1
Learning about Return and Risk
from The Historical Record
Chapter 5
Bodi Kane Marcus Ch 5
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Bodi Kane Marcus Ch 5
Determinants of the level of Interests
•
•
•
•
Real and Nominal rates of Interest
The equilibrium Real Rate of Interest
The equilibrium Nominal Rate of Interest
Taxes and Real Rate of Interest
more…
Bodi Kane Marcus Ch 5
Real vs. Nominal Rates
Fisher effect: Approximation
nominal rate = real rate + inflation premium
R = r + i or r = R - i
Example r = 3%, i = 6%
R = 9% = 3% + 6% or 3% = 9% - 6%
Fisher effect: Exact
r = (R - i) / (1 + i)
2.83% = (9%-6%) / (1.06)
Empirical Relationship:
Inflation and interest rates move closely together
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Bodi Kane Marcus Ch 5
Factors Influencing Rates
• Supply
▫ Households
• Demand
▫ Businesses
• Government’s Net Supply and/or Demand
▫ Federal Reserve Actions
4
The equilibrium Real Rate of Interest
Interest Rates
Bodi Kane Marcus Ch 5
Supply
r1
r0
Demand
Q0 Q1
Funds
Pergeseran kurva Demand ke kanan dapat terjadi karena pemerintah menerapkan
budget deficit  permintaan akan uang meningkat  interest rate naik
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6
Bodi Kane Marcus Ch 5
Risk and Risks Premium
• Holding Period Returns
• Expected Return and Standard Deviation
• Excess Returns and Risk Premiums
more……
Bodi Kane Marcus Ch 5
Holding Period Returns
Rates of Return: Single Period
P D
P
HPR 
P
1
0
1
0
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one
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Bodi Kane Marcus Ch 5
Rates of Return:
Single Period Example
Ending Price ($) =
Beginning Price ($) =
Dividend ($) =
48
40
2
HPR = (48 - 40 + 2 )/ (40) = 25%
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Bodi Kane Marcus Ch 5
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
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Bodi Kane Marcus Ch 5
Characteristics of Probability Distributions
1) Mean:
The simple mathematical average of a set of
two or more numbers
2) Variance : A measure of the dispersion of a set
of data points around their mean value.
▫ Variance is a mathematical expectation of the
average squared deviations from the mean.
3) Skewness : an asymmetry in the distribution of
the data values
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Bodi Kane Marcus Ch 5
The Coefficient of Variation (CV)
• A statistical measure of the dispersion of data
points in a data series around the mean. It is
calculated as follows:
The coefficient of variation represents the ratio
of the standard deviation to the mean.
In the investing world, CV determine how much
volatility (risk) in comparison to the amount of
return you can expect from your investment.
Source: Investopedia
Bodi Kane Marcus Ch12
5
Normal Distribution
s.d.
s.d.
r
Symmetric distribution
Bodi Kane Marcus Ch13
5
Measuring Mean:
Scenario or Subjective Returns
Subjective returns
E(r) =  p(s) r(s)
s
p(s) = probability of a state
r(s) = return if a state occurs
1 to s states
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Bodi Kane Marcus Ch 5
Numerical Example:
Subjective or Scenario Distributions
State
Prob. of State r in State
1
.1
-.05
2
.2
.05
3
.4
.15
4
.2
.25
5
.1
.35
E(r) = (.1)(-.05) + (.2)(.05)...+ (.1)(.35)
E(r) = .15
Bodi Kane Marcus Ch 5
Measuring Variance or
Dispersion of Returns
Subjective or Scenario
2

Variance=
p(s) [rs - E(r)]
s
Standard deviation = [variance]1/2
Using Our Example:
Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2]
Var= .01199
S.D.= [ .01199] 1/2 = .1095
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Bodi Kane Marcus Ch 5
Annual Holding Period Returns
Geom.
Series Mean%
Large Stock
10.5
Small Stock
12.6
LongT Gov
5.0
T-Bills
3.7
Inflation
3.1
Arith.
Stan.
Mean%
Dev.%
12.5
19.0
5.3
3.8
3.2
20.4
40.4
8.0
3.3
4.5
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Bodi Kane Marcus Ch 5
Expected Return and Standard
Deviation
• Spreadsheet 5.1
Purchase Price
$
100
T-Bill Rate
6%
Ending Price Dividends HPR
Probability
($)
Boom
0.3
129.5
4.5
0.34
Normal growth
0.5
110
4
0.14
Recession
0.2
80.5
3.5
-0.16
Expected value (mean)
Standard Deviations
p* HPR
0.10
0.07
-0.03
0.14
2
Dev
0.04
0
0.09
Excess
p*Dev
Returns
0.012
0.28
0
0.08
0.018
-0.22
2
0.1732
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Teknis Penghitungan
Bodi Kane Marcus Ch 5
• Expected Value
▫ Jumlahkan hasil perkalian probabilitas dengan HPR
• Standard Deviation of HPR
▫ Deviasi= HPR dikurangi mean
▫ Kuadratkan Deviasi
▫ Kalikan probabilitas dg Dev2
▫ Jumlahkan (p* Dev2 ), kemudian pangkat-kan (0.5) atau diakar
• Excess Return
▫ HPR dikurangi RFR
• Squared Deviations (Dev2 ) Excess Return
▫ Excess Return dikurangi risk premium, kemudian dikuadratkan
• Standard Deviation of Excess Return
▫ Kalikan probabilitas dg Dev2 Excess Return
▫ Jumlahkan (p* Dev2 ), kemudian pangkat-kan (0.5) atau diakar
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Bodi Kane Marcus Ch 5
Expected Return and Standard Deviation
• Problem 5-7 page 151
Ending Price HPR (incl
Probability
($)
Dividends)
Boom
0.35
140
44.50%
Normal growth
0.3
110
14%
Recession
0.35
80
-16.50%
Expected value (mean)
p* HPR
0.16
0.04
-0.06
0.14
Dev2
0.093025
0
0.093025
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Bodi Kane Marcus Ch 5
Excess Returns and Risk Premiums
• Excess Returns : Returns in excess of the risk-free rate or
in excess of a market measure, such as an index fund.
▫ When you have excess returns you are making more money
than if you put your money into an index fund like the Dow
Jones Industrial Average (DJIA).
• Risk Premiums : The return in excess of the risk-free rate
of return that an investment is expected to yield.
▫ An asset's risk premium is a form of compensation for
investors who tolerate the extra risk - compared to that
of a risk-free asset - in a given investment.
Source: Investopedia
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Bodi Kane Marcus Ch 5
Time Series Analysis of Past Rates of Return
• Time Series versus Scenario Analysis
• Expected Returns and the arithmetic Average
• The Geometric (Time Weighted) Average
Return
• The Reward to Volatility (Sharpe) Ratio
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Bodi Kane Marcus Ch 5
Time Series Analysis of Past Rates of Return
• Time Series versus Scenario Analysis
• Time Series Analysis
▫ useful to see how a given asset, security or economic
variable changes over time or how it changes
compared to other variables over the same time period
• Scenario Analysis: The process of estimating the
expected value of a portfolio after a given period of time,
assuming specific changes in the values of the portfolio's
securities or key factors that would affect security values,
such as changes in the interest rate.
Source: Investopedia
23
the arithmetic mean of a stock's closing price = $ 74.00 / 5 = $14.80.
Bodi Kane Marcus Ch 5
Time Series Analysis of Past Rates of Return
• The arithmetic Average: A mathematical representation of the
typical value of a series of numbers, computed as the sum of all the
numbers in the series divided by the count of all numbers in the
series.
Source: Investopedia
Day
1
2
3
4
5
Sum
Closing
Price
$14.50
$14.80
$15.20
$14.00
$15.50
$74.00
The arithmetic mean of a stock's closing price =
$ 74.00 / 5 = $14.80.
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Bodi Kane Marcus Ch 5
Time Series Analysis of Past Rates of Return
• The Geometric (Time Weighted) Average Return: The
average of a set of products, the calculation of which
is commonly used to determine the performance results
of an investment or portfolio.
• The Geometric = {(1+ r1)*(1+r2)*…*(1+rn)} 1/n -1
Average Return
more……
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Illustration Geometric Mean
Year
Return
1
0.15
2
0.20
3
-0.20
• Geometric Mean
• [(1.15) x(1.20) x (0.80)]1/3 – 1
• = (1.104) 1/3 -1 =0.03353 = 3.353%
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Bodi Kane Marcus Ch 5
Time Series Analysis of Past Rates of Return
• The Reward to Volatility (Sharpe) Ratio: A ratio
developed by Nobel laureate William F. Sharpe to
measure risk-adjusted performance.
▫ The Sharpe ratio is calculated by subtracting the risk-free
rate from the rate of return for a portfolio and dividing the
result by the standard deviation of the portfolio returns
Source: Investopedia
Sharpe Ratio 
Risk Premium
SD of Excess Return
(5.18)
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Bodi Kane Marcus Ch 5
The Normal Distribution
Problem 5.6/ CFA Problem/p.153
Jawaban: Probabilitas perekonomian dalam keadaan neutral dan saham pada kondisi kinerja poor=
0.15
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Bodi Kane Marcus Ch 5
Deviations from Normality
Skew 
E[r(s)- E(r)]3

3
(5.19)
Positively skewed
Negatively skewed
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Bodi Kane Marcus Ch 5
The Historical Record of Returns on
Equities and Long Term Bonds
•
•
•
•
•
•
•
Average Returns and Standard Deviations
Other statistics of Risky Portfolios
Sharpe Ratios
Serial Correlation
Skewness and Kurtosis
Estimates of Historical Risk Premiums
A Global View of the Historical Record
more……
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Bodi Kane Marcus Ch 5
Average Returns and Standard Deviation
Answer CFA Problem 3 dan 4/ p.153
stock X
Exp Return
Probability
Bull market
Normal
Market
Bear Market
0.3
0.5
0.2
50%
18%
Dev
Dev2 Dev2*p
30.00%
0.090 0.0270
-2.00%
0.000 0.0002
-40.00%
0.160 0.0320
15.00%
9.00%
-20%
-4.00%
Exp Return
20.00%
Variance
St Dev
0.0592
0.2433
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Bodi Kane Marcus Ch 5
Measurement of Risk with non Normal
Distributions
• Value at Risk (VaR) : A technique used to estimate the
probability of portfolio losses based on the statistical
analysis of historical price trends and volatilities.
• Conditional Tail Expectation (CTE) : an important
actuarial risk measure and a useful tool in financial risk
assessment.
• Lower Partial Standard Deviation (LPSD) : Compute
expected lower partial moments for normal asset returns
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Lower Partial Standard Deviation
(LPSD)
Bodi Kane Marcus Ch 5
• Measure of risk non normal distributions
• The LPSD for the large and small stock
portfolios are not very different from value from
the normal distributions because the skews are
similar to those from the normal (see Table 5.5)
Large US Stocks
Small US Stocks
Lower Partial Standard
Deviation (%)
History
Normal
History
Normal
LPSD for 25 year HPR
4.34
4.23
7.09
7.14
LPSD for 1 year HPR
21.71
21.16
35.45
35.72
Average 1 –year HPR
12.13
12.15
17.97
17.95
Bodi Kane Marcus Ch 5
Annual Holding Period Risk Premiums
and Real Returns
Risk
Series
Premiums%
Lg Stk
8.7
Sm Stk
15.2
LT Gov
1.5
T-Bills
0
Inflation
0.6
Real
Returns%
9.3
15.8
2.1
0.6
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