Transcript Slide 1
Chapter 5
Risk and
Return: Past
and Prologue
Measuring Ex-Post (Past) Returns
One period investment: regardless of the length of the
period. Must be in %. Holding period return (HPR):
HPR
= [P1 – P0 + CF] / P0
5-2
Annualizing HPRs
Q: Why would you want to annualize returns?
1. Annualizing HPRs for holding periods of greater
than one year:
– Without compounding (Simple or APR):
HPRann = HPR/n
–
–
With compounding: EAR
HPRann = [(1+HPR)1/n]-1
where n = number of years held
5-3
Measuring Ex-Post (Past)
Returns
• Must be Geometric Means for multiperiods
• Must be weighted means for portfolios
Dollar-Weighted Return
i. Dollar-weighted return procedure (DWR):
= IRR we learned
e.g. a stock pays $2 dividend was purchased for $50. The
prices at the end of the year 1 and year 2 are $53 and $54,
respectively. What’s DWR (or IRR)?
5-5
Time-Weighted Returns
AAR or GAR
ii. Time-weighted returns (TWR):
TWRs assume you buy one
___ share of
the stock at the beginning of each
one share at
interim period and sell ___
the end of each interim period. TWRs are thus
___________
independent of the amount invested in a given period.
To calculate TWRs:
Calculate the return for each time period, typically a year.
calculate either an arithmetic (AAR) or a geometric
Then
average (GAR) of the returns.
5-6
Measuring Ex-Post (Past) Returns
HPR for year 1:
[$53 + $2 - $50] / $50 = 10%
HPR for year 2:
[$54 - $53 +$2] / $53 = 5.66%
AAR = [0.10 + 0.0566] / 2 = 7.83% or 7.81% for GAR
5-7
Measuring Ex-Post (Past) Returns
Q: When should you use the GAR and when should you use
the AAR?
A1: When you are evaluating PAST RESULTS (ex-post):
Use the AAR (average without compounding) if you ARE
NOT reinvesting any cash flows received before the end of
the period.
Use the GAR (average with compounding) if you
ARE reinvesting any cash flows received before the end of
the period.
A2: When you are trying to estimate an expected return (exante return):
Use the AAR
5-8
Characteristics of Probability
Distributions
Arithmetic average & usually most likely _
1. Mean: __________________________________
2. Median:
Middle
observation
_________________
3. Variance or standard deviation:
Dispersion of returns about the mean
4. Skewness:_______________________________
Long tailed distribution, either side
5. Leptokurtosis: ______________________________
Too many observations in the tails
If a distribution is approximately normal, the distribution
1 and 3
is fully described by Characteristics
_____________________
5-9
Value at Risk (VaR)
VaR versus standard deviation:
• For normally distributed returns VaR is equivalent to
standard deviation (although VaR is typically
reported in dollars rather than in % returns)
• VaR adds value as a risk measure when return
distributions are not normally distributed.
– Actual 5% probability level will differ from 1.68445
standard deviations from the mean due to
kurtosis and skewness.
5-10
5.3 The Historical Record
You read
5-11
Inflation, Taxes and Returns
Real vs. Nominal Rates
Fisher effect: Approximation
real rate nominal rate - inflation rate
rreal rnom - i
rreal = real interest rate
Example rnom = 9%, i = 6%
rnom = nominal interest rate
rreal 3%
i = expected inflation rate
Fisher effect: Exact
rreal = [(1 + rnom) / (1 + i)] – 1
or
rreal = (rnom - i) / (1 + i)
rreal = (9% - 6%) / (1.06) = 2.83%
The exact real rate is less than the approximate
real rate.
5-12
Historical Real Returns &
Sharpe Ratios (p121)
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.
5-13
5.5 Asset Allocation Across
Risky and Risk Free
Portfolios
5-14
Example
rf = 5%
srf = 0%
E(rp) = 14%
srp = 22%
y = % in rp
(1-y) = % in rf
5-15
Expected Returns for Combinations
E(rC) = yE(rp) + (1 - y)rf
sc = ysrp + (1-y)srf
E(rC) = Return for complete or combined portfolio
For example, let y = 0.75
____
E(rC) = (.75 x .14) + (.25 x .05)
E(rC) = .1175 or 11.75%
sC = ysrp + (1-y)srf
sC = (0.75 x 0.22) + (0.25 x 0) = 0.165 or 16.5%
5-16
Quantifying Risk Aversion
E rp rf 0.5 A s p
p121
2
E(rp) = Expected return on portfolio p
rf = the risk free rate
0.5 = Scale factor
A x sp2 = Proportional risk premium
The larger A is, the larger will be the
_________________________________________
investor’s added return required to bear risk
Many studies have concluded that investors’ average
risk aversion is between 2 and 4 (or 1 and 2 w/o
scaling)
5-17
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.
5-18
Excess Returns and Sharpe Ratios
implied by the CML
Excess Return or Risk
Premium
Time
Period
1926-2008
1926-1955
1956-1984
1985-2008
Average
7.86
11.67
5.01
5.95
s
20.88
25.40
17.58
18.23
Sharpe
Ratio
0.37
0.46
0.28
0.33
The average risk premium implied by the CML for
large common stocks over the entire time period is
7.86%.
• How much confidence do we have that this
historical data can be used to predict the risk
premium now?
5-19
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 investor’s
risk aversion.
5-20