5.1 Rates of Return

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Measuring Ex-Post (Past) Returns

•An example: Suppose you buy one share of a stock today for $45 and you hold it for one year and sell it for $52. You also received $8 in dividends at the end of the year.•(PB = , PS = , CF = ): •HPR =

$45 $52 $8(52 - 45 + 8) / 45 = 33.33%

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Arithmetic Average

Finding the average HPR for a time series of returns:• i. Without compounding (AAR or Arithmetic Average

Return):

• n = number of time periods

n

1T

Tavg n

HPRHPR

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Arithmetic Average

AAR =

An example: You have the following rates of return on a stock: 2000 -21.56% 2001 44.63% 2002 23.35% 2003 20.98% 2004 3.11% 2005 34.46% 2006 17.62%

n

1T

Tavg n

HPRHPR

7

.1762).3446.0311.2098.2335.4463(-.2156HPRavg

17.51%

17.51%

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Geometric Average

•With compounding (geometric average or GAR: Geometric Average Return):

GAR =

An example: You have the following rates of return on a stock: 2000 -21.56% 2001 44.63% 2002 23.35% 2003 20.98% 2004 3.11% 2005 34.46% 2006 17.62%

1 )HPR(1HPR/1n

1TTavg

n

11.1762)1.34461.03111.20981.23351.4463(0.7844HPR 1/7avg 15.61%

15.61%

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Q: When should you use the GAR and when should you use the AAR?

A1: When you are evaluating PAST RESULTS (ex-post):

A2: When you are trying to estimate an expected return (ex-ante return):

Use the AAR (average without compounding) if you ARE NOT reinvesting any cash flows received before the end of the period.Use the GAR (average with compounding) if you ARE reinvesting any cash flows received before the end of the period.

Use the AAR

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Measuring Ex-Post (Past) Returns for a portfolio

•Finding the average HPR for a portfolio of assets for a given time period:

•where VI = amount invested in asset I, •J = Total # of securities•and TV = total amount invested;•thus VI/TV = percentage of total investment invested in asset I

J

1IIavg HPRHPR

TVVI

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•For example: Suppose you have $1000 invested in a stock portfolio in September. You have $200 invested in Stock A, $300 in Stock B and $500 in Stock C. The HPR for the month of September for Stock A was 2%, for Stock B the HPR was 4% and for Stock C the HPR was - 5%.

•The average HPR for the month of September for this portfolio is:

J

1IIavg HPRHPR

TVVI

)(500/1000)(-.05 )(300/1000)(.04 )(200/1000)(.02HPRavg -0.9%

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5.2 Risk and Risk Premiums

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Subjective expected returns

E(r) = Expected Returnp(s) = probability of a stater(s) = return if a state occurs1 to s states

Measuring Mean: Scenario or Subjective Returnsa. Subjective or Scenario

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

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= [= [22]]1/21/2

E(r) = Expected Returnp(s) = probability of a staters = return in state “s”

Measuring Variance or Dispersion of Returns

a. Subjective or ScenarioVariance

s

2s

2 E(r)][rp(s)σ

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Numerical Example: Subjective or Scenario Distributions

State Prob. of State Return1 .2 - .052 .5 .053 .3 .15

E(r) = (.2)(-0.05) + (.5)(0.05) + (.3)(0.15) = 6%

2 = [(.2)(-0.05-0.06)2 + (.5)(0.05- 0.06)2 + (.3)(0.15-0.06)2]2 = 0.0049%2

= [ 0.0049]1/2 = .07 or 7%

s

2s

2 E(r)][rp(s)σ

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Expost Expected Return &

Annualizing the statistics:

n

ii rr

n 1

2)(1

1 : VarianceExpost 2

periods # periodannual

periods # rr periodannual

2σσ :Deviation Standard Expost

n

1T

Tn

HPRr HPR averager

nsobservatio #n

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Using Ex-Post Returns to estimate Expected HPR

Estimating Expected HPR (E[r]) from ex-post data.

Use the arithmetic average of past returns as a forecast of expected future returns and,

Perhaps apply some (usually ad-hoc) adjustment to past returns

Problems?• Which historical time period?

• Have to adjust for current economic situation

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Characteristics of Probability Distributions

1. Mean: __________________________________ _

2. Median: _________________

3. Variance or standard deviation:

4. Skewness:_______________________________

5. Leptokurtosis: ______________________________

If a distribution is approximately normal, the distribution is fully described by _____________________

Arithmetic average & usually most likelyArithmetic average & usually most likely

Dispersion of returns about the meanDispersion of returns about the mean

Long tailed distribution, either sideLong tailed distribution, either side

Too many observations in the tailsToo many observations in the tails

Characteristics 1 and 3Characteristics 1 and 3

Middle observationMiddle observation

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Normal Distribution

E[r] = 10%

= 20%Average = Median

Risk is the Risk is the possibility of getting possibility of getting returns different returns different from expected.from expected.

measures deviations measures deviations above the mean as well as above the mean as well as below the mean. below the mean.

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5.3 The Historical Record

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Frequency distributions of annual HPRs, 1926-2008

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Rates of return on stocks, bonds and bills, 1926-2008

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Annual Holding Period Returns Statistics 1926-2008From Table 5.3

• Geometric mean: Best measure of compound historical return

• Arithmetic Mean:Expected return

• Deviations from normality?

Geom. Arith. ExcessSeries Mean% Mean% Return% Kurt. Skew.World Stk 9.20 11.00 7.25 1.03 -0.16 US Lg. Stk 9.34 11.43 7.68 -0.10 -0.26 Sm. Stk 11.43 17.26 13.51 1.60 0.81 World Bnd 5.56 5.92 2.17 1.10 0.77 LT Bond 5.31 5.60 1.85 0.80 0.51

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Historical Real Returns & Sharpe Ratios

Real Returns% Sharpe RatioSeriesWorld Stk 6.00 0.37US Lg. Stk 6.13 0.37Sm. Stk 8.17 0.36

World Bnd 2.46 0.24LT Bond 2.22 0.24

• Real returns have been much higher for stocks than for bonds• Sharpe ratios measure the excess return relative to standard

deviation.• The higher the Sharpe ratio the better.• Stocks have had much higher Sharpe ratios than bonds.

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5.4 Inflation and Real Rates of Return

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Inflation, Taxes and ReturnsThe average inflation rate from 1966 to 2005 was _____.This relatively small inflation rate reduces the terminal value of $1 invested in T-bills in 1966 from a nominal value of ______ in 2005 to a real value of _____.

Taxes are paid on _______ investment income. This reduces _____ investment income even further.

You earn a ____ nominal, pre-tax rate of return and you are in a ____ tax bracket and face a _____ inflation rate. What is your real after tax rate of return?

rreal [6% x (1 - 0.15)] – 4.29% 0.81%; taxed on nominal

4.29%

$10.08 $1.63

nominalreal

6%15% 4.29%

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Real vs. Nominal RatesFisher effect: Approximationreal rate nominal rate - inflation rate

rreal rnom - iExample rnom = 9%, i = 6%

rreal 3% Fisher effect: Exact

rreal = or rreal = rreal = The exact real rate is less than the approximate real rate.

[(1 + rnom) / (1 + i)] – 1 (rnom - i) / (1 + i) (9% - 6%) / (1.06) = 2.83%

rreal = real interest rate

rnom = nominal interest rate

i = expected inflation rate

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