return and risk returns – nominal vs. real holding period return multi-period return return...
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Return and Risk
Returns – Nominal vs. RealHolding Period Return
Multi-period ReturnReturn DistributionHistorical RecordRisk and Return
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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:
i
iR
i
RrirR
1
11
1)1()1(1
iRr
%86.2%51
%5%8
1
i
iRr
%3%5%8 iRr
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Single Period Return Holding Period Return:
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 = 10
011
P
PDPHPR
%2520
20124
0
011
P
PDPHPR
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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
1)1( /1
THPREAR
T
HPRAPR
T
EARAPR
T 1)1(
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Multi-period Return - Examples Example 1
25-year zero-coupon Treasury Bond
Example 2 What’s the APR and EAR if monthly return is 1%
%606.01)2918.31(
%17.131317.025
18.329
%18.329
25/1
EAR
APR
HPR
%68.121%)11(1)1(
%12%11212
NrEAR
rNAPR
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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
Return Variance
Standard Deviation (“Risk”)
%14%)]16(25.0%145.0%4425.0[
)()(][
s
srsprE
045.0)14.16.(25.0)14.14(.5.0)14.44(.25.0
])[)()((][
222
22
rEsrsprVars
%21.212121.0045.0][][ rVarrSD
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Risk and Return Measures More Numerical Analysis
Using ExcelState (s) Prob: p(s) HPR: r(s) p(s)*r(s) p(s)*(r(s)-E[r])^2
1 0.10 -5% -0.005 0.0042 0.20 5% 0.01 0.0023 0.40 15% 0.06 04 0.20 25% 0.05 0.0025 0.10 35% 0.035 0.004
E[r] = 15.00%Var[r] = 0.012SD[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 1926-2005
Asset ClassGeometric
MeanArithmetic
MeanStandard Deviation
Risk Premium
Real 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 S&P 500 Returns 1970 – 2005
How do they compare* ? Mean 0.0341*260 = 8.866% Std. Dev. 1.0001*260 = 260.026%
SURPRISED???
Daily Yearly
Mean 0.0341% Mean 8.9526%
Std. Dev. 1.0001% Std. Dev. 15.4574%
* There is approximately 260 working days in a year
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Consecutive ReturnsIt 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:
Bottom line:
Short-term risks are big, but they “cancel out” in the long run!
%.260.052 )(ryear 12616
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Accounting for Risk - Sharpe Ratio Reward-to-Variability (Sharpe) Ratio
E[r] – rf - Risk Premium
r – rf - Excess Return
Sharpe ratio for a portfolio:
orreturnexcessof
premiumRiskSR
p
fp rrESR
][
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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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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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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?