Time Value of Money



CHAPTER 9 Capital Market

I. Review of Basic Concepts

1. Variance [or STDEV] are appropriate measures of risk for assets held by themselves.

2. Variance and standard deviation are not good measures of an asset’s risk once it is a part of a

portfolio since the individual asset’s risks may offset one another.

3. Beta is the appropriate measure of an asset’s risk if held as part of a portfolio

Investors will only invest in riskier assets (as measured by beta) if compensated for taking

this additional risk as follows:

E(ri) = rf + β[E(rM) - rf]

NOTE: this is the basic result of the Capital Asset Pricing Model (CAPM)

known as the security market line (SML)

II. Estimating Risk and Return using Historical Data

KEY ( past return and risk on an asset may indicate return and risk will face in future

NOTE: danger ( future return and risk may be unlike past returns and risk

Ex. Suppose have collected returns for New Honda Dealership Inc. and for Harry’s Automotive

Repair Inc. for past 4 years.

Return on:

Year Dealership Repair

1991 19 14

1992 39 8

1993 15 11

1994 0 19

A. Average historical return

( best estimate of what will earn in any one year (if past returns representative)

( add up historical returns and divide by number of observations

[pic]

where:

[pic]= average return

rt = return for period t

T = number of observations

Ex. [pic]

[pic]

B. Variance ((2) and standard deviation (()

NOTES:

1) measures dispersion around the average return

2) measures the risk of an asset when held by itself

( the higher the std. deviation, the greater the uncertainty about the return

3) usually work with standard deviation since same unit of measurement as average return

KEY ( Add up squared deviations from average and divide by T-1

( the financial calculator ?

[pic]

[pic]

Return on:

Year Dealership Repair

1991 19 14

1992 39 8

1993 15 11

1994 0 19

average 18.25 13

Ex. [pic]

[pic]

[pic]

( more uncertainty about the return on the dealership

Covariance

Return on:

Year Dealership Repair

1991 19 14

1992 39 8

1993 15 11

1994 0 19

average 18.25 13

[pic]

where:

ri(t) = return on asset i at time t

ri = average return on asset i

( positive covariance

( when one asset’s return is above average, other asset’s return tends to be above average also

( negative covariance : Interpretation ?

(zero covariance : Interpretation

[pic] ?

Correlation

[pic]

Ex. [pic] ?

1) Interpretation :( same as covariance

2) Correlation ranges between -1 and +1, unlike covariance.

( = +1 ( perfectly positively corelated

( = -1 ( perfectly negatively corelated

0 < ( < 1 ( positively correlated

-1 < ( < 0 ( negatively correlated

III. Estimating Risk and Return using Forecasted Data

Key ( based on forecasts of possible returns and associated probabilities

( invest in the future, not in the past

(danger : difficult to forecast the future

Ex. Suppose have estimated possible returns for New Honda Dealership Inc. and Harry’s Automotive Repair for the coming year based on how the economy does

Return on:

Economy Prob. Dealership Repair

Boom .25 40% 6%

Average .55 15% 15%

Bust .20 -1% 17%

Expected Return

( best estimate of future return on the asset

( probability weighted average of asset returns across all states

( actual return may be higher or lower than expected return

[pic]

where:

Pr(s) = probability of scenario s

r(s) = return in scenario s

Example. E(rDealership) = .25(40) + .55(15) + .2(-1) = 18.05%

E(rRepair) = 13.15%

( expect to earn a higher return on dealership than repair shop for the coming year

Variance and Standard Deviation

1) measures dispersion around the average return

2) measures the risk of an asset when held by itself

( the higher the standard deviation, the greater the uncertainty about the return

3) usually work with standard deviation since same unit of measurement as average return

[pic] [pic]

Example. [pic]

(Dealership = 14.08%

(Repair = 4.20%

( return is more uncertain for the dealership

IV. Risk and Return for Portfolios using Forecasted Data

( Portfolio ( collection of assets

A. Expected Return

( probability weighted avg. of portfolio returns across all states : just like individual assets

( determining portfolio return for each state

[pic]

where: Xi = % of portfolio value invested in asset i

Ex. Assume invest $8000 in dealership and $12,000 in the repair shop

Return on:

Economy Prob. Dealership Repair

Boom .25 40% 6%

Average .55 15% 15%

Bust .20 -1% 17%

rP(boom) = .4(40) + .6(6) = 19.6%

rP(average) = .4(15) + .6(15) = 15%

rP(bust) = 9.8%

E(rP) = .25(19.6) + .55(15) + .2(9.8) = 15.11%

Variance and Standard Deviation

Return on:

Economy Prob. Dealership Repair

Boom .25 40% 6%

Average .55 15% 15%

Bust .20 -1% 17%

1. weighted average of squared deviations from E(r)( same as individual assets

[pic]

NOTE: when combined dealership and repair shop, std. deviation of portfolio (3.27) less than either the dealership (14.08) or the repair shop (4.19)

2. what happened to the risk?

( diversification : risk reduction from offsetting variability

( key to diversification : more closely related returns, less diversification possible

NOTE: The relationship between asset returns and diversification benefits will be more obvious with a following portfolio example.

Covariance

[pic]

where:

Pr(s) = probability of scenario s

ri(s) = return on asset i in scenario s

E(ri) = expected return on asset i

Ex. (D,R = ?

V. Relationship Between Risk and Return

A. Systematic vs. Unsystematic Risk

systematic risk - risk that affects all assets in the market portfolio

unsystematic risk - risk that only affects a single asset or a small group of assets

Impact of Diversification

1. unsystematic risk

( as combine assets, unsystematic risk begins to cancel out

( after combine 20 or 30 randomly chosen assets, almost all unsystematic risk gone

reason: very unlikely that all 20 have positive surprises at same time

( once part of market portfolio, all is gone

2. systematic risk

( not affected by diversification

( affects all assets

( should only be compensated for systematic risk since only it remains once part

of market portfolio

C. Measuring Systematic Risk

KEY ( systematic risk of an asset measured by its beta

[pic]

where:

[pic]= covariance between asset and the market

[pic] = variance of returns on the market

NOTE: discussion of how to calculate covariance will be discussed in class.

Ex. Suppose that the variance of returns on the market is 124.3275. Assume also that the covariance between New Honda Dealership and the market is 144.1825 and that the covariance between Harry’s Automotive Repair and the market is -36.6025. What are the betas for New Honda Dealership and Harry’s Automotive Repair?

( [pic]

( [pic]

NOTE : beta is the slope of least squares regression line of the return on the asset

against the return on the market

( called characteristic line

[pic]

( dots represent period returns

( slope of line is the beta for asset i

( slope = 1.5 (( = 1.5) ( stock tends to be 1.5 times as volatile as the market

( slope = .8 ((= 0.8) ( stock tends to be 80% as volatile as the market

( deviations from line = unique risk

NOTE: an easier approach ( look up ( in value line or S&P

D. Portfolio Betas

KEY ( weighted average of betas in portfolio

[pic]

Ex. Invest $8000 in the the dealership with a beta of 1.1597 and $12,000 in the repair shop with a beta of -0.2944.

( (P = .4(1.1597) + .6(-.2944) = 0.2872

E. The Capital Asset Pricing Model (CAPM)

KEY ( since beta is relevant measure of risk, there should be a relationship

between beta and return

[pic]

NOTE:

1) this equation is known as the security market line

2) the model has very strong intuition

If ( = 2, the asset has two “units” of market risk

( twice as volatile as the market (as part of a portfolio)

( should earn twice as much of a premium (above the risk free rate)

If ( = .5, the asset has one-half of a “unit” of market risk

( one-half as volatile as the market (as part of a portfolio)

( should earn one-half of a premium

Ex. The return on T-bills (risk-free rate) is 5% and the market risk premium is 8%.

What does the CAPM tell us is the required return on New Honda and Harry’s

Automotive Repair?

E(rDealership) = .05 + (.08)(1.1597) = .1428

E(rRepair) = .05 + (.08)(-.2944) = .0265

( Solve the following problems in the textbook.

1, 2, 3, 4, 7, 8, 10, 11, 15

EXERCISE PROBLEMS FOR CHAPTER 9.

1. Find the expected return on Bio-Pharma stock when there are two possible states of economy. The probability that the high state will occur is 32% and that the low state will occur is 68%. The return on Bio-Pharma stock is 16.83% in the high state and 7.91% in the low state. [[10.76%]]

2. With two possible states of the economy, find the variance of returns on Texas Eastern stock. The probability that State 1 will occur is 28% and that State 2 will occur is 72%. The return in State 1 is 16.49% and in State 2 is 7.04%. [[18.0034]]

3. Your budget manager demands new calculations of covariance of returns based on the best case and worst case possibilities. You estimate that your firm will receive returns given in the following table in the two states. You also estimate returns to the Dow Jones 30 in these two states. Find the covariance of returns. [[0.2855]]

State of Nature Probability Texas Eastern Return Index Return

The best state 52% 16.79% 16.02%

The worst state 48% 5.35% 6.02%

4. You are considering investing in a portfolio of two stocks, with 57% of your money in the first stock and 43% in the second stock. You calculate that the expected returns for the two stocks are 9.60% and 16.25%, respectively. Find the portfolio expected return.[[12.46%]]

5. Find the variance of the portfolio when the variance of expected returns is 0.10262 for Asset 1 and 0.11397 for Asset 2. The covariance of returns is 0.01925. Funds are invested with 53% of the money in Asset 1 and 47% in Asset 2.[[0.0636]]

6. The stock analyst for US East has determined that if the best state occurs in the economy, US East stock will yield a return of 16.17%. If the worst state occurs, the return will be 6.96%. If the normal state occurs, the return will be 12.56%. With probabilities of 17% on the best state, 31% on the worst state, and 52% on the normal state, what is the expected return on the stock? [[11.44%]]

7. Before approving acquisition of new machinery, the treasurer of Sip-n-Smoke wants to know the standard deviation of returns on it. You tell him that the firm should receive a return of 15.15% with a probability of 49%, and a return of 5.21% with a probability of 51%. What is the standard deviation of returns on the machinery?[[4.97%]]

8. The project manager at O'Reilly Corp figures the states of nature and returns for O'Reilly Corp from a new project and also the returns for the value weighted stock index. These appear in the following table. What is the covariance of returns of the project with the value weighted stock index? [[0.1348]]

State of Nature Probability O'Reilly Corp Return Index Return

The boom state 73% 15.47% 14.64%

The recession 27% 6.15% 7.30%

9. You have invested 29% of your money in Stock X, 50% in Stock Y, and the rest in Stock Z. The portfolio expected return is 7.48%, and the expected returns of X and Y are 7.83% and 6.83%. What is the expected return of Stock Z? [[8.54%]]

10. You are considering a portfolio of two stocks, with 90% of your money in the first stock, and 10% in the second stock. The variance of returns of Stock 1 is 0.10065 and the variance of returns of Stock 2 is 0.13753. The correlation coefficient of returns is 0.598. Find the variance of the portfolio. [[0.0956]]

CHAPTER 10. RISK AND RETURN [CAPM]

I. Examples from Chapter 9.

Ex. Historical Returns for Dealership Inc. and for Repair Inc. for past 4 years.

How to compute by using financial calculator ?

Return on:

Year Dealership Repair

1992 19 14

1993 39 8

1994 15 11

1995 0 19

[pic] 18.25 13.00

( 16.07 4.69

Ex. Estimated possible future returns for Dealership Inc. and Repair Inc for the coming year

based on how the economy does

Return on:

Economy Prob. Dealership Repair

Boom .25 40% 6%

Average .55 15% 15%

Bust .2 -1% 17%

E(r) 18.05 13.15

( 14.08 4.20

II. Risk and Return for Portfolios using Forecasted Data

( weighted average of variances and covariances

[pic]

where:

[pic]variance of returns on portfolio

Xi = % of portfolio invested in asset i [=weight or proportion]

[pic]variance of returns on asset i

[pic] covariance between asset 1 and asset 2

III. Estimating Risk and Return for Portfolios using Historical Data

( essentially the same as w/ forecasted data except use historical data except for calculation of covariance

IV. The relationship between risk and return

[ A.] Combinations of risky assets

1. Portfolios of two risky assets

Ex. Combinations of New Honda Dealership and Harry’s Auto Repair based on forecasted data.

XD E(rP) (P

.0 13.15% 4.20%

.2 14.13 0.98

.4 15.11 3.27

.6 16.09 6.84

.8 17.07 10.45

1.0 18.05 14.08

Note: Curve due to diversification : Correlation negative.

What if perfect positive correlation ?

2. Portfolios of many risky assets

a. Feasible set – the set of all portfolios that could be formed from a group of N securities.

[pic]

The efficient set – An investor will choose his or her optimal portfolio from the set of portfolios that

1. Offer maximum expected return for varying levels of risk and

2. Offer minimum risk for varying levels of return.

[pic]

note: optimal portfolio depends on risk preferences

[ B.] Combinations of a risky asset and risk-free borrowing or lending

1. relationship between risk and return

note: std. deviation of risk-free borrowing/lending is zero and covariance between risk-free borrowing/lending and any other asset is zero

a. E(rp) = XAE(rA) + Xrf rf

b. (P = [pic] = XA(A

( second and third terms drop out since zero

( can combine 1st & 2nd equations to obtain relationship between risk and return

( solve for XA in eq. b, substitute into a, and simplify

( E(rp) = rf + [pic]

Ex. Assume combine risky asset “A” from the efficient set with riskfree borrowing/lending

E(rA) = 14%; (A = 10%; rf = 9% ( E(rp) =

[pic]

Assume initial wealth is $5000 and invest $3500 in “A” and lend the remainder at riskfree rate.

( E(rp) = ?

Assume initial wealth is $5000 and that invest $7500 in “A” by borrowing

( E(rp) = ?

2. The optimal risky asset

[pic]

Q: Is “A” the best risky asset to combine with riskfree-borrowing and lending?

(

Point M deserves some attention. Why ?

Because there is no other portfolio consisting purely of risky assets that, when connected by a straight line to the riskfree asset, lies northwest of it. In other words, of all the lines that can be drawn emanating from the riskfree asset and connecting with either risky asset or risky portfolio, none has a greater slope than the line that goes to M.

( Above statement is important. Why ?

: Efficient set has been changed when we introduce the riskfree lending and borrowing.

( The crucial point is that every investor holds the optimal risky portfolio P, regardless of his/her risk aversion.

3) The equation of the capital market line is E(rp) = rf + [pic]

[ C.] Homogeneous expectations

( assume investors have homogeneous expectations

[ D.] The Capital Asset Pricing Model (CAPM)

1. relationship between risk and return

( assume combine some risky asset “A” with risk-free borrowing or lending

( beta of risk-free borrowing/lending is zero

a. E(rp) = XA(E(rA) + Xrf( rf

b. (P = XA(A + Xrf(0) = XA(A

( can combine a and b to obtain relationship between risk and return

( solve for XA in eq. b, substitute into a, and simplify

( E(rp) = rf + [pic]

1) slope of line = [pic] ( called the reward to risk ratio

Ex. E(rA) = 14%, rf = 9%, (A = 0.5 ( reward to risk ratio = ?

[pic]

2. Market portfolio

RTRM =

( [pic]

( called security market line (SML)

Ex. Suppose asset A has a beta of 0.5. How can we combine the market with risk-free borrowing and lending to achieve the same risk?

Ex. Suppose asset B has a beta of 1.3. How can we combine the market with risk-free borrowing and lending to achieve the same risk?

TAKE-HOME EXERCISE [CHAPTER 10]

1. John Beere's economists have determined that the best state in the market will occur with a 20% probability, that the worst state will occur with a 25% probability, and that the normal state will occur with a 55%probability. If the return on John Beere will be 14.73% in the best state, 7.46% in the worst state, and 12.57% in the normal state, what is the expected return on John Beere stock?

2. With three possible states of the economy, find the standard deviation of returns on Quicken stock. The probability that the best state will occur is 17%, that the worst state will occur is 26% and that the normal state will occur is 57%. The return in the best state is 14.40%, with 5.70% in the worst state and 10.31% in the normal state.

3. The project manager at Dell figures the states of nature and returns for Dell from a new project and also the returns for for the S&P500. These appear in the floating table. What is the covariance of returns of the project with the S&P500?

State of Nature Probability Dell Return Index Return

Boom 76% 17.24% 17.20%

Recession 24% 7.42% 7.23%

4. Suppose the expected return of a three-stock portfolio is 13.95%. The expected returns of the first and second stocks are 17.83% and 6.84%. With investment proportions of 37%, 27%, and 36%, find the expected return of the third stock.

5. Find the variance of returns of the two-asset portfolio where 67% is invested in Asset 1 and 33% is invested in Asset 2. The standard deviation of Asset 1 is 30.41%and the standard deviation of Asset 2 is 24.36%. The covariance of returns is -0.00126.

6. You are considering a portfolio of two stocks, with 87% of your money in the first stock, and 13% in the second stock. The variance of returns of Stock 1 is 0.05653 and the variance of returns of Stock 2 is 0.04421. The correlation coefficient of returns is 0.188. Find the variance of the portfolio.

7. Find the variance of returns of the two-asset portfolio where 64% is invested in Asset 1 and 36% is invested in Asset 2. The standard deviation of Asset 1 is 31.94%and the standard deviation of Asset 2 is 29.24%. The covariance of returns is -0.03306.

8. You've been told that the beta of a three-stock portfolio is 1.465. The betas of the first and second stocks are 2.161 and 2.071. With investment proportions of 24%, 37%, and 39%, what is the beta of the third stock?

9. Your investment analyst has determined the expected return for Wacom Tech stock is 17.66%. The stock's beta is 1.525 and the current T-bill rate is 3.92%. What is the expected return on the market portfolio?

10. The expected market risk premium is 9.81% and the expected return on a stock with a beta of 1.460 is 17.94%. What is the risk-free rate?

ANSWERS

1. 11.72% 2.) 2.85% 3.) 0.1786

4.) 15.29% 5.) 0.0474 6.) 0.0457

7. 0.0376 8.) 0.463 9.) 12.93%

10. 3.62%

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