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Risk and Return
Riccardo Colacito
Foundations of Financial Markets 2
Roadmap
1. Rates of Return– Holding Period Return– Arithmetic and Geometric Averages– Annual Percentage Rate and Effective Annual Rate
2. Summary Statistics of rates of return– Probability Distribution– Expected Return– Variance, Covariance and Standard Deviation– Other properties
3. Historical record of Bills, Bonds, and Stocks– Risk premia from 1926-2003?– Inflation and Real Rates of Return
Foundations of Financial Markets 3
Holding Period Return
0
101
P
DPPHPR
DividendCash
Price Ending
Price Beginning
1
1
0
D
P
P
Foundations of Financial Markets 4
Rates of Return: Single Period Example
Ending Price = 24
Beginning Price = 20
Dividend = 1
HPR = ( 24 - 20 + 1 )/ ( 20) = 25%
Foundations of Financial Markets 5
Roadmap
1. Rates of Return– Holding Period Return– Arithmetic and Geometric Averages– Annual Percentage Rate and Effective Annual Rate
2. Summary Statistics of rates of return– Probability Distribution– Expected Return– Variance, Covariance and Standard Deviation– Other properties
3. Historical record of Bills, Bonds, and Stocks– Risk premia from 1926-2003?– Inflation and Real Rates of Return
Foundations of Financial Markets 6
Returns Using Arithmetic and Geometric Averaging
Time 1 2 3 4
HPR .1 .25 -.20 .25
Arithmeticra = (r1 + r2 +... rn) / nra = (.10 + .25 - .20 + .25) / 4 = .10 or 10%Geometricrg = [(1+r1) (1+r2) .... (1+rn)]1/n - 1rg = [(1.1) (1.25) (.8) (1.25)]1/4 - 1 = (1.5150) 1/4 -1 = .0829 = 8.29%
Foundations of Financial Markets 7
Roadmap
1. Rates of Return– Holding Period Return– Arithmetic and Geometric Averages– Annual Percentage Rate and Effective Annual Rate
2. Summary Statistics of rates of return– Probability Distribution– Expected Return– Variance and Standard Deviation– Other properties
3. Historical record of Bills, Bonds, and Stocks– Risk premia from 1926-2003?– Inflation and Real Rates of Return
Foundations of Financial Markets 8
Quoting Conventions
• Annual Percentage Rate
APR = (periods in year) X (rate for period)
• Effective Annual Rate
EAR = ( 1+ rate for period)Periods per yr – 1
• Example: monthly return of 1%
APR = 1% X 12 = 12%
EAR = (1.01)12 - 1 = 12.68%
Foundations of Financial Markets 9
Roadmap
1. Rates of Return– Holding Period Return– Arithmetic and Geometric Averages– Annual Percentage Rate and Effective Annual Rate
2. Summary Statistics of rates of return– Probability Distribution– Expected Return– Variance, Covariance and Standard Deviation– Other properties
3. Historical record of Bills, Bonds, and Stocks– Risk premia from 1926-2003?– Inflation and Real Rates of Return
Foundations of Financial Markets 10
Probability distribution
• Definition: list of possible outcomes with associated probabilities
• Example:
State Outcome Prob1 -2 .1
2 -1 .2
3 0 .4
4 1 .2
5 2 .1
Foundations of Financial Markets 11
Probability distribution: figure
-3 -2 -1 0 1 2 30
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Outcomes
Pro
babi
lity
Foundations of Financial Markets 12
Normal distribution
Foundations of Financial Markets 13
Notation
• Let p(i) denote the probability with which state i occurs
• Then– p(1)=0.1– p(2)=0.2– p(3)=0.4– p(4)=0.2– p(5)=0.1
State Outcome Prob
1 -2 .1
2 -1 .2
3 0 .4
4 1 .2
5 2 .1
Foundations of Financial Markets 14
Roadmap
1. Rates of Return– Holding Period Return– Arithmetic and Geometric Averages– Annual Percentage Rate and Effective Annual Rate
2. Summary Statistics of rates of return– Probability Distribution– Expected Return– Variance, Covariance and Standard Deviation– Other properties
3. Historical record of Bills, Bonds, and Stocks– Risk premia from 1926-2003?– Inflation and Real Rates of Return
Foundations of Financial Markets 15
Expected Return
Definition:Definition:
• p(s) = probability of a state
• r(s) = return if a state occurs
• 1 to s states
E(r) = p(s) r(s)s
Foundations of Financial Markets 16
Numerical Example
E(r) = (.1)(-2) + (.2)(-1) + (.4)(0) + (.2)(1) + (.1)(2) = 0E(r) = (.1)(-2) + (.2)(-1) + (.4)(0) + (.2)(1) + (.1)(2) = 0
State Prob Return1 .1 -2
2 .2 -1
3 .4 0
4 .2 1
5 .1 2
Foundations of Financial Markets 17
Roadmap
1. Rates of Return– Holding Period Return– Arithmetic and Geometric Averages– Annual Percentage Rate and Effective Annual Rate
2. Summary Statistics of rates of return– Probability Distribution– Expected Return– Variance, Covariance and Standard Deviation– Other properties
3. Historical record of Bills, Bonds, and Stocks– Risk premia from 1926-2003?– Inflation and Real Rates of Return
Foundations of Financial Markets 18
Why do we need the variance?
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Outcomes
Pro
babi
lity
•Two variables with the same mean.
•What do we know about their dispersion?
Foundations of Financial Markets 19
Measuring Variance or Dispersion of Returns
Standard deviation = varianceStandard deviation = variance1/21/2
Variance = s
p(s) [rs - E(r)]2
Why do we take squared deviations?
Foundations of Financial Markets 20
Numerical example
Var = .1 (-2-0)Var = .1 (-2-0)22 + .2 (-1-0) + .2 (-1-0)22 + .4 (0-0) + .4 (0-0)22 + .2 (1-0) + .2 (1-0)22 + .1 (2-0) + .1 (2-0)22 = 1.2 = 1.2
Std dev= (1.2)Std dev= (1.2)1/21/2 = 1.095 = 1.095
State Prob Return1 .1 -2
2 .2 -1
3 .4 0
4 .2 1
5 .1 2
Foundations of Financial Markets 21
One important property of variance and standard deviation
• Let w be a constant
Var(wxr) = w2 x Var(r)
• Similarly
Std Dev(wxr) = w x Std Dev(r)
Foundations of Financial Markets 22
Covariance: Preliminaries
• Covariance– The extent at which two assets tend to move
together– Can be positive or negative
• Correlation– Same idea of covariance, but bounded
between -1 and 1
Foundations of Financial Markets 23
Covariance: definition
221121,cov rEsrrEsrsprrs
2asset of valueexpected :
1asset of valueexpected :
occurs s state when 2asset ofreturn :
occurs s state when 1asset ofreturn :
occurs s statech y with whiprobabilit :
2
1
2
1
rE
rE
sr
sr
sp
Foundations of Financial Markets 24
Correlation: definition
21
2121
,cov,
rVarrVar
rrrrcorr
2asset of variance:
1asset of variance:
and between covariance :,cov
2
1
2121
rVar
rVar
rrrr
Foundations of Financial Markets 25
Correlation (cont’d)
212121 ,,
notation theuseoften willWe
rrrrrrcorr
assets two theof
deviations standard theare and where
,cov
thatNote
21
212121
rr
rrrrrr
Foundations of Financial Markets 26
Other properties
2121212
221
212211
21
2
:constants twobe and Let
rrrrwwrVarwrVarwrwrwVar
ww
11 :for valuesof Range2121 rrrr ρρ-
-
Foundations of Financial Markets 27
Correlation=-1
0 1 2 3 4 5 60
1
2
3
4
5
6
r1
r 2r1 r2 probability
1 5 .2
2 4 .2
3 3 .2
4 2 .2
5 1 .2
Foundations of Financial Markets 28
Correlation=+1
0 1 2 3 4 5 60
1
2
3
4
5
6
r1
r 2r1 r2 probability
1 1 .2
2 2 .2
3 3 .2
4 4 .2
5 5 .2
Foundations of Financial Markets 29
Correlation=0
r1 r2 probability
2 2 .2
2 4 .2
3 3 .2
4 4 .2
4 2 .2 0 1 2 3 4 5 60
1
2
3
4
5
6
r1
r 2
Foundations of Financial Markets 30
Roadmap
1. Rates of Return– Holding Period Return– Arithmetic and Geometric Averages– Annual Percentage Rate and Effective Annual Rate
2. Summary Statistics of rates of return– Probability Distribution– Expected Return– Variance, Covariance and Standard Deviation– Other properties
3. Historical record of Bills, Bonds, and Stocks– Risk premia from 1926-2003?– Inflation and Real Rates of Return
Foundations of Financial Markets 31
Characteristics of Probability Distributions
1) Mean: most likely value
2) Variance or standard deviation
3) Skewness
* If a distribution is approximately normal, the distribution is described by characteristics 1 and 2
Foundations of Financial Markets 32
rrNegativeNegative PositivePositive
Skewed Distribution: Large Negative Returns Possible
Median
Foundations of Financial Markets 33
rrNegativeNegative PositivePositive
Skewed Distribution: Large Positive Returns Possible
Median
Foundations of Financial Markets 34
Roadmap
1. Rates of Return– Holding Period Return– Arithmetic and Geometric Averages– Annual Percentage Rate and Effective Annual Rate
2. Summary Statistics of rates of return– Probability Distribution– Expected Return– Variance, Covariance and Standard Deviation– Other properties
3. Historical record of Bills, Bonds, and Stocks– Risk premia from 1926-2003?– Inflation and Real Rates of Return
Foundations of Financial Markets 35
Risk premium
• An expected return in excess of that of a risk free rate
• Example– The expected return on the S&P500 is 9%– The return on a 1-month T-bill is 3%– The risk premium is 6% (9%-3%)
Foundations of Financial Markets 36
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
Foundations of Financial Markets 37
Risk Premia
Arith. Stan.
Series Mean% Dev.%
World Stk 7.37 18.69
US Lg Stk 8.46 20.80
US Sm Stk 14.64 38.72
Wor Bonds 2.34 8.98
LT Treas 1.85 8.00
Foundations of Financial Markets 38
Figure 5.1 Frequency Distributions of Holding Period Returns
Foundations of Financial Markets 39
Figure 5.2 Rates of Return on Stocks, Bonds and Bills
Foundations of Financial Markets 40
Roadmap
1. Rates of Return– Holding Period Return– Arithmetic and Geometric Averages– Annual Percentage Rate and Effective Annual Rate
2. Summary Statistics of rates of return– Probability Distribution– Expected Return– Variance, Covariance and Standard Deviation– Other properties
3. Historical record of Bills, Bonds, and Stocks– Risk premia from 1926-2003?– Inflation and Real Rates of Return
Foundations of Financial Markets 41
Real vs. Nominal Rates
• Notation:– R=nominal return– i =inflation rate– r =real return
• Exact relationship
• Approximate relationship
• Example R = 9%, i = 6%: what is r?
Rir 111
iRr
Foundations of Financial Markets 42
Figure 5.4 Interest, Inflation and Real Rates of Return
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