risk and return riccardo colacito. foundations of financial markets 2 roadmap 1.rates of return...

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


Holding Period Return

0

101

P

DPPHPR

DividendCash

Price Ending

Price Beginning

1

1

0

D

P

P


Rates of Return: Single Period Example

Ending Price = 24

Beginning Price = 20

Dividend = 1

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


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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%


Roadmap


2. Summary Statistics of rates of return– Probability Distribution– Expected Return– Variance and Standard Deviation– Other properties



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%


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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


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


Normal distribution


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


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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


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


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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?


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?


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


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)


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


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


Correlation: definition

21

2121

,cov,

rVarrVar

rrrrcorr

2asset of variance:

1asset of variance:

and between covariance :,cov

2

1

2121

rVar

rVar

rrrr


Correlation (cont’d)

212121 ,,

notation theuseoften willWe

rrrrrrcorr

assets two theof

deviations standard theare and where

,cov

thatNote

21

212121

rr

rrrrrr


Other properties

2121212

221

212211

21

2

:constants twobe and Let

rrrrwwrVarwrVarwrwrwVar

ww

11 :for valuesof Range2121 rrrr ρρ-

-


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


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


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


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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


rrNegativeNegative PositivePositive

Skewed Distribution: Large Negative Returns Possible

Median


rrNegativeNegative PositivePositive

Skewed Distribution: Large Positive Returns Possible

Median


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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%)


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


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


Figure 5.1 Frequency Distributions of Holding Period Returns


Figure 5.2 Rates of Return on Stocks, Bonds and Bills


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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


Figure 5.4 Interest, Inflation and Real Rates of Return

risk and return riccardo colacito. foundations of financial markets 2 roadmap 1.rates of return...

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