Download Investments 7

Return and Risk
Returns – Nominal vs. Real
Holding Period Return
Multi-period Return
Return Distribution
Historical Record
Risk and Return
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
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8% nominal rate, 5% inflation, real rate?

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R  i 8%  5%
r

 2.86%
1 i
1  5%
Approximation: r  R  i  8%  5%  3%
Exact:
2
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

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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
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Multi-period Return: APR vs. EAR


APR – arithmetic average
EAR – geometric average

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
HPR
APR 
T
EAR  (1  HPR )1/ T  1
T: length of a holding period (in years)
HPR: holding period return
APR and EAR relationship
(1  EAR)T  1
APR 
T
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Multi-period Return - Examples

Example 1

25-year zero-coupon Treasury Bond
HPR  329.18%
329.18
APR 
 0.1317  13.17%
25
EAR  (1  3.2918)1/ 25  1  0.06  6%

Example 2

What’s the APR and EAR if monthly return is 1%
APR  N  r  12 1%  12%
EAR  (1  r ) N  1  (1  1%)12  1  12.68%
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Return (Probability) Distribution

Moments of probability distribution

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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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Risk and Return Measures


You decide to invest in IBM, what will be
your return over next year?
Scenario Analysis vs. Historical Record
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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
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More Numerical Analysis
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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
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Example
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Current stock price $23.50.
Forecast by analysts:
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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
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Annual HPR of different securities
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Risk premium = asset return – risk free return
Real return = nominal return – inflation
From historical record 1926-2005
Geometric Arithmetic Standard
Risk
Real
Asset Class
Mean
Mean
Deviation Premium Return
Small Stocks
12.01%
17.95% 38.71% 14.20% 14.82%
Large Stocks
10.17%
12.15% 20.26%
8.40% 9.02%
LT Gov Bond
5.38%
5.68%
8.09%
1.93% 2.55%
T-bills
3.70%
3.75%
3.15%
0.00% 0.62%
Inflation
2.99%
3.13%
4.29%
N/A
N/A
Risk Premium and Real Return are based on APR, i.e. arithmetic average
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Risk and Horizon
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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* ?
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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
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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!
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S&P 500 Calculations
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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
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Reward-to-Variability (Sharpe) Ratio
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E[r] – rf
r – rf
- Risk Premium
- Excess Return
Sharpe ratio for a portfolio:
E[rp ]  r f
Risk premium
or SR 
SR 
p
 of excess return
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Normality Assumption
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The normality assumption for simple returns is
reasonable if the horizon is not too short (less than a
month) or too long (decades).
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Other Measures of Risk - Value at Risk
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Term coined at J.P. Morgan in late 1980s
Alternative risk measurement to variance, focusing on
the potential for large losses
• VaR statements are
typically made in $ and
pertain to a particular
investment horizon, e.g.
–“Under normal market
conditions, the most the
portfolio can lose over a
month is $2.5 million at the
95% confidence level”
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Wrap-up
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What is the holding period return?
What are the major ways of calculating
multi-period returns?
What are the important moments of a
probability distribution?
How do we measure risk and return?
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