Risk and Return
Holding Period Return
Return Distribution
Historical Record
Risk and Return
Single Period Return

Holding Period Return:

Percentage gain during a period
P0
P1  D1  P0
HPR 
P0
t=0
 HPR: holding period return
 P0: beginning price
 P1: ending price
 D1: cash dividend

P1+D1
t=1
Example

Investments 7
You bought a stock at $20. A year later, the stock price
appreciates to $24. You also receive a cash dividend of
$1 during the year. What’s the HPR?
P  D1  P0 24  1  20
HPR  1

 25%
P0
20
2
Return (Probability) Distribution

Moments of probability distribution




Mean: measure of central tendency
Variance or Standard Deviation (SD):
measure of dispersion – measures RISK
Median: measure of half population point
Return Distribution

Describe frequency of returns falling to
different levels
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3
Risk and Return Measures


You decide to invest in IBM, what will be
your return over next year?
Scenario Analysis vs. Historical Record

Scenario Analysis:
Economy State (s) Prob: p(s) HPR: r(s)
Boom
1
0.25
44%
Normal
2
0.50
14%
Bust
3
0.25
-16%
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Risk and Return Measures

Scenario Analysis and Probability Distribution

Expected Return
E[r ]     p( s)r ( s)
s
 [0.25  44%  0.5 14%  0.25  (16%)]  14%

Return Variance
Var[r ]   2   p ( s )( r ( s )  E[r ]) 2
s
 0.25  (.44  .14) 2  0.5  (.14  .14) 2  0.25  (.16  .14) 2  0.045

Standard Deviation (“Risk”)
SD[r ]    Var[r ]  0.045  0.2121  21.21%
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Risk and Return Measures

More Numerical Analysis

Using Excel
State (s) Prob: p(s) HPR: r(s)
1
0.10
-5%
2
0.20
5%
3
0.40
15%
4
0.20
25%
5
0.10
35%
p(s)*r(s) p(s)*(r(s)-E[r])^2
-0.005
0.004
0.01
0.002
0.06
0
0.05
0.002
0.035
0.004
E[r] =
15.00%
Var[r] =
0.012
SD[r] = 10.95%
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Risk and Return Measures

Example


Current stock price $23.50.
Forecast by analysts:




optimistic analysts (7): $35 target and $4.4 dividend
neutral analysts (6): $27 target and $4 dividend
pessimistic analysts (7): $15 target and $4 dividend
Expected HPR? Standard Deviation?
Economy State (s) Prob: p(s) Target P Dividend HPR: r(s)
Optimist
1
0.35
35.00
4.40 67.66%
Neutral
2
0.30
27.00
4.00 31.91%
Pessimist
3
0.35
15.00
4.00 -19.15%
E[HPR] = 26.55%
Std Dev = 36.48%
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Historical Record

Annual HPR of different securities



Risk premium = asset return – risk free return
Real return = nominal return – inflation
From historical record 1926-2006
Geometric Arithmetic Standard
Risk
Real
Asset Class
Mean
Mean
Deviation Premium Return
Small Stocks
12.43%
18.14% 36.93% 14.37% 15.01%
Large Stocks
10.23%
12.19% 20.14%
8.42% 9.06%
LT Gov Bond
5.35%
5.64%
8.06%
1.87% 2.51%
T-bills
3.72%
3.77%
3.11%
0.00% 0.64%
Inflation
3.04%
3.13%
4.27%
N/A
N/A
Risk Premium and Real Return are based on APR, i.e. arithmetic average
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Real vs. Nominal Rate

Real vs. Nominal Rate – Exact Calculation:
1 R
R i
1  R  (1  r )  (1  i )  r 
1 
1 i
1 i




R: nominal interest rate (in monetary terms)
r: real interest rate (in purchasing powers)
i: inflation rate
Approximation (low inflation):
r  R i

Example

8% nominal rate, 5% inflation, real rate?


Investments 7
R  i 8%  5%
r

 2.86%
1 i
1  5%
Approximation: r  R  i  8%  5%  3%
Exact:
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Risk and Horizon

S&P 500 Returns 1970 – 2005
Daily
Mean
0.0341%
Std. Dev.
1.0001%

Yearly
Mean
8.9526%
Std. Dev. 15.4574%
How do they compare* ?


Mean
Std. Dev.
0.0341*260 = 8.866%
1.0001*260 = 260.026%
SURPRISED???
* There is approximately 260 working days in a year
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Consecutive Returns
It is accepted that stock returns are
independent across time



Consider 260 days of returns r1,…, r260
Means:
E(ryear) = E(r1) + … + E(r260)
Variances vs. Standard Deviations:
(ryear)  (r1) + … + (r260)
Var(ryear) = Var(r1) + … + Var(r260)
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Consecutive Returns Volatility
Daily volatility seems to be disproportionately
huge!

S&P 500 Calculations



Daily: Var(rday) = 1.0001^2 = 1.0002001
Yearly: Var(ryear) = 1.0002001*260 = 260.052
Yearly:  (ryear )  260.052  16.126%
Bottom line:
Short-term risks are big, but they “cancel out”
in the long run!

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Accounting for Risk - Sharpe Ratio

Reward-to-Variability (Sharpe) Ratio



E[r] – rf
r – rf
- Risk Premium
- Excess Return
Sharpe ratio for a portfolio:
Risk premium
SR 
 of excess return
Investments 7
or SR 
E[rp ]  r f
p
13
Wrap-up



What is the holding period return?
What are the important moments of a
probability distribution?
How do we measure risk and return?
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