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

Risk and Return

Holding Period Return

Multi-period Return

Return Distribution

Historical Record

Risk and Return

Investments 7 2

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 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?

1Q 2Q 3Q 4QAssets at the start ($M) 1.0 1.2 2.0 0.8HPR 10.0% 25.0% -20.0% 25.0%Assets before net inflow 1.1 1.5 1.6 1.0Net Inflow 0.1 0.5 -0.8 0.0Assets in the end 1.2 2.0 0.8 1.0

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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

N

ii

Na r

NN

rrrr

1

21 1...

%104

%25%20%25%10...21

N

rrrr N

a

Investments 7 5

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

1)1(1)1(...)1()1(

1

1

1

21

NN

iiNNg rrrrr

%29.81%)251(%)201(%)251(%)101( 4

1

gr

Investments 7 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)

N

ii

iN

N

IRR

CF

IRR

CF

IRR

CF

IRR

CFCF

02

210 )1()1(

...)1(1

0

Investments 7 7

Multi-period Return: Dollar-weighted Example: IRR = ? (assets in million dollars)

By definition

Using Excel

1Q 2Q 3Q 4QAssets at the start 1.0 1.2 2.0 0.8HPR 10.0% 25.0% -20.0% 25.0%Assets before net inflow 1.1 1.5 1.6 1.0Net Inflow 0.1 0.5 -0.8 0.0Assets in the end 1.2 2.0 0.8 1.0

t =1 t =2 t =3 t =4

CF0 = -1

t =0

CF1 = -.1 CF2 = -.5 CF3 = .8 CF4 = 1.0

432 )1(

01

)1(

8

)1(

5

1

1010

IRR

.

IRR

.

IRR

.

IRR

.

Time 0 1 2 3 4 IRRCF -1.0 -0.1 -0.5 0.8 1.0 4.17%

Investments 7 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

1)1( /1

THPREAR

T

HPRAPR

T

EARAPR

T 1)1(

Investments 7 9

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%

Investments 7 12

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 13

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%

Investments 7 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?

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

Asset ClassGeometric

MeanArithmetic

MeanStandard Deviation

Risk Premium

Real 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

Investments 7 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:

i

iR

i

RrirR

1

11

1)1()1(1

iRr

%86.2%51

%5%8

1

i

iRr

%3%5%8 iRr

Investments 7 17

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

Investments 7 18

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 19

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 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:

orreturnexcessof

premiumRiskSR

p

fp rrESR

][

Investments 7 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?