CHAPTER 5 -...
Transcript of CHAPTER 5 -...
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Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin
CHAPTER 5
Introduction to Risk, Return, and
the Historical Record
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Interest Rate Determinants
• Supply
– Households
• Demand
– Businesses
• Government’s Net Supply and/or Demand
– Federal Reserve Actions
5-2
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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
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Fig 5.1: Real Rate of Interest Equilibrium
5-4
Determined by supply, demand, government actions,
expected rate of inflation
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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
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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
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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
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Time Does Matter
5-8
Use Annualized Rates of Return
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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
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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?
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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%
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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
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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
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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%
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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
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Table 5.2 Statistics for T-Bill Rates, Inflation Rates and Real Rates, 1926-2012
5-16
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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
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Fig 5.3: Interest Rates and Inflation1926-2009
5-18
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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
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Rates of Return: Single Period Example
• Ending Price = 110
• Beginning Price = 100
• Dividend = 4
• HPR = (110 - 100 + 4 ) / (100) = 14%
5-20
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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
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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
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Variance and Standard Deviation
5-23
22 ( ) ( ) ( )
s
p s r s E r
2STD
Standard Deviation (STD):
Variance (VAR):
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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
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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?
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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
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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
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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
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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
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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
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Figure 5.4 The Normal Distribution
5-31
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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
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Skew and Kurtosis
5-33
3
3
RRaverageskew
3
ˆ 4
4
RRaveragekurtosis
onsdistributi symmetricfor zero is this
ondistributi Normal afor 3 equals this
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Fig.5.5A Normal and Skewed Distributions
5-34
Mean = 6%
SD = 17%
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Fig 5.5B Normal & Fat-Tailed Distributions
5-35
Mean = 0.1
SD = 0.2
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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
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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
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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
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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
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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
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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
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A lot more coins
• Let’s play a game: flip 30 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?
5-42
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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 Variance?
• Q. What is the portfolio Standard Deviation?
5-43
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A Portfolio of 3 stocks
• Portfolio = 𝑤𝐴 × 𝐴 + 𝑤𝐵 × 𝐵 + 𝑤𝐶 × 𝐶
• Q. What is the portfolio expected return?
• Q. What is the portfolio Variance?
• Q. What is the portfolio Standard Deviation?
• Q. What is if you have N stocks?
5-44
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Q. Which portfolio has best Sharpe?
(A)
(B) (C)
(D) (E)
30% (A)50% (B)20% (D)
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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
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Figure 5.7 Nominal and Real Equity Returns Around the World, 1900-2000
5-47
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Figure 5.8 Standard Deviations of Real Equity and Bond Returns Around the World, 1900-2000
5-48
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Figure 5.9 Probability of Investment Outcomes After 25 Years with a Lognormal Distribution
5-49
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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