risk and return holding period return multi-period return return distribution historical record risk...
TRANSCRIPT
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
Investments 7 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
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
Investments 7 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
Investments 7 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:
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%
Investments 7 15
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?