher risk



returns.

(k ) (k )i ii

N

k P2




xxxx

((((

– 10.5%)2 = 24.025%2

– 10.5%)2 = 6.05%2

– 10.5%)2 = 0.10%2

– 10.5%)2 = 27.075%2

2 = 57.25%2

= 57.25%2

= 7.57%

NOTE:NOTE: The standard deviation of the T-Bill is 0%

NOTE:NOTE: The standard deviation of the T-Bill is 0%

Higher standard deviation, higher riskHigher standard deviation, higher risk



returns.

(k ) (k )i ii

N

k P2




xxxx

((((

– 10.5%)2 = 24.025%2

– 10.5%)2 = 6.05%2

– 10.5%)2 = 0.10%2

– 10.5%)2 = 27.075%2

2 = 57.25%2

= 57.25%2

= 7.57%

Can compare the of 7.57 to another stock with expected return of 10.5%

Can compare the of 7.57 to another stock with expected return of 10.5%


Measuring RiskStandard Deviation () for historical data can be used to measure the dispersion of historical returns.

N

ii kk

n 1

2)(_)1(

1


Use the following data to calculate the historical return Use the following data to calculate the historical return of XYZof XYZ

YearYear ReturnReturn19921992 12%12%19931993 16%16%19941994 -8%-8%1995 6% 1995 6%


Risk and DiversificationRisk and DiversificationRisk of a company's stock can be separated into two

parts:



parts:Firm Specific Risk - Risk due to factors within the firm



parts:Firm Specific Risk - Risk due to factors within the firm

Stock price will most likely fall if a major government contract is discontinued unexpectedly.

Stock price will most likely fall if a major government contract is discontinued unexpectedly.



parts:Firm Specific Risk - Risk due to factors within the firmMarket related Risk - Risk due to overall market

conditions




conditions

Stock price is likely to rise if overall stock market is doing well.

Stock price is likely to rise if overall stock market is doing well.




conditionsDiversification: If investors hold stock of many

companies, the firm specific risk will be canceled out: Investors diversify portfolio.




conditionsDiversification: If investors hold stock of many

companies, the firm specific risk will be canceled out: Investors diversify portfolio.

Firm specific risk also called diversifiable risk or unsystematic risk

Firm specific risk also called diversifiable risk or unsystematic risk


Risk and DiversificationRisk and DiversificationRisk of a company's stock can be separated into two parts:

Firm Specific Risk - Risk due to factors within the firmMarket related Risk - Risk due to overall market conditions

Diversification: If investors hold stock of many companies, the firm specific risk will be canceled out: Investors diversify portfolio.

Even if hold many stocks, cannot eliminate the market related risk


Risk and DiversificationRisk and DiversificationRisk of a company's stock can be separated into two parts:

Firm Specific Risk - Risk due to factors within the firmMarket related Risk - Risk due to overall market conditions

Diversification: If investors hold stock of many companies, the firm specific risk will be canceled out: Investors diversify portfolio.

Even if hold many stocks, cannot eliminate the market related risk

Market related risk is also called non-diversifiable risk or systematic risk

Market related risk is also called non-diversifiable risk or systematic risk


Risk and DiversificationRisk and DiversificationIf an investor holds enough stocks in portfolio (about

20) company specific (diversifiable) risk is virtually eliminated




Number of stocks in Portfolio

Variability of Returns

Market Related Risk






Firm Specific Risk






Total Risk






20




Holding a general stock mutual fund (not a specific industry fund) is similar to holding a well-diversified portfolio.



20


Measuring Market RiskMeasuring Market RiskMarket risk is the risk of the overall market. To

measure the market risk we need to compare individual stock returns to the overall market returns.


Measuring Market RiskMeasuring Market RiskMarket risk is the risk of the overall market. To

measure the market risk we need to compare individual stock returns to the overall market returns.

A proxy for the market is usually used: An index of stocks such as the S&P 500


Measuring Market RiskMeasuring Market RiskMarket risk is the risk of the overall market, so to

measure need to compare individual stock returns to the overall market returns.


Market risk measures how individual stock returns are affected by this market


Measuring Market RiskMeasuring Market RiskMarket risk is the risk of the overall market, so to

measure need to compare individual stock returns to the overall market returns.


Market risk measures how individual stock returns are affected by this market

Regress individual stock returns on Market index


Measuring Market RiskMeasuring Market RiskRegress individual stock returns on Market index

S&PReturn

PepsiCoReturn

-15% 15%-10% -5% 10%5%

5%

10%

15%

-5%

-10%

-15%



S&PReturn

PepsiCoReturn

-15% 15%-10% -5% 10%5%

5%

10%

15%

-5%

-10%

-15%

Jan 1992PepsiCo -0.37%S&P -1.99%



S&PReturn

PepsiCoReturn

-15% 15%-10% -5% 10%5%

5%

10%

15%

-5%

-10%

-15%

Plot Remaining Points



S&PReturn

PepsiCoReturn

-15% 15%-10% -5% 10%5%

5%

10%

15%

-5%

-10%

-15%

Fit Regression Line



S&PReturn

PepsiCoReturn

-15% 15%-10% -5% 10%5%

5%

10%

15%

-5%

-10%

-15%

Slope =riserun

5.5%5%

= = 1.1


Measuring Market RiskMeasuring Market RiskMarket Risk is measured by Beta


Measuring Market RiskMeasuring Market RiskMarket Risk is measured by BetaBeta is the slope of the characteristic line

S&PReturn

PepsiCoReturn

-15% 15%-10% -5% 10%5%

5%

10%

15%

-5%

-10%

-15%

Slope =riserun

5.5%5%

= = 1.1 = Beta ()


Measuring Market RiskMeasuring Market RiskMarket Risk is measured by BetaBeta is the slope of the characteristic lineInterpreting Beta

Beta = 1Market Beta = 1Company with a beta of 1 has average risk




Beta < 1Low Risk CompanyReturn on stock will be less affected by the market than average




Beta < 1Low Risk CompanyReturn on stock will be less affected by the market than average

Beta > 1High Market Risk CompanyStock return will be more affected by the market than average


RequiredRate ofReturn

Minimum rate of return necessary to attract investors to buy funds=




Required rate of return, K, depends on the risk-free rate(Krf) and the risk premium(Krp)





Using the capital asset pricing model (CAPM) the risk premium(Krp) depends on market risk





Using the capital asset pricing model (CAPM) the risk premium(Krp) depends on market risk

Kj = Krf + j ( Km – Krf )

Security Market LineSecurity Market Line

where:Kj = required rate of return on the jth securityj = Beta for the jth security


Example:Example:If the expected return on the market is 12% and the risk free rate is 5%:





Kj = 5% + j (12% – 5% )





Kj = 5% + j (12% – 5% )



Beta1.51.0.50

15%

10%

5%Risk Free Rate



Kj = 5% + j (12% – 5% )



Beta1.51.0.50

15%

10%

5%

12%

Risk & Returnon market



Kj = 5% + j (12% – 5% )



Beta1.51.0.50

15%

10%

5%

SML

Connect Points forSecurity Market Line

Market



Kj = 5% + j (12% – 5% )



Beta1.51.0.50

15%

10%

5%

SMLIf of security j =1.2

Market



Kj = 5% + j (12% – 5% )



Beta1.51.0.50

15%

10%

5%


1.2

Market

j

Kj = 5%+1.2(12% – 5%)



Kj = 5% + j (12% – 5% )



Beta1.51.0.50

15%

10%

5%


1.2

13.4%

Market

j

Kj = 5%+1.2(12% – 5%)=13.4%



Kj = 5% + j (12% – 5% )



Beta1.51.0.50

15%

10%

5%


1.2

13.4%

Market

j

Kj = 5%+1.2(12% – 5%)=13.4%

If = 1.2, investors will require a 13.4% return on the stock

If = 1.2, investors will require a 13.4% return on the stock

108Risk and Rates of Return Risk and Rates of Return

ki : Expected (or required) rate of return from an ki : Expected (or required) rate of return from an investment i.investment i.

KRF : Risk free rate of return (e.g., 3 moth T-Bill rate)KRF : Risk free rate of return (e.g., 3 moth T-Bill rate)kM : Expected return from a market (e.g., S&P500) kM : Expected return from a market (e.g., S&P500)

portfolioportfolio(kM - kRF) : Market Risk Premium(kM - kRF) : Market Risk Premium(kM - kRF) : Risk Premium on asset i(kM - kRF) : Risk Premium on asset i


Portfolio Return = Portfolio Return = w wii x k x kii

Return of a portfolio is the weighted average return of Return of a portfolio is the weighted average return of individual securities in the portfolio.individual securities in the portfolio.

Portfolio beta = Portfolio beta = w wii x x ii

Beta of a portfolio is the weighted average beta of Beta of a portfolio is the weighted average beta of individual securities in the portfolio.individual securities in the portfolio.

risk and rates of return

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