Transcript Chapter 5 - Risk and Return: Past and Prologue

Fina2802: Investments and Portfolio Analysis
Spring, 2010
Dragon Tang
Lecture 8
Risk and Return: Past and Prologue
February 4, 2010
Readings: Chapter 5
Practice Problem Sets: 3,4,5,14,15-17
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
1
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
2
Risk and Return in Chinese
Risk: 危机
Danger | Opportunity
Return: 回报
Come back | Gratitude
Fin 2802, Spring 10
08 - Tang
Chapter 5: Risk and Return
3
Risk and Return
Objectives:
1. Characterize the risk and return on stocks (risky) and
bonds (risk-free).
2. Historical risk and return of various securities
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
4
Return over One Period:
Holding Period Return (HPR)
HPR: Rate of return over a given investment period
HPR 
Fin 2802, Spring 10 - Tang
Ending Price - Beginning Price  Dividends
Beginning Price
Chapter 5: Risk and Return
5
Rates of Return: Single Period Example
Beginning Price =
Ending Price =
Dividend =
100
110
4
HPR = ( 110 - 100 + 4 )/ ( 100) = 14%
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
6
Real vs. Nominal Rates
• Notation:
– R=nominal return
– i =inflation rate
– r =real return
• Exact relationship
1  r 1  i   1  R
• Approximate relationship
r  R i
• Example R = 9%, i = 6%: what is r?
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
7
Quoting Conventions
APR = annual percentage rate
(periods in year) X (rate for period)
EAR = effective annual rate
( 1+ rate for period)Periods per yr - 1
Example: monthly return of 1%
APR = 1% X 12 = 12%
EAR = (1.01)12 - 1 = 12.68%
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
8
Return over Multiple Periods
$X
t= 0
$Y
r1
1
$Z
r2
2
$X, $Y, $Z: Cash Flows; r1, r2: one-period HPR
• Dollar-weighting: Internal Rate of Return (IRR)
$Y
$Z
$X 

0
1
2
1  IRR 1  IRR
• Time-weighting:
– Arithmetic Average: rA = (r1+r2)/2
– Geometric Average: rG = [(1+r1)(1+r2)]1/2 – 1
– rA ? rG always
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
9
Example
1
Assets(Beg.) 1.0
HPR
.10
TA (Before
Net Flows)
1.1
Net Flows
0.1
End Assets
1.2
Fin 2802, Spring 10 - Tang
2
1.2
.25
3
4
2.0
.8
(.20) .25
1.5
0.5
2.0
1.6
1.0
(0.8) 0.0
.8
1.0
Chapter 5: Risk and Return
10
Returns Using Arithmetic
and Geometric Averaging
Arithmetic
ra = (r1 + r2 + r3 + ... rn) / n
ra = (.10 + .25 - .20 + .25) / 4
= .10 or 10%
Geometric
rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1
rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1
= (1.5150) 1/4 -1 = .0829 = 8.29%
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
11
Dollar Weighted Average Example
Net CFs
$ (mil)
0
-1.0
1
2
- 0.1 - 0.5
3
0.8
4
1.0
Solving for IRR
1.0 = -.1/(1+r)1 + -.5/(1+r)2 + .8/(1+r)3 +1.0/(1+r)4
r = .0417 or 4.17%
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
12
Which One to Use?
T
T
Inflowt
Outflowt

Dollar-weighted return(IRR): 
t
t




1

IRR
1

IRR
t 1
t 1
• Use if focus is total amount of money at some terminal date
(wealth)
Time-weighted return:
- Arithmetic Average: rA  r1  r2    rn , ignore compounding
n
- Geometric Average: rG  1 r1 1 r2    1 rn 1/ n 1,
compounding over time.
• Use if there is no control over timing
• Used most by money management industry
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
13
HPR - Expected Return
Er    ps r s 
s
p s  the probability of each scenario
r  s  the HPR in each scenario
s
Fin 2802, Spring 10 - Tang
indexation variable for scenarios
Chapter 5: Risk and Return
14
Normal distribution
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
15
HPR - Risk Measure
Variance or standard deviation:
   ps r s   Er 
2
2
s

Fin 2802, Spring 10 - Tang

2
Chapter 5: Risk and Return
16
Why do we need the variance?
0.4
0.35
•Two variables
with the same
mean.
0.3
•What do we
know about their
dispersion?
Probability
0.25
0.2
0.15
0.1
0.05
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
Outcomes
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
17
Example
Suppose your expectations regarding the stock market are as
follows:
State of the economy
Scenario(s)
Probability(p(s))
Boom
1
0.3
Normal Growth
2
0.4
Recession
3
0.3
Compute the mean and standard deviation of the HPR on stocks.
HPR
44%
14%
-16%
E( r ) = 0.3*44% + 0.4*14%+0.3*(-16%)=14%
Sigma^2=0.3*(44%-14%)^2+0.4*(14%-14%)^2
+0.3*(-16%-14%)^2=0.54
Sigma=0.7348=73.48%
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
18
Historical Mean and Variance
Data in the n-point time series are treated as realization
of a particular scenario each with equal probability 1/n
n
rt
r 
t 1 n

rt  r 
n
 

n  1 t 1
n
n
2
2
Fin 2802, Spring 08
10 - Tang
Chapter 5: Risk and Return
19
Annual Holding Period Returns
Historical Returns: 1926-2003
Series
World Stk
US Lg Stk
US Sm Stk
Wor Bonds
LT Treas
T-Bills
Inflation
Fin 2802, Spring 08
10 - Tang
Geom.
Mean%
9.41
10.23
11.80
5.34
5.10
3.71
2.98
Arith.
Mean%
11.17
12.25
18.43
6.13
5.64
3.79
3.12
Chapter 5: Risk and Return
Stan.
Dev.%
18.38
20.50
38.11
9.14
8.19
3.18
4.35
20
Skewed Distribution: Large Negative Returns Possible
Median
Negative
Fin 2802, Spring 10 - Tang
r
Chapter 5: Risk and Return
Positive
21
Skewed Distribution: Large Positive Returns Possible
Median
Negative
Fin 2802, Spring 10 - Tang
r
Chapter 5: Risk and Return
Positive
22
Table 5.5 Risk Measures for Non-Normal Distributions
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
23
Summary
Definition of Returns: HPR, APR and AER.
Risk and expected return
Next: Asset Allocation
Fin 2802, Spring 10 - Tang
Chapter 5: Risk and Return
24