5
Risk Chapter and Return: Past &
Prologue
Measuring Past Returns
One period investment: regardless of the length of the
period.
Holding period return (HPR):
HPR = (PS PB + CF)/PB where
PS
= Sale price (or P1)
PB
= Buy price ($ you put up) (or P0)
CF
= Cash flow during holding period
5-2
Measuring Past Returns
An example: Suppose you buy one share of a stock today for
$45 and you hold it for two years and sell it for $52. You also
received $8 in dividends at the end of the two years.
(PB = $45, PS = $52, CF = $8):
HPR = (52 - 45 + 8) / 45 = 33.33%
HPRann = 0.3333/2 = 16.66%
5-3
Arithmetic Average
Finding the average HPR for a time series of returns:
AAR or Arithmetic Average Return:
n
HPR avg =
HPR T
n
T =1
n = number of time periods
5-4
Arithmetic Average
An example: You have the following rates of return on a stock:
2000 -21.56%
2001 44.63%
2002 23.35%
2003 20.98%
2004
3.11%
2005 34.46%
2006 17.62%
n
HPR T
n
T =1
HPR avg =
HPR avg =
(-.2156 + .4463 + .2335 + .2098 + .0311 + .3446 + .1762)
= 17.51%
7
AAR = 17.51%
5-5
Geometric Average
An example: You have the following rate s of return on a stock:
2000 -21.56%
2001 44.63%
2002 23.35%
2003 20.98%
2004
3.11%
2005 34.46%
2006 17.62%
Geometric average or GAR: Geometric Average
Return:
1/ n
n
HPR avg
=
(1 + HPR T )
T =1
1
HPR avg = (0.7844 1.4463 1.2335 1.2098 1.0311 1.3446 1.1762)1/7 1 =
GAR = 15.61%
5-6
Dollar-Weighted Return
Dollar-weighted return procedure (DWR):
Find the internal rate of return for the cash
flows (i.e. find the discount rate that makes the
NPV of the net cash flows equal zero.)
5-7
Tips on Calculating Dollar Weighted
Returns
This measure of return considers both changes in investment
and security performance
Initial Investment is an outflow
Ending value is considered as an inflow
Additional investment is an outflow
Security sales are an inflow
5-8
Measuring Mean: Scenario or Subjective
Returns
Subjective expected returns
E(r) = p(s) r(s)
s
E(r) = Expected Return
p(s) = probability of a state
r(s) = return if a state occurs
1 to s states
5-9
Measuring Variance or Dispersion of
Returns
Variance
2 = p(s) [rs E(r)] 2
s
= [ 2]1/2
E(r) = Expected Return
p(s) = probability of a state
rs = return in state s
5-10
Numerical Example: Subjective or
Scenario Distributions
State Prob. of State Return
1
.2
- .05
2
.5
.05
3
.3
.15
E(r) = (.2)(-0.05) + (.5)(0.05) + (.3)(0.15) = 6%
2 = p(s) [rs E(r)] 2
s
2 = [(.2)(-0.05-0.06)2 + (.5)(0.05- 0.06)2 + (.3)(0.15-0.06)2]
2 = 0.0049%2
= [ 0.0049]1/2 = .07 or 7%
5-11
Expected Return &
n HPR
T
r=
T =1 n
r = average HPR
n = # observatio ns
1n
Variance : 2 =
(ri r ) 2
n 1 i =1
Standard Deviation : = 2
5-12
Normal Distribution
Risk is the possibility
Risk
of getting returns
different from
expected.
expected.
measures deviations above
measures deviations above
the mean as well as below
the mean as well as below
the mean.
the mean.
Returns > E[r] may not be
Returns > E[r] may not be
considered as risk, but with
considered as risk, but with
symmetric distribution, it is ok
symmetric distribution, it is ok
to use to measure risk.
to use to measure risk.
Average = Median
E[r] = 10%
= 20%
5-13
Risk Premium & Risk Aversion
The risk free rate is the rate of return that can be
earned with certainty.
The risk premium is the difference between the
expected return of a risky asset and the risk-free rate.
Excess Return or Risk Premiumasset = ] rf
E[rasset
Risk aversion is an investors reluctance to accept
risk.
How is the aversion to accept risk overcome?
By offering investors a higher risk premium.
5-14
Annual Holding Period Returns Statistics 1926-2008
Geom.
Excess
Mean%
Series
Arith.
Mean%
Return%
World Stk
9.20
11.00
7.25
US Lg. Stk
9.34
11.43
7.68
11.43
17.26
13.51
5.56
5.92
2.17
5.60
1.85
Sm. Stk
World Bnd
LT Bond
5.31
Geometric mean:
Best measure of
compound historical
return
Arithmetic Mean:
Expected return
5-15
Annual Holding Period Excess Returns 19262008
Series
World Stk
US Lg Stk
US Sm Stk
World Bonds
US LT Bonds
Arith.
Avg%
7.25
7.68
13.51
2.17
1.85
Required
Return%
10.25
10.68
16.51
5.17
4.85
If the risk free rate is currently 3%, then what return
should an investor require for each asset class?
Historical data
Assumes all securities in the
category are equally risky
5-16
Real vs. Nominal Rates
Fisher effect: Approximation
real rate nominal rate - inflation rate
rreal rnom - i
r
Example rnom = 9%, i = 6%
rreal 3%
real
= real interest rate
rnom = nominal interest = rate
i expected inflation rate
Fisher effect: Exact
rreal = [(1 + rnom) / (1 + i)] 1
r = (rnom - i) / (1 + i)
real
rreal = (9% - 6%) / (1.06) = 2.83%
The exact real rate is less than the approximate real
rate.
5-17
Historical Real Returns & Sharpe
Ratios
Series
World Stk
US Lg. Stk
Sm. Stk
World Bnd
LT Bond
Real Returns%
6.00
6.13
8.17
Sharpe Ratio
0.37
0.37
0.36
2.46
2.22
0.24
0.24
Real returns have been much higher for stocks than for bonds
Sharpe ratios measure the excess return to standard deviation.
The higher the Sharpe ratio the better.
Stocks have had much higher Sharpe ratios than bonds.
5-18
Allocating Capital Between Risky &
Risk-Free Assets
Possible to split investment funds between safe and
risky assets
Risk free asset rf : proxy; T-bills or money market fund
Risky asset or portfolio rp: risky portfolio
Example. Your total wealth is $10,000. You put $2,500
in risk free T-Bills and $7,500 in a stock portfolio
invested as follows:
Stock A you put $2,500
Stock B you put $3,000
Stock C you put $2,000
5-19
Allocating Capital Between Risky &
Risk-Free Assets
Weights in rp
$2,500 / $7,500 = 33.33%
WA =
$3,000 / $7,500 = 40.00%
WB =
$2,000 / $7,500 = 26.67%
WC =
100.00%
Stock A $2,500
Stock A $2,500
Stock B $3,000
Stock B $3,000
Stock C $2,000
Stock C $2,000
The complete portfolio includes the riskless investment
and rp. W = 25%; W = 75%
rf
rp
In the complete portfolio
WA = 0.75 x 33.33% = 25%; WB = 0.75 x 40.00% = 30%
WC = 0.75 x 26.67% = 20%; Wrf = 25%
5-20
Example
rf = 5%
rf = 0%
E(rp) = 14%
rp = 22%
y = % in rp
(1-y) = % in rf
5-21
Expected Returns for Combinations
E(rC) = yE(rp) + (1 - y)rf
c = y rp + (1-y) rf
E(rC) = Return for complete or combined portfolio
rf = 5%
0.75
For example, let y = ____
E(rC) = (.75 x .14) + (.25 x .05)
E(rC) = .1175 or 11.75%
C = y rp + (1-y) rf
rf = 0%
E(rp) = 14%
rp = 22%
y = % in rp
(1-y) = % in rf
C = (0.75 x 0.22) + (0.25 x 0) = 0.165 or 16.5%
5-22
Complete portfolio
E(rc) = yE(rp) + (1 - y)rf
c = y rp + (1-y) rf
Varying y results in E[rC] and C that are linear
combinations of E[rp] and rf and rp and rf
respectively.
This is NOT generally the case for
the of combinations of two or
more risky assets.
5-23
E(r)
Possible Combinations
E(rp) = 14%
P
E(rp) = 11.75%
y=1
y =.75
rf = 5%
F
y=0
0
16.5%
22%
5-24
Combinations Without Leverage
rf = 5%
rf = 0%
E(rp) = 14%
rp = 22%
y = % in rp
(1-y) = % in rf
Since rf = 0
E(rc) = yE(rp) + (1 - y)rf
c= y p
y = .75
If y = .75, then
(.75)(.14) + (.25)(.05) = 11.75%
75(.22) = 16.5% E(rc) =
c=
If y = 1
1(.22) = 22%
c=
If y = 0
0(.22) = 0%
c=
y=1
E(rc) = (1)(.14) + (0)(.05) = 14.00%
y=0
E(rc) = (0)(.14) + (1)(.05) = 5.00%
5-25
Using Leverage with Capital Allocation
Line
Borrow at the Risk-Free Rate and invest in stock
(Using 50% Leverage)
y = 1.5
E(rc) =(1.5) (.14) + (-.5) (.05) = 0.185 = 18.5%
rf = 5%
= (1.5) (.22) = 0.33 or 33%
c
E(r)
Possible Combinations
E(rp) = 14%
y = 1.5
E(rC) =18.5%
E(rp) = 14%
rf = 0%
rp = 22%
y = % in rp
(1-y) = % in rf
P
E(rp) = 11.75%
y=1
y =.75
rf = 5%
0
F
16.5%
22%
33%
5-26
Risk Aversion and Allocation
Greater levels of risk aversion lead investors to
choose larger proportions of the risk free rate
Lower levels of risk aversion lead investors to
choose larger proportions of the portfolio of risky
assets
Willingness to accept high levels of risk for high
levels of returns would result in
Possible Combinations
leveraged combinations
E(r)
E(rC) =18.5%
y = 1.5
E(rp) = 14%
P
E(rp) = 11.75%
y=1
y =.75
r=
y =f 0 5%
0
F
16.5%
22%
33%
5-27
E(r)
E(r)
P or combinations of P
or
& Rf offer a return per
unit of risk of 9/22.
unit
CAL
(Capital
Allocation
Line)
P
E(rp) = 14%
E(rp) - rf = 9%
r f = 5%
0
) Slope = 9/22
Slope
F
rp = 22%
22%
5-28
A Passive Strategy
Investing in a broad stock index and a risk free
investment is an example of a passive strategy.
The investor makes no attempt to actively find
undervalued strategies nor actively switch their
asset allocations.
The CAL that employs the market (or an index
that mimics overall market performance) is called
the Capital Market Line or CML.
5-29
Active versus Passive Strategies
Active strategies entail more trading costs than
passive strategies.
Passive investor free-rides in a competitive
investment environment.
Passive involves investment in two passive portfolios
Short-term T-bills
Fund of common stocks that mimics a broad market
index
Vary combinations according to investors risk
aversion.
5-30