Return and Risk
Returns – Nominal vs. Real
Holding Period Return
Multi-period Return
Return Distribution
Historical Record
Risk and Return

Real vs. Nominal Rate

Real vs. Nominal Rate – Exact Calculation:

1

R

( 1
 r )

( 1
 i )
 r

1

R
1
 i

1

R
1

 i 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

Investments 7
8% nominal rate, 5% inflation, real rate?


Exact: r

R
1
Approximation:

 r i i

8 %
1


R
 i

5 %
5 %

8 %


2 .
86
5 %
%

3 %
2

Single Period Return

Holding Period Return:


Percentage gain during a period
HPR

P
1

D
1
P
0

P
0
P
0 t = 0

HPR : holding period return



P
0
: beginning price
P
1
: ending price
D
1
: cash dividend
Example
Investments 7

P
1
+D
1 t = 1
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?
HPR

P
1

D
P
0
1

P
0

24

1
20

20

25 %
3

Multi-period Return: APR vs. EAR


APR – arithmetic average
EAR – geometric average
APR
EAR


HPR
T
( 1

HPR )
1 / T 
1



T : length of a holding period (in years)
HPR : holding period return
APR and EAR relationship
APR

( 1

EAR )
T 
1
T
Investments 7 4

Multi-period Return - Examples

Example 1

25-year zero-coupon Treasury Bond
HPR

329 .
18 %

APR
EAR


329 .
18

0 .
1317
25
( 1

3 .
2918 )
1 / 25 
1


13 .
17 %
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 5

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 6

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
Normal
Bust
1
2
3
0.25
0.50
0.25
44%
14%
-16%
Investments 7 7

Risk and Return Measures

Scenario Analysis and Probability Distribution

Expected Return
E [ r ]
    s

[ 0 .
25

44 %
 p ( s ) r ( s )
0 .
5

14 %

0 .
25

(

16 %)]

14 %

Return Variance
Var [ r ]
 
2   s p ( s )( r ( s )

E [ r ])
2

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 8

Risk and Return Measures

More Numerical Analysis

Using Excel
State (s) Prob: p(s) HPR: r(s) p(s)*r(s) p(s)*(r(s)-E[r])^2
1 0.10
-5% -0.005
0.004
2
3
0.20
0.40
5%
15%
0.01
0.06
0.002
0
4
5
0.20
0.10
25%
35%
0.05
0.035
0.002
0.004
E[r] = 15.00%
Var[r] = 0.012
SD[r] = 10.95%
Investments 7 9

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
Neutral
1
2
0.35
0.30
35.00
27.00
4.40
4.00
67.66%
31.91%
Pessimist 3
E[HPR] = 26.55%
0.35
15.00
4.00
-19.15%
Std Dev = 36.48%
Investments 7 10

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 Class
Small Stocks
Large Stocks
LT Gov Bond
T-bills
Inflation
Geometric
Mean
Arithmetic
Mean
Standard
Deviation
Risk
Premium
Real
Return
12.01% 17.95% 38.71% 14.20% 14.82%
10.17% 12.15% 20.26% 8.40% 9.02%
5.38% 5.68% 8.09% 1.93% 2.55%
3.70%
2.99%
3.75% 3.15% 0.00% 0.62%
3.13% 4.29% N/A N/A
Risk Premium and Real Return are based on APR, i.e. arithmetic average
Investments 7 11

Risk and Horizon

S&P 500 Returns 1970 – 2005
Mean
Daily
0.0341% Mean
Yearly
8.9526%
Std. Dev.
1.0001% 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 12

Consecutive Returns
It is accepted that stock returns are independent across time



Consider 260 days of returns r
1
,…, r
260
Means:
E( r year
) = E( r
1
) + … + E( r
260
)
Variances vs. Standard Deviations:

( r year
)
 
( r
1
) + … + 
( r
260
)
Var( r year
) = Var( r
1
) + … + Var( r
260
)
Investments 7 13

Consecutive Returns Volatility
Daily volatility seems to be disproportionately huge!

S&P 500 Calculations



Daily: Var( r day
) = 1.0001^2 = 1.0002001
Yearly: Var( r year
) = 1.0002001*260 = 260.052
Yearly:

(r year
)

260.052

16 .
126 %

Bottom line:
Shortterm risks are big, but they “cancel out” in the long run!
Investments 7 14

Accounting for Risk - Sharpe Ratio

Reward-to-Variability (Sharpe) Ratio


E[r] – r f r – r f
- Risk Premium
- Excess Return

Sharpe ratio for a portfolio:
SR

Risk
 of premium excess return or SR

E [ r p

] p
 r f
Investments 7 15

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).
Investments 7 16

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 17

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 18