Return and Risk

Returns – Nominal vs. RealHolding Period Return

Multi-period ReturnReturn DistributionHistorical RecordRisk and Return

Investments 7 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:

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

Investments 7 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

1)1( /1

THPREAR

T

HPRAPR

T

EARAPR

T 1)1(

Investments 7 5

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

Investments 7 9

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%

Investments 7 11

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)

Investments 7 14

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

Investments 7 15

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

][

Investments 7 16

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”

Investments 7 18

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?