Risk and Return风险与回报

Interpreting the Standard Deviation
40%
Stock and Bond Indices:
United States 1926-1995
10000
S&P 500 Small Company Stocks
S&P 500 monthly returns 1926-1995
_______
σ(r1) σ(r2)
-10%
Stock 1
Covariance = .0068 / (5-1) = .0017
Always between -1 and +1
6
Correlation Between Returns
Correlation +1
10%
Correlation -1
10%
The log return is calculated as ln Pt - ln Pt-1 .
The next graph is a time series plot of log returns.
3
Time Series of Log Returns
10 0% 5% 0 3 .4 4 .3 3 % 2 % -8 % .3 -2 .9 9 % 9 % 5 .5 2 % 4 .7
U (W0 − x)
W0 − x
W0
W0 + x
W
The issues we have considered today: How do you measure risk? How do you compare risk and return?
8
Relationship Between Wealth and Utility
Utility Utility Function Suppose an individual has: (i) current wealth of W0 and (ii) the opportunity to undertake an investment which has a 50% chance of earning £X and a 50% chance of earning -£X. Is this an investment the individual would undertake voluntarily? Wealth
(.091-.093)2 (.334-.093)2 (.423-.093)2 (-.083-.093)2 (-.299-.093)2 Total =
0 % 9 % .1 -5 % 0 -1 0 0% 12 96
-4 % .9 -6 .4 4 %
-1 .8 5 %
13 95
Variance = .3516 / (5-1) = 0.0879 Standard deviation = √Var = .2965
10%
Correlation
S toc k 2
(.04-.02)(.02-.03) (-.02-.02)(.03-.03) (.08-.02)(.06-.03) (-.04-.02)(-.04-.03) (.04-.02)(.08-.03) Total =
ρ12 =
-10% 10%
Cov (r1 r2)
30%
Log Index Values (1926 = 1) 1000
Corporate Bonds LT Govt Bond
20% 10% 0% -10% -20% -30%
100
30Day TBills
10
1 1926 0.1
1950
1974
1998
This graph shows one standard deviation (5.7%) above and below the monthly mean return (1.0%). For normal data, about one third of the spikes are outside the lines.
Risky Investments
S&P500, 1926-1995
Risk and Return
Returns
Return on an investment comes in two forms: A) income i.e. dividends in the case of stocks and shares B) Capital Gain or loss based on change in price
Month Stock 1 Stock 2 t=1 t=2 t=3 t=4 t=5 1 .04 .02 2 -.02 .03 3 .08 .06 4 -.04 -.04 5 .04 .08 -.0002 .0000 .0018 .0042 .0010 .0068 Mean .02 .03
Correlation
4
Volatility
S&P 500 Returns 70% Small Company Stock Returns
Comparing Risk and Return
Mean annual return Small Company Stocks S&P 500 11.5% 9.7% 5.3% 4.7% 3.6% Standard deviation 31.7% 19.7% 7.5% 7.3% 3.1%
1
2
Log Returns
Index 1926 1927 1928 1929 1930 1931 1.0000 1.0947 +0.0947 +9.47% +0.091 +0.334 +0.423 -0.083 -0.299 Change % return Log return
1.5286 +0.4338 +39.63% 2.3324 +0.8038 +52.59% 2.1474 1.5920 -0.1850 -0.5554 -7.93% -25.86%
-10%
10%
-10%
10%
-10%
-10%
Perfect substitute
No correlation
10%
Direct competitor
-10%
10%
-10%
Completely unrelated
Risky or Certain Outcome?
Toss two coins: Outcome Gain H&H +£6.00 H&T +£1.00 T&T - £4.00 Total Probability .25 .50 .25 Exp. gain £1.50 £0.50 -£1.00 £1.00
Each series starts at an index value of 1 in January 1926. The graph shows how much an investor who invests one dollar in January 1926 would have received in 1995.
Long-term average 9.3%
Mean and Standard Deviation
Month 1 2 3 .423 4 5 Mean .093 Returns .091 .334 1 2 3 4 5 -.083 -.299 0.0000 0.0580 0.1088 0.0310 0.1538 0.3516 29.65% 9.30%
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Equities
Bonds
Cash
5
Remember: Standard Deviation
Month 1 2 3 .423 4 5 Mean .093 Returns .091 .334 1 2 3 4 5 -.083 -.299 0.0000 0.0580 0.1088 0.0310 0.1538 0.3516 29.65% 9.30%
%
40 30 20 10 0 -10
5%
0% 0% 5% 10% 15% 20% 25% Standard deviation of returns
-20 -30 Dec-85 Dec-86 Dec-87 Dec-88 Dec-89 Dec-90 Dec-91 Dec-92 Dec-93 Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec-00
Which distribution do you prefer, safe or risky?
120 100 80 60 40 20 0
50 45 40 35 30 25 20 15 10 5 0 T T H T H H
7
Risk Aversion
An individual is said to be risk averse if he/she prefers less risk for the same expected return. Individuals are generally risk averse when it comes to situations in which a large fraction of their wealth is at risk. Insurance Investing Given a choice between £X with certainty, or a risky gamble in which the expected payoff is also £X, a risk averse individual will choose the certain payoff. What does this imply about the relationship between an individual’s wealth and the utility derived from such wealth?
Risk and Return:
The basic relationship
15%
Risk and Return in the UK
70 60 50
Small Company Stocks
M ean return 10%
S&P 500 Corporate Bonds T-Bills Govt Bonds
Utility Curve
U
U (W0 + x ) U (W0 )
u d
Implications of Risk Aversion
Risk averse individuals will try to avoid “fair bets.” require higher expected returns on riskier investments. Whether an individual undertakes a risky investment will depend upon three things: The individual’s utility function. The individual’s initial wealth. The payoffs on the risky investment relative to those on a risk-free investment.
35%
0% 1940
1950
1960
1970
1980
1990
Corporate Bonds LT Government Bonds 30Day T-Bills
-35%
This graph compares large and small capitalisation shares. Which offers the higher return? Which is the more volatile?
(.091-.093)2 (.334-.093)2 (.423-.093)2 (-.083-.093)2 (-.299-.093)2 Total =
Variance = .3516 / (5-1) = 0.0879 Standard deviation = √Var = .2965
Covariance