Transcript Risk and Return: Past and Prologue Risk Aversion and Capital

Risk and Return: Past and
Prologue
Risk Aversion and Capital
Allocation to Risk Assets
HPR: Rate of return over a given investment period
HPR 
Ending Price - Beginning Price  Dividends
Beginning Price
Ending Price =
110
Beginning Price = 100
Dividend =
4
$100
$50
r1
t= 0
1
r1, r2: one-period HPR

$100
r2
2
What is the average return of your investment per
period?
◦ Arithmetic Average: rA = (r1+r2)/2
◦ Geometric Average: rG = [(1+r1)(1+r2)]1/2 – 1

Arithmetic return: return earned in an average period
over multiple period r  r  r    r
1
A
◦
◦
◦
◦
2
n
n
It is the simple average return.
It ignores compounding effect
It represents the return of a typical (average) period
Provides a good forecast of future expected return
rG  1  r1 1  r2    1  rn  1
1/ n

Geometric return
◦ Average compound return per period
◦ Takes into account compounding effect
◦ Provides an actual performance per year of the investment over the
full sample period
◦ Geometric returns <= arithmetic returns
Quarter
HPR
1
.10
2
.25
3
4
(.20) .25
What are the arithmetic and geometric return of
this mutual fund?
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%

Invest $1 into 2 investments: one gives 10% per
year compounded annually, the other gives 10%
compounded semi-annually. Which one gives
higher return
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%
m
APR 

EAR  1 
 1
m 


Risk in finance: uncertainty related to outcomes
of an investment
◦ The higher uncertainty, the riskier the investment.
◦ How to measure risk and return in the future


Probability distribution: list of all possible
outcomes and probability associated with each
outcome, and sum of all prob. = 1.
For any distribution, the 2 most important
characteristics
◦ Mean
◦ Standard deviation
s.d.
s.d.
r or E(r)
n
E r    pi ri
i 1
pi  the probabilit y of each scenario
ri  the HPR in each scenario
i indexation variable for scenarios
Variance or standard deviation:
n
   pi (ri  E (r ))
2
i 1


2
2
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.
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=540
Sigma=23.24%
HPR
44%
14%
-16%
0.4
0.35
0.25
Probability
•Two variables
with the same
mean.
0.3
0.2
0.15
•What do we
know about their
dispersion?
0.1
0.05
0
-5
-4
-3
-2
-1
0
Outcomes
1
2
3
4
5
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

n
2
1
2
rt  r 


n  1 t 1


Year
Ri(%)
1988
1989
1990
1991
1992
16.9
31.3
-3.2
30.7
7.7
Compute the mean and variance of this sample
Series
World Stk
US Lg Stk
US Sm Stk
Wor Bonds
LT Treas.
T-Bills
Inflation
Geom.
Arith.
Mean% Mean%
9.80
11.32
10.23
12.19
12.43
18.14
5.80
6.17
5.35
5.64
3.72
3.77
3.04
3.13
Stan.
Dev.%
18.05
20.14
36.93
9.05
8.06
3.11
4.27
Risk
Premium
7.56
8.42
14.37
2.40
2.07
0
18%
Small-Company Stocks
Annual Return Average
16%
14%
Large-Company Stocks
12%
10%
8%
6%
T-Bonds
4%
T-Bills
2%
0%
5%
10%
15%
20%
25%
Annual Return Standard Deviation
30%
35%
Figure 5.2 Rates of Return on Stocks, Bonds and Bills

A large enough sample drawn from a normal
distribution looks like a bell-shaped curve.
Probability
The probability that a yearly return
will fall within 20.2 percent of the
mean of 12.2 percent will be
approximately 2/3.
– 3
– 48.3%
– 2
– 28.1%
– 1
– 7.9%
0
12.2%
68.26%
95.44%
99.74%
+ 1
32.5%
+ 2
52.7%
+ 3
72.9%
Return on
large company common
stocks

The 20.14% standard deviation we found for
large stock returns from 1926 through
2005 can now be interpreted in the
following way: if stock returns are
approximately normally distributed, the
probability that a yearly return will fall
within 20.14 percent of the mean of 12.2%
will be approximately 2/3.
• Risk aversion: higher risk requires higher return, risk
averse investors are rational investors
• Risk-free rate: the rate you can earn by leaving the money
in risk-free assets such as T-bills.
• Risk premium
(=Risky return –Risk-free return)
• It is the reward for investor for taking risk involved in
investing risky asset rather than risk-free asset.
•The Risk Premium is the added return (over and above the riskfree rate) resulting from bearing risk.






Historically, stock is riskier than bond, bond is riskier
than bill
Return of stock > bond > bill
More risk averse, put more money on bond
Less risk averse, put more money on stock
This decision is asset allocation
John Bogle, chairman of the Vanguard Group of
Investment Companies
◦ “The most fundamental decision of investing is the allocation of
your assets: how much should you own in stock, how much in
bond, how much in cash reserves. That decision accounts for an
astonishing 94% difference in total returns achieved by
institutionally managed pension funds. ... There is no reason to
believe that the same relationship does not hold true for individual
investors.”
The complete portfolio is composed of:
• The risk-free asset: Risk can be reduced by allocating
more to the risk-free asset
• The risky portfolio: Composition of risky portfolio does
not change
•This is called Two-Fund Separation Theorem.
The proportions depend on your risk aversion.
Total portfolio value
= $300,000
Risk-free value
=
90,000
Risky (Vanguard & Fidelity) = 210,000
Vanguard (V) = 54%
Fidelity (F)
= 46%
Vanguard
Fidelity
Portfolio P
Risk-Free Assets F
Portfolio C
113,400/300,000 = 0.378
96,600/300,000 = 0.322
210,000/300,000 = 0.700
90,000/300,000 = 0.300
300,000/300,000 = 1.000


Only the government can issue defaultfree bonds
◦ Guaranteed real rate only if the
duration of the bond is identical to the
investor’s desire holding period
T-bills viewed as the risk-free asset
◦ Less sensitive to interest rate
fluctuations



It’s possible to split investment funds
between safe and risky assets.
Risk free asset: proxy; T-bills
Risky asset: stock (or a portfolio)
Example: Let the expected return on the risky portfolio, E(rP),
be 15%, the return on the risk-free asset, rf, be 7%. What is the
return on the complete portfolio if all of the funds are invested
in the risk-free asset? What is the risk premium?
7%
0
What is the return on the portfolio if all of the funds are
invested in the risky portfolio?
15%
8%
Example: Let the expected return on the risky portfolio, E(rP),
be 15%, the return on the risk-free asset, rf, be 7%. What is the
return on the complete portfolio if 50% of the funds are
invested in the risky portfolio and 50% in the risk-free asset?
What is the risk premium?
0.5*15%+0.5*7%=11%
4%
In general:
E rC   yE (rp )  (1  y )rf
y  fraction of funds invested in the risky asset
E rC   rf  risk premium on the complete portfolio
E rP   rf  risk premium on the risky asset
 C2  y 2 P2  1  y 2  r2  2 y1  y  Pr  P r
f
f
f
 C  y P if  r  0
f
where
c - standard deviation of the complete portfolio
P - standard deviation of the risky portfolio
rf - standard deviation of the risk-free rate
y - weight of the complete portfolio invested in the
risky asset
Example: Let the standard deviation on the risky portfolio,
P, be 22%. What is the standard deviation of the complete
portfolio if 50% of the funds are invested in the risky
portfolio and 50% in the risk-free asset?
22%*0.5=11%
We know that given a risky asset (p) and a risk-free asset, the
expected return and standard deviation of any complete
portfolio (c) satisfy the following relationship:
E (rc )  rf  y ( E (rp )  rf )
 c  y p
Where y is the fraction of the portfolio invested in the risky asset

E (rc )  rf
c
E (rc )  rf 

E (rp )  rf
p
E (rp )  rf
p
c
for every complete portfolio c
Risk Tolerance and Asset Allocation:
•More risk averse - closer to point F
•Less risk averse - closer to P
S
E  rP   r f
P
S is the increase in expected return per unit
of additional standard deviation
S is the reward-to-variability ratio or
Sharpe Ratio
Example: Let the expected return on the risky portfolio, E(rP),
be 15%, the return on the risk-free asset, rf, be 7% and the
standard deviation on the risky portfolio, P, be 22%. What is
the slope of the CAL for the complete portfolio?
S = (15%-7%)/22% = 8/22
•So far, we only consider 0<=y<=1, that means we use only our own
money.
•Can y > 1?
•Borrow money or use leverage
•Example: budget = 300,000. Borrow additional 150,000 at the risk-free
rate and invest all money into risky portfolio
•y = 450,000/150,000 = 1.5
•1-y = -0.5
•Negative sign means short position.
•Instead of earning risk-free rate as before, now have to pay riskfree rate
E rC   yE (rp )  (1  y )rf
 1.5 *15  (0.5) * 7
 19%
 c  1.5 * 22  0.33
S 

E rP   r f
P
19  7
 0.36
.33
The slope = 0.36 means the portfolio c is still in the CAL but on the
right hand side of portfolio P
Example: Let the expected return on the risky portfolio, E(rP),
be 15%, the return on the risk-free asset, rf, be 7%, the
borrowing rate, rB, be 9% and the standard deviation on the risky
portfolio, P, be 22%.
Suppose the budget = 300,000. Borrow additional 150,000 at the
borrowing rate and invest all money into risky portfolio
What is the slope of the CAL for the complete portfolio for
points where y > 1,
y = 1.5; E(Rc) = 1.5(15) + (-0.5)*9 = 18%
 c  1.5 * 22  0.33
Slope = (0.18-0.09)/0.33 = 0.27
Note: For y  1, the slope is as indicated above if the lending
rate is rf.
SPECIAL CASE OF CAL (I.e., P=MKT)
The line provided by one-month T-bills and a
broad index of common stocks (e.g. S&P500)
Consequence of a passive investment strategy
based on stocks and T-bills
E(r)
P3?
E(Rm) = 12%
M
P2?
P1?
S=0.45
rf = 3%
F
0

20%
48

Risk Preference
◦ Risk averse
 Require compensation for taking risk
◦ Risk neutral
 No requirement of risk premium
◦ Risk loving
 Pay to take risk

Utility Values: A is risk aversion parameter
U  E (r )  0.5 A 2
FIN 8330 Lecture 7 10/04/07
49
1
2
U  E (r )  A
2
Where
U = utility
E ( r ) = expected return on the asset
or portfolio
A = coefficient of risk aversion
2 = variance of returns



Greater levels of risk aversion lead to larger
proportions of the risk free rate.
Lower levels of risk aversion lead to larger
proportions of the portfolio of risky assets.
Willingness to accept high levels of risk for high
levels of returns would result in leveraged
combinations.
55

Solve the maximization problem:
Max U  E (r )  0.5 A 2
 rf  y  ( E ( R p )  rf )  0.5 A( y p ) 2

Two approaches:
1. Try different y
2. Use calculus:

Solution:
y* 
dU  E R   r  Ay 2  0
p
f
p
dy
E ( R p )  rf
A
2
p
56
If A = 4, rf = 7%,
E(Rp) = 15%,
 p  22%
y* 
E ( R p )  rf
A p2
.15  .07

 .41
2
4  .22
58



If CAL is from 1-month T-bills and a broad index of
common stocks, then CAL is also called Capital
Market Line (CML)
Why passive strategy: (1) strategies are costly; (2)
market is competitive.
Mutual fund separation theorem: capital should be
invested in the (same optimal) risky portfolio and riskfree asset.
FIN 8330 Lecture 7 10/04/07
66


Passive strategy involves a decision that
avoids any direct or indirect security
analysis
Supply and demand forces may make such a
strategy a reasonable choice for many
investors


A natural candidate for a passively held
risky asset would be a well-diversified
portfolio of common stocks
Because a passive strategy requires
devoting no resources to acquiring
information on any individual stock or
group we must follow a “neutral”
diversification strategy
•Definition of Returns: HPR, APR and AER.
•Risk and expected return
•Shifting funds between the risky portfolio to the risk-free asset
reduces risk
•Examples for determining the return on the risk-free asset
•Examples of the risky portfolio (asset)
•Capital allocation line (CAL)
All combinations of the risky and risk-free asset
Slope is the reward-to-variability ratio
•Risk aversion determines position on the capital allocation line