Risk and Return: Past Risk and Return: Past and Prologueand Prologue

CHAPTER 5CHAPTER 5CHAPTER 5CHAPTER 5

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Holding Period Return

ndCashDivide

eEndingPricP

riceBeginningPP

D

PDPPHPR

1

0

1

0

101

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Rates of Return: Single Period Example

Ending Price = 24Beginning Price = 20Dividend = 1

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

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Data from Table 5.1 1 2 3 4

Assets(Beg.) 1.0 1.2 2.0 .8HPR .10 .25 (.20) .25TA (Before Net Flows 1.1 1.5 1.6

1.0Net Flows 0.1 0.5 (0.8) 0.0End Assets 1.2 2.0 .8 1.0

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Returns Using Arithmetic and Geometric Averaging

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

ra = (.10 + .25 - .20 + .25) / 4

= .10 or 10%Geometricrg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1

rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1

= (1.5150) 1/4 -1 = .0829 = 8.29%

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Dollar Weighted Returns

Internal Rate of Return (IRR) - the discount rate that results in present value of the future cash flows being equal to the investment amount

• Considers changes in investment• Initial Investment is an outflow• Ending value is considered as an inflow• Additional investment is a negative flow• Reduced investment is a positive flow

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Dollar Weighted Average Using Text Example

Net CFs 1 2 3 4$ (mil) - .1 - .5 .8 1.0

Solving for IRR1.0 = -.1/(1+r)1 + -.5/(1+r)2 + .8/(1+r)3

+ 1.0/(1+r)4

r = .0417 or 4.17%

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Quoting ConventionsAPR = annual percentage rate

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

( 1+ rate for period)Periods per yr - 1Example: monthly return of 1%

APR = 1% X 12 = 12%EAR = (1.01)12 - 1 = 12.68%

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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

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rr

Symmetric distributionSymmetric distribution

Normal Distribution

s.d. s.d.

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rrNegativeNegative PositivePositive

Skewed Distribution: Large Negative Returns Possible

Median

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rrNegativeNegative PositivePositive

Skewed Distribution: Large Positive Returns Possible

Median

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Subjective returnsSubjective 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)s

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Numerical Example: Subjective or Scenario Distributions

StateState Prob. of StateProb. of State rrinin State State11 .1.1 -.05-.0522 .2.2 .05.0533 .4.4 .15.1544 .2.2 .25.2555 .1.1 .35.35

E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35)E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35)E(r) = .15E(r) = .15

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Standard deviation = [variance]Standard deviation = [variance]1/21/2

Measuring Variance or Dispersion of Returns

Subjective or ScenarioVariance =

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

Var =[(.1)(-.05-.15)Var =[(.1)(-.05-.15)22+(.2)(.05- .15)+(.2)(.05- .15)22...+ .1(.35-.15)...+ .1(.35-.15)22]]Var= .01199Var= .01199S.D.= [ .01199] S.D.= [ .01199] 1/2 1/2 = .1095= .1095

Using Our Example:Using Our Example:

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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 Stk11.80 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

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Annual Holding Period Excess Returns

From Table 5.3 of Text

Arith. Stan.Series Mean% Dev.%World Stk 7.37 18.69US Lg Stk 8.46 20.80US Sm Stk14.64 38.72Wor Bonds 2.34 8.98LT Treas 1.85 8.00

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Figure 5.1 Frequency Distributions of Holding Period Returns

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Figure 5.2 Rates of Return on Stocks, Bonds and Bills

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Figure 5.3 Normal Distribution with Mean of 12.25% and St Dev of 20.50%

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

R = r + i or r = R - i

Example r = 3%, i = 6%R = 9% = 3% + 6% or 3% = 9% - 6%

Fisher effect: Exactr = (R - i) / (1 + i) 2.83% = (9%-6%) / (1.06)

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Figure 5.4 Interest, Inflation and Real Rates of Return

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• Possible to split investment funds between safe and risky assets

• Risk free asset: proxy; T-bills• Risky asset: stock (or a portfolio)

Allocating Capital Between Risky & Risk-Free Assets

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Allocating Capital Between Risky & Risk-Free Assets (cont.)

• Issues– Examine risk/ return tradeoff– Demonstrate how different degrees of

risk aversion will affect allocations between risky and risk free assets

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

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

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

Example Using the Numbers in Chapter 5 (pp 146-148)

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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

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Figure 5.5 Investment Opportunity Set with a Risk-Free Investment

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ppcc ==

SinceSince rfrf

yy

Variance on the Possible Combined Portfolios

= 0, then= 0, then

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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

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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

c = (1.5) (.22) = .33

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Figure 5.6 Investment Opportunity Set with Differential Borrowing and

Lending Rates

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Risk Aversion and Allocation

• Greater 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

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Table 5.5 Average Rates of Return, Standard Deviation and Reward to

Variability