Investments 7
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Transcript Investments 7
Risk and Return
Holding Period Return
Multi-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
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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
$1 during the year. What’s the HPR?
P D1 P0 24 1 20
HPR 1
25%
P0
20
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Multi-period Return
What’s the return over a few periods?
Consider a mutual fund story
Assets at the start ($M)
HPR
Assets before net inflow
Net Inflow
Assets in the end
1Q
2Q
3Q
4Q
1.0
1.2
2.0
0.8
10.0% 25.0% -20.0% 25.0%
1.1
1.5
1.6
1.0
0.1
0.5
-0.8
0.0
1.2
2.0
0.8
1.0
Net inflow when the fund does well
Net outflow when the fund does poorly
Question:
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How would we characterize the fund’s performance over
the year?
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Multi-period Return
Arithmetic Average
Sum of each period return scaled by the number
of periods
r1 r2 ... rN 1 N
ra
ri
N
N i 1
ra: arithmetic return
ri: HPR in the ith period
N: number of periods
Example:
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Calculate the arithmetic return of the fund
r r ... rN 10% 25% 20% 25%
ra 1 2
10%
N
4
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Multi-period Return
Geometric Average
Single period return giving the same cumulative
performance as the sequence of actual returns
1
N
rg (1 r1 ) (1 r2 ) ... (1 rN ) 1 (1 ri ) 1
i 1
rg: geometric return
ri: HPR in the ith period
N: number of periods
1
N
N
Example:
Calculate the geometric return of the fund
rg (1 10%) (1 25%) (1 20%) (1 25%) 1 8.29%
1
4
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Multi-period Return: Dollar-weighted
Internal Rate of Return (IRR)
The discount rate that sets the present value of
the future cash flows equal to the amount of initial
investment
N
CFN
CFi
CF1
CF2
0 CF0
...
1 IRR (1 IRR) 2
(1 IRR) N i 0 (1 IRR)i
Considers change in the initial investment
Conventions (from investor’s viewpoint)
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Initial investment as outflow (negative)
Ending value as inflow (positive)
Additional investment as outflow (negative)
Reduced investment as inflow (positive)
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Multi-period Return: Dollar-weighted
Example: IRR = ? (assets in million dollars)
Assets at the start
HPR
Assets before net inflow
Net Inflow
Assets in the end
t =0
CF0 = -1
t =1
CF1 = -.1
By definition
Using Excel
1Q
2Q
3Q
4Q
1.0
1.2
2.0
0.8
10.0% 25.0% -20.0% 25.0%
1.1
1.5
1.6
1.0
0.1
0.5
-0.8
0.0
1.2
2.0
0.8
1.0
t =2
t =3
CF2 = -.5
CF3 = .8
t =4
CF4 = 1.0
0.1
.5
.8
1.0
0 1
2
3
1 IRR (1 IRR ) (1 IRR ) (1 IRR ) 4
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Time 0
1
2
3
4
IRR
CF
-1.0 -0.1 -0.5 0.8 1.0 4.17%
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Multi-period Return: APR vs. EAR
APR – arithmetic average
EAR – geometric average
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 121% 12%
EAR (1 r ) N 1 (1 1%)12 1 12.68%
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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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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?
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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:
E[rp ] rf
Risk prem ium
or SR
SR
p
of excessreturn
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Wrap-up
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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