Holding Period Return

ndCashDivide

eEndingPricP

riceBeginningPP

1

1

0

0

101

D

PDPPHPR

Rates of Return: Single Period ExampleEnding Price = 24Beginning Price = 20Dividend = 1

HPR = ( 24 - 20 + 1 )/ ( 20) = 25%

Example 43You purchased a share of stock for $20. One year later you received $2 as

dividend and sold the share for $23. Your holding-period return was __________. A) 5 percent B) 10 percent C) 20 percent D) 25 percent

The holding period return on a stock was 25%. Its ending price was $18 and its beginning price was $16. Its cash dividend must have been __________. A) $0.25 B) $1.00 C) $2.00 D) $4.00

Returns Using Arithmetic and Geometric Averaging

Arithmeticra = (r1 + r2 + r3 + ... rn) / n

ra =.10 or 10%

Geometricrg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1

rg = .0829 = 8.29%

Example 12The arithmetic average of 12%, 15% and 20% is _________.

A) 15.7% B) 15% C) 17.2% D) 20%

The geometric average of 10%, 20% and 25% is __________. A) 15% B) 18.2% C) 18.3% D) 23%

Quoting ConventionsAPR = annual percentage rate

(periods in year) X (rate for period)EAR = effective annual rate

( 1+ rate for period)Periods per yr - 1

Example: monthly return of 1%APR = EAR =

Characteristics of Probability Distributions

1) Mean: most likely value2) Variance or standard deviation3) Skewness

* If a distribution is approximately normal, the distribution is described by characteristics 1 and 2

Return Variability: The Second Lesson

1 - 8

Example

What range of return would you expect to see in 95% of time?

Is it possible you can earn 65% return annually at 1% significant level?

Subjective expected returns

p(s) = probability of a stater(s) = return if a state occurs1 to s states

p(s) = probability of a stater(s) = return if a state occurs1 to s states

Measuring Mean: Scenario or Subjective Returns

E (r) = p(s ) r(s )Ss

Numerical Example: Subjective or Scenario Distributions

State Prob. of State rin State1 .1 -.052 .2 .053 .4 .154 .2 .255 .1 .35

E(r) =.15

Standard deviation = [variance]1/2

Measuring Variance or Dispersion of Returns

Subjective or ScenarioVariance = S

s p(s ) [rs - E (r)]2

Var=.01199S.D.= [ .01199] 1/2 = .1095

Using Our Example:

Historical mean and standard deviation

)()(

1

))(()(

)(

1

2

1

ii

T

tiit

i

T

tit

i

RVarRSD

T

RERRVar

T

RRE

Risk Premiums and Risk Aversion

Degree to which investors are willing to commit fundsRisk aversion

If T-Bill denotes the risk-free rate, rf, and variance, , denotes volatility of returns then:

The risk premium of a portfolio is:

E(Rp) - Rf

2P

( )P fE r r

Risk Premiums and Risk Aversion

To quantify the degree of risk aversion with parameter A:

E(Rp) – Rf = (1/2) A σp2

2

( )

12

P f

P

E r rA

The Sharpe (Reward-to-Volatility) Measure

Sharpe ratio = Portfolio risk premium/standard deviation of the

excess returns = (E(Rp) – Rf )/σp

Annual Holding Period ReturnsFrom Table 5.3 of Text

Geom.Arith. Stan.Series Mean% Mean% Dev.%World Stk 9.41 11.17 18.38US Lg Stk 10.23 12.25 20.50US Sm Stk 11.80 18.43 38.11Wor Bonds 5.34 6.13 9.14LT Treas 5.10 5.64 8.19T-Bills 3.71 3.79 3.18Inflation 2.98 3.12 4.35

Figure 5.1 Frequency Distributions of Holding Period Returns

Figure 5.2 Rates of Return on Stocks, Bonds and Bills

Real vs. Nominal RatesFisher effect: Approximationnominal rate = real rate + inflation premium

R = r + i or r = R - iExample r = 3%, i = 6%

R = 9% = 3% + 6% or 3% = 9% - 6%Fisher effect: Exact

r = (1+R) / (1 + i) - 1 2.83% = (9%-6%) / (1.06)

Figure 5.4 Interest, Inflation and Real Rates of Return

Possible to split investment funds between safe and risky assets

Risk free asset: proxy; T-billsRisky asset: stock (or a portfolio)Risk premium: risk asset return-risk-free rate Issues

Examine risk/ return tradeoffDemonstrate how different degrees of risk aversion will

affect allocations between risky and risk free assets

Allocating Capital Between Risky & Risk-Free Assets

rf = 7%rf = 7% srf = 0%srf = 0%

E(rp) = 15%E(rp) = 15% sp = 22%sp = 22%

y = % in py = % in p (1-y) = % in rf(1-y) = % in rf

Example:Given:

E(rc) = yE(rp) + (1 - y)rfE(rc) = yE(rp) + (1 - y)rf

rc = complete or combined portfoliorc = complete or combined portfolio

For example, y = .75For example, y = .75E(rc) = .75(.15) + .25(.07)E(rc) = .75(.15) + .25(.07)

= .13 or 13%= .13 or 13%

Expected Returns for Combinations

ppcc ==

SinceSince rfrf

yy

Variance on the Possible Combined Portfolios

= 0, then= 0, thenss

ssss

cc = .75(.22) = .165 or 16.5%= .75(.22) = .165 or 16.5%

If y = .75, thenIf y = .75, then

cc = 1(.22) = .22 or 22%= 1(.22) = .22 or 22%

If y = 1If y = 1

cc = 0(.22) = .00 or 0%= 0(.22) = .00 or 0%If y = 0If y = 0

Combinations Without Leverage

ss

ss

ss

Using Leverage with Capital Allocation Line

Borrow at the Risk-Free Rate and invest in stock

Using 50% Leveragerc = (-.5) (.07) + (1.5) (.15) = .19

sc = (1.5) (.22) = .33

Reward-to-variability ratio = risk premium/standard deviation

Figure 5.5 Investment Opportunity Set with a Risk-Free Investment

Figure 5.6 Investment Opportunity Set with Differential Borrowing and Lending Rates

Risk Aversion and AllocationGreater levels of risk aversion lead to larger

proportions of the risk free rate

Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets

Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

Example 22Consider the following two investment alternatives. First, a risky portfolio that pays 15%

rate of return with a probability of 60% or 5% with a probability of 40%. Second, a treasury bill that pays 6%. The risk premium on the risky investment is __________. A) 1% B) 5% C) 9% D) 11%

Consider the following two investment alternatives. First, a risky portfolio that pays 20% rate of return with a probability of 60% or 5% with a probability of 40%. Second, a treasury that pays 6%. If you invest $50,000 in the risky portfolio, your expected profit would be __________. A) $3,000 B) $7,000 C) $7,500 D) $10,000

Example 41You have $500,000 available to invest. The risk-free rate as well as your borrowing

rate is 8%. The return on the risky portfolio is 16%. If you wish to earn a 22% return, you should __________. A) invest $125,000 in the risk-free asset B) invest $375,000 in the risk-free asset C) borrow $125,000 D) borrow $375,000

The Manhawkin Fund has an expected return of 12% and a standard deviation return of 16%. The risk free rate is 4%. What is the reward-to-volatility ratio for the Manhawkin Fund? A) 0.5 B) 1.3 C) 3.0 D) 1.0

Example 422You invest $100 in a complete portfolio. The complete portfolio is composed of a risky asset with an expected rate of

return of 12% and a standard deviation of 15% and a treasury bill with a rate of return of 5%. __________ of your money should be invested in the risky asset to form a portfolio with an expected rate of return of 9% A) 87% B) 77% C) 67% D) 57%

An investor invests 40% of his wealth in a risky asset with an expected rate of return of 15% and a variance of 4% and 60% in a treasury bill that pays 6%. Her portfolio's expected rate of return and standard deviation are __________ and __________ respectively. A) 8.0%, 12% B) 9.6%, 8% C) 9.6%, 10% D) 11.4%, 12%

The expected return of portfolio is 8.9% and the risk free rate is 3.5%. If the portfolio standard deviation is 12.0%, what is the reward to variability ratio of the portfolio? A) 0.0 B) 0.45 C) 0.74 D) 1.35