1. Risk and Return Holding Period Return Multi-period Return
Return Distribution
Historical Record
Risk and Return

2. Single Period Return Holding Period Return: Example
Percentage gain during a period
HPR: holding period return
P0: beginning price
P1: ending price
D1: cash dividend
Example
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? P0
P1+D1
t = 0
t = 1
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3. Multi-period Return What’s the return over a few periods?
Consider a mutual fund story
Net inflow when the fund does well
Net outflow when the fund does poorly
Question:
How would we characterize the fund’s performance over the year?
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4. Multi-period Return Arithmetic Average
Sum of each period return scaled by the number of periods
ra: arithmetic return
ri: HPR in the ith period
N: number of periods
Example:
Calculate the arithmetic return of the fund
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5. Multi-period Return Geometric Average
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Multi-period Return
Geometric Average
Single period return giving the same cumulative performance as the sequence of actual returns
rg: geometric return
ri: HPR in the ith period
N: number of periods
Example:
Calculate the geometric return of the fund
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6. 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
Considers change in the initial investment
Conventions (from investor’s viewpoint)
Initial investment as outflow (negative)
Ending value as inflow (positive)
Additional investment as outflow (negative)
Reduced investment as inflow (positive)
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7. Multi-period Return: Dollar-weighted
Example: IRR = ? (assets in million dollars)
By definition
Using Excel
t =0
t =1
t =2
t =3
t =4
CF0 = -1
CF1 = -.1
CF2 = -.5
CF3 = .8
CF4 = 1.0
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8. Multi-period Return: APR vs. EAR
APR – arithmetic average
EAR – geometric average
T: length of a holding period (in years)
HPR: holding period return
APR and EAR relationship
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9. Multi-period Return - Examples
25-year zero-coupon Treasury Bond
Example 2
What’s the APR and EAR if monthly return is 1%
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10. 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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11. 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:
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12. Risk and Return Measures
Scenario Analysis and Probability Distribution
Expected Return
Return Variance
Standard Deviation (“Risk”)
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13. Risk and Return Measures
More Numerical Analysis
Using Excel
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14. 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?
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15. Historical Record Annual HPR of different securities
Risk premium = asset return – risk free return
Real return = nominal return – inflation
From historical record
Risk Premium and Real Return are based on APR, i.e. arithmetic average
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16. Real vs. Nominal Rate Real vs. Nominal Rate – Exact Calculation:
R: nominal interest rate (in monetary terms)
r: real interest rate (in purchasing powers)
i: inflation rate
Approximation (low inflation):
Example
8% nominal rate, 5% inflation, real rate?
Exact:
Approximation:
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17. Risk and Horizon S&P 500 Returns 1970 – 2005 How do they compare* ?
Mean *260 = 8.866%
Std. Dev *260 = %
SURPRISED???
Daily
Yearly
Mean
0.0341%
8.9526%
Std. Dev.
1.0001% %
* There is approximately 260 working days in a year
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18. It is accepted that stock returns are independent across time
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:
s(ryear) ¹ s(r1) + … + s(r260)
Var(ryear) = Var(r1) + … + Var(r260)
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19. Consecutive Returns Volatility
Daily volatility seems to be disproportionately huge!
S&P 500 Calculations
Daily: Var(rday) = ^2 =
Yearly: Var(ryear) = *260 =
Yearly:
Bottom line:
Short-term risks are big, but they “cancel out” in the long run!
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20. Accounting for Risk - Sharpe Ratio
Reward-to-Variability (Sharpe) Ratio
E[r] – rf - Risk Premium
r – rf Excess Return
Sharpe ratio for a portfolio: or
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21. 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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