1. Returns – Nominal vs. Real
Return and Risk
Returns – Nominal vs. Real
Holding Period Return
Multi-period Return
Return Distribution
Historical Record
Risk and Return

2. 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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3. 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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4. 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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5. 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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6. 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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7. Measuring Risk and Return
You decide to invest in IBM, what will be your return over next year?
Scenario Analysis vs. Historical Record
Scenario Analysis:
Historical Record:
What time period historical data should you use?
What data is relevant now? 1930s? 1980s? 2008?
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8. Risk and Return Measures
Scenario Analysis and Probability Distribution
Expected Return
Return Variance
Standard Deviation (“Risk”)
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9. Risk and Return Measures
More Numerical Analysis
Using Excel
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10. 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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11. Accounting for Risk - Sharpe Ratio
Reward-to-Variability (Sharpe) Ratio
E[r] – rf - Risk Premium
r – rf Excess Return
rf - Risk-free rate, i.e. 1 month T-Bill rate
Sharpe ratio for a portfolio: or
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12. 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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13. 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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14. 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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15. Normality Assumption
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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16. Other Measures of Risk - Value at Risk
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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17. 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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