CHAPTER 5 - leeds-courses.colorado.eduleeds-courses.colorado.edu/FNCE4030/MISC/slides/FNCE4030-Fall-201… · INVESTMENTS | BODIE, KANE, MARCUS Fig 5.1: Real Rate of Interest Equilibrium

INVESTMENTS | BODIE, KANE, MARCUS

Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

CHAPTER 5

Introduction to Risk, Return, and

the Historical Record


Interest Rate Determinants

• Supply

– Households

• Demand

– Businesses

• Government’s Net Supply and/or Demand

– Federal Reserve Actions

5-2


Real and Nominal Rates of Interest

• Nominal interest rate: Growth rate of your money

• Real interest rate: Growth rate of your purchasing power(how many Big Macs can I buy with my money?)*

*The Big Mac Index is a different thing

5-3

Let rn = nominal rate,

rr = real rate and

i = inflation rate. Then:

𝑟𝑟 ≈ 𝑟𝑛 − 𝑖

More precisely:

1 + 𝑟𝑟 =1 + 𝑟𝑛1 + 𝑖

solve

𝑟𝑟 =𝑟𝑛 − 𝑖

1 + 𝑖

http://www.economist.com/content/big-mac-index


Fig 5.1: Real Rate of Interest Equilibrium

5-4

Determined by supply, demand, government actions,

expected rate of inflation


Equilibrium Nominal Rate of Interest

• As the inflation rate increases, investors will demand higher nominal rates of return

• If E(i) denotes current expectations of inflation, then we get the Fisher Equation:

• Nominal rate = real rate + expected inflation

5-5

( )R r E i


Taxes and the Real Rate of Interest

• Tax liabilities are based on nominal income

– Given a tax rate (t) and nominal interest rate (R),

the real after-tax rate of return is:

5-6

• As intuition suggests, the after-tax, real rate

of return falls as the inflation rate rises.

titritiritR 11 1


Rates of Returnfor Different Holding Periods

• Zero Coupon Bond

• Par = $100

• T = maturity

• P = price

• rf(T) = total risk free return

5-7

TrP

f

1

100 1

100

PTrf


Time Does Matter

5-8

Use Annualized Rates of Return


Effective Annual Rate (EAR)

• Time matters → use EAR to annualize

• EAR definition: percentage increase in funds invested over a 1-year horizon

5-9

Tf EARTr 11

Tf TrEAR

1

11


Equation 5.8 APR

• Annual Percentage Rate (APR): annualizing using simple interest

5-10

TEARTAPR 11

T

EARAPR

T11

Q. You invest $1 for 30 years. Do you prefer [A] 5% APR, or [B] 5% EAR?

INVESTMENTS | BODIE, KANE, MARCUS 5-11

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0 5 10 15 20 25 30

Inve

stm

en

t En

d V

alu

e

(years)

End Value with APR=5.0%

End Value with EAR=5.0%


Table 5.1 APR vs. EAR

1-12

Hold the EAR fixed at 5.8%

and solve for APR

for each holding period

Hold the APR fixed at 5.8%

and solve for EAR

for each holding period


Continuous Compounding

• Frequency of compounding matters

• At the limit to (compounding time)→0:

5-13

ccreEAR 1

Q. You invest $1 for 30 years. Which interest rate do you prefer?

A. 5% EAR

B. 5% Rcc


1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0 5 10 15 20 25 30

Inve

stm

en

t En

d V

alu

e

(years)

End Value with APR=5.0%

End Value with EAR=5.0%

End Value with Rcc=5.0%


S

xTNxNT /Let r=rate and

x=compounding time →

Nxrxrxr 111 Value End

timesN gcompoundin

NxrNexr

1ln

0x0x lim1lim

How to derive Rcc

Substitute

N=T/x

x

xrT

e

1ln

0xlim

xdx

d

xrTdx

d

e

1ln

0xlim

rT

rxr

T

ee

1

1

1

0xlim

Looks like 0/0.

Use de l’Hôpital

Q.E.D.

Make x very

small. Then

use A=eln(A)

Checks: r=0 →End Value=1

T=0 →End Value=1


Table 5.2 Statistics for T-Bill Rates, Inflation Rates and Real Rates, 1926-2012

5-16


Bills and Inflation, 1926-2009

• Moderate inflation can offset most of the nominal gains on low-risk investments.

• One dollar invested in T-bills from1926–2012 grew to $20.25, but with a real value of only $1.55.

• Negative correlation between real rate and inflation rate means the nominal rate doesn’t fully compensate investors for increased in inflation

5-17


Fig 5.3: Interest Rates and Inflation1926-2009

5-18


Risk and Risk Premiums

5-19

P

DPPHPR

0

101

HPR = Holding Period Return

P0 = Beginning price

P1 = Ending price

D1 = Dividend during period one

Rates of Return: Single Period


Rates of Return: Single Period Example

• Ending Price = 110

• Beginning Price = 100

• Dividend = 4

• HPR = (110 - 100 + 4 ) / (100) = 14%

5-20


Expected Return and Standard Deviation

5-21

Expected (or mean) returns

s = state

p(s)= probability of a state

r(s) = return if a state occurs

( ) ( ) ( )s

E r p s r s

Q. What is the expected value of rolling a die?

A. 1

B. Sqrt(6)

C. Pi

D. 3.5

E. 6


Scenario Returns: Example

5-22

State Prob. of state r for that state

Excellent 0.25 0.3100

Good 0.45 0.1400

Poor 0.25 -0.0675

Crash 0.05 -0.5200

E(r) = (0.25)(0.31)

+ (0.45)(0.14)

+ (0.25)(-0.0675)

+ (0.05)(-0.52)

= 0.0976

= 9.76% (think of a probability-weighted avg)

NOTE: use decimals instead of percentages to be safe


Variance and Standard Deviation

5-23

22 ( ) ( ) ( )

s

p s r s E r

2STD

Standard Deviation (STD):

Variance (VAR):


Scenario VARiance and STD

• Example VARiance calculation:

σ2 = 0.25(0.31 - 0.0976)2 +

0.45(0.14 - 0.0976)2 +

0.25(-0.0675 - 0.0976)2 +

0.05(-0.52 - 0.0976)2 =

= 0.038

• Example STD calculation:

5-24

1949.0038.0


Time Series Analysis of Past Rates of Return

n

s

n

s

srn

srsprE11

1)()(

5-25

The Arithmetic Average of historical

rate of return as an estimator of the

expected rate of return

Q. What assumptions are we implicitly making?


Geometric Average Return

1/1 1 TVggTV nn

5-26

TV = Terminal Value of the Investment

g = geometric average rate of return

)1)...(1)(1( 21 nn rrrTV

Solve for a rate g that, if compounded n

times, gives you the same TV


EstimatingVariance and Standard Deviation

• Estimated Variance = expected value of squared deviations (from the mean)

5-27

2

1

2 1ˆ

n

s

rsrn

22 ( ) ( ) ( )

s

p s r s E r

Recall the definition of variance


Geometric Variance and Standard Deviation Formulas

Using the estimated ravg instead of the real E(r) introduces a bias:

– we already used the n observations to estimate ravg

– we really have only (n-1) independent observations

– correct by multiplying by n/(n-1)

When eliminating the bias, Variance and Standard Deviation become*:

5-28

2

11

1ˆ

n

j

rsrn

* More at http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation

http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation


The Reward-to-Volatility (Sharpe) Ratio

• Excess Return

• The difference in any particular period between

the actual rate of return on a risky asset and the

actual risk-free rate

• Risk Premium

• The difference between the expected HPR on a

risky asset and the risk-free rate

• Sharpe Ratio

5-29

Returns Excess of SD

PremiumRisk


The Normal Distribution

• Investment management math is easier when returns are normal

– Standard deviation is a good measure of risk

when returns are symmetric

– If security returns are symmetric, portfolio returns

will be, too

– Assuming Normality, future scenarios can be

estimated using just mean and standard

deviation

5-30


Figure 5.4 The Normal Distribution

5-31


Normality and Risk Measures

• What if excess returns are not normally distributed?

– Standard deviation is no longer a complete

measure of risk

– Sharpe ratio would not be a complete measure of

portfolio performance

– Need to consider higher moments, like skew and

kurtosis

5-32


Skew and Kurtosis

5-33

3

3

RRaverageskew

3

ˆ 4

4

RRaveragekurtosis

onsdistributi symmetricfor zero is this

ondistributi Normal afor 3 equals this


Fig.5.5A Normal and Skewed Distributions

5-34

Mean = 6%

SD = 17%


Fig 5.5B Normal & Fat-Tailed Distributions

5-35

Mean = 0.1

SD = 0.2


Value at Risk (VaR)

• A measure of loss most frequently associated with extreme negative returns

• VaR is the quantile of a distribution below which lies q% of the possible values of that distribution– The 5% VaR, commonly estimated in practice, is

the return at the 5th percentile when returns are

sorted from high to low.

Also referred to as 95%-ile (depends on

perspective)

5-36


0

0.5

1

1.5

2

2.5

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Normal Distribution and VaR

VaR

The area is

the percentile


Expected Shortfall (ES)

• a.k.a. Conditional Tail Expectation (CTE)

• More conservative measure of downside risk than VaR:

– VaR = highest return from the worst cases

– Real life distributions are asymmetric and have

fat tails

– ES = average return of the worst cases

5-38


0

0.5

1

1.5

2

2.5

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Normal Distribution, VaR, and Expected Shortfall

The area is

the percentile

VaRExpected

Shortfall


A game with a coin

• Let’s play a game: flip one coin, and receive $1 if heads

• Assume Pr[Heads]= p (for example p=50%)

• Q. What is the game’s expected outcome?

• Q. What is the Variance?

• Q. What is the St.Dev?

5-40


A game with two coins

• Let’s play a game: flip 2 fair coins, and receive $1 for each head

• Q. What is the portfolio expected return?

• Q. What is the portfolio Variance?

• Q. What is the portfolio St.Dev?

5-41


A lot more coins

• Let’s play a game: flip 30 fair coins, and receive $1 for each head.



• Q. What is the portfolio St.Dev?

5-42


A Portfolio of 2 stocks

• Portfolio = 0.5 * A + 0.5 * B

• A: rA = 0.08 StDevA = 0.1

• B: rB = 0.10 StDevB = 0.1

• Q. What is the portfolio Expected Return?


• Q. What is the portfolio Standard Deviation?

5-43


A Portfolio of 3 stocks

• Portfolio = 𝑤𝐴 × 𝐴 + 𝑤𝐵 × 𝐵 + 𝑤𝐶 × 𝐶



• Q. What is the portfolio Standard Deviation?

• Q. What is if you have N stocks?

5-44


Q. Which portfolio has best Sharpe?

(A)

(B) (C)

(D) (E)

30% (A)50% (B)20% (D)


Historic Returns on Risky Portfolios

• Normal distribution is generally a good approximation of portfolio returns

– VaR indicates no greater tail risk than is

characteristic of the equivalent normal

– The ES does not exceed 0.41 of the monthly SD,

presenting no evidence against the normality

• However

– Negative skew is present in some of the

portfolios some of the time, and positive kurtosis

is present in all portfolios all the time5-46


Figure 5.7 Nominal and Real Equity Returns Around the World, 1900-2000

5-47


Figure 5.8 Standard Deviations of Real Equity and Bond Returns Around the World, 1900-2000

5-48


Figure 5.9 Probability of Investment Outcomes After 25 Years with a Lognormal Distribution

5-49


Terminal Value with Continuous Compounding

5-50

When the continuously compounded rate of

return on an asset is normally distributed, the

effective rate of return will be lognormally

distributed. Remember:

2

2

5.0 so

5.0Avg Arithm.Avg Geom.

gm

EE

The Terminal Value will then be:

1 + 𝐸𝐴𝑅 𝑇 = 𝑒𝑔+0.5 𝜎2 𝑇

= 𝑒𝑇𝑔+0.5𝑇𝜎2

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CHAPTER 5 - leeds-courses.colorado.eduleeds-courses.colorado.edu/FNCE4030/MISC/slides/FNCE4030-Fall-201… · INVESTMENTS | BODIE, KANE, MARCUS Fig 5.1: Real Rate of Interest Equilibrium