Risk and Return
Holding Period Return
Multi-period Return
Return Distribution
Historical Record
Risk and Return
Single Period Return

Holding Period Return:

Percentage gain during a period
P0
P1  D1  P0
HPR 
P0
t=0
 HPR: holding period return
 P0: beginning price
 P1: ending price
 D1: cash dividend

P1+D1
t=1
Example

Investments 7
You bought a stock at $20. A year later, the stock price
appreciates to $24. You also receive a cash dividend of
$1 during the year. What’s the HPR?
P  D1  P0 24  1  20
HPR  1

 25%
P0
20
2
Multi-period Return

What’s the return over a few periods?

Consider a mutual fund story
Assets at the start ($M)
HPR
Assets before net inflow
Net Inflow
Assets in the end



1Q
2Q
3Q
4Q
1.0
1.2
2.0
0.8
10.0% 25.0% -20.0% 25.0%
1.1
1.5
1.6
1.0
0.1
0.5
-0.8
0.0
1.2
2.0
0.8
1.0
Net inflow when the fund does well
Net outflow when the fund does poorly
Question:

Investments 7
How would we characterize the fund’s performance over
the year?
3
Multi-period Return

Arithmetic Average

Sum of each period return scaled by the number
of periods
r1  r2  ...  rN 1 N
ra 
  ri
N
N i 1




ra: arithmetic return
ri: HPR in the ith period
N: number of periods
Example:

Investments 7
Calculate the arithmetic return of the fund
r  r  ...  rN 10%  25%  20%  25%
ra  1 2

 10%
N
4
4
Multi-period Return

Geometric Average

Single period return giving the same cumulative
performance as the sequence of actual returns
1
N


rg  (1  r1 )  (1  r2 )  ... (1  rN )  1   (1  ri )  1
 i 1

 rg: geometric return
 ri: HPR in the ith period
 N: number of periods
1
N

N
Example:

Calculate the geometric return of the fund
rg  (1  10%)  (1  25%)  (1  20%)  (1  25%)  1  8.29%
1
4
Investments 7
5
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
N
CFN
CFi
CF1
CF2
0  CF0 


...



1  IRR (1  IRR) 2
(1  IRR) N i 0 (1  IRR)i


Considers change in the initial investment
Conventions (from investor’s viewpoint)




Investments 7
Initial investment as outflow (negative)
Ending value as inflow (positive)
Additional investment as outflow (negative)
Reduced investment as inflow (positive)
6
Multi-period Return: Dollar-weighted

Example: IRR = ? (assets in million dollars)
Assets at the start
HPR
Assets before net inflow
Net Inflow
Assets in the end
t =0
CF0 = -1
t =1
CF1 = -.1

By definition

Using Excel
1Q
2Q
3Q
4Q
1.0
1.2
2.0
0.8
10.0% 25.0% -20.0% 25.0%
1.1
1.5
1.6
1.0
0.1
0.5
-0.8
0.0
1.2
2.0
0.8
1.0
t =2
t =3
CF2 = -.5
CF3 = .8
t =4
CF4 = 1.0
 0.1
 .5
.8
1.0
0  1 



2
3
1  IRR (1  IRR ) (1  IRR ) (1  IRR ) 4
Investments 7
Time 0
1
2
3
4
IRR
CF
-1.0 -0.1 -0.5 0.8 1.0 4.17%
7
Multi-period Return: APR vs. EAR


APR – arithmetic average
EAR – geometric average



HPR
APR 
T
EAR  (1  HPR)1/ T  1
T: length of a holding period (in years)
HPR: holding period return
APR and EAR relationship
(1  EAR)T  1
APR 
T
Investments 7
8
Multi-period Return - Examples

Example 1

25-year zero-coupon Treasury Bond
HPR  329.18%
329.18
APR 
 0.1317 13.17%
25
EAR  (1  3.2918)1/ 25  1  0.06  6%

Example 2

What’s the APR and EAR if monthly return is 1%
APR  N  r  121%  12%
EAR  (1  r ) N  1  (1  1%)12  1  12.68%
Investments 7
9
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
10
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
11
Risk and Return Measures

Scenario Analysis and Probability Distribution

Expected Return
E[r ]     p( s)r ( s)
s
 [0.25 44%  0.5 14%  0.25 (16%)]  14%

Return Variance
Var[r ]   2   p(s)(r (s)  E[r ])2
s
 0.25 (.44  .14) 2  0.5  (.14  .14) 2  0.25 (.16  .14) 2  0.045

Standard Deviation (“Risk”)
SD[r]    Var[r]  0.045  0.2121 21.21%
Investments 7
12
Risk and Return Measures

More Numerical Analysis

Using Excel
State (s) Prob: p(s) HPR: r(s)
1
0.10
-5%
2
0.20
5%
3
0.40
15%
4
0.20
25%
5
0.10
35%
p(s)*r(s) p(s)*(r(s)-E[r])^2
-0.005
0.004
0.01
0.002
0.06
0
0.05
0.002
0.035
0.004
E[r] =
15.00%
Var[r] =
0.012
SD[r] = 10.95%
Investments 7
13
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
14
Historical Record

Annual HPR of different securities



Risk premium = asset return – risk free return
Real return = nominal return – inflation
From historical record 1926-2006
Geometric Arithmetic Standard
Risk
Real
Asset Class
Mean
Mean
Deviation Premium 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
15
Real vs. Nominal Rate

Real vs. Nominal Rate – Exact Calculation:
1 R
R i
1  R  (1  r )  (1  i )  r 
1 
1 i
1 i




R: nominal interest rate (in monetary terms)
r: real interest rate (in purchasing powers)
i: inflation rate
Approximation (low inflation):
r  R i

Example

8% nominal rate, 5% inflation, real rate?


Investments 7
R  i 8%  5%
r

 2.86 %
1 i
1  5%
Approximation: r  R  i  8%  5%  3%
Exact:
16
Risk and Horizon

S&P 500 Returns 1970 – 2005
Daily
Mean
0.0341%
Std. Dev.
1.0001%

Yearly
Mean
8.9526%
Std. Dev. 15.4574%
How do they compare* ?


Mean
Std. Dev.
0.0341*260 = 8.866%
1.0001*260 = 260.026%
SURPRISED???
* There is approximately 260 working days in a year
Investments 7
17
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:
(ryear)  (r1) + … + (r260)
Var(ryear) = Var(r1) + … + Var(r260)
Investments 7
18
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:  (ryear )  260.052 16.126%
Bottom line:
Short-term risks are big, but they “cancel out”
in the long run!

Investments 7
19
Accounting for Risk - Sharpe Ratio

Reward-to-Variability (Sharpe) Ratio



E[r] – rf
r – rf
- Risk Premium
- Excess Return
Sharpe ratio for a portfolio:
E[rp ]  rf
Risk prem ium
or SR 
SR 
p
 of excessreturn
Investments 7
20
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?
Investments 7
21