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Unformatted text preview: Outlines of Lecture 2 Objectives
1. Discuss if asset returns are normally
distributed
2. Measure the expected return and risk
3. Discuss if time diversification works References
Quant_Review.pdf; Chapter 5 & 7.5, BKM;
Handouts
Hsiulang Chen Portfolio Analysis 1 Are assets returns normally
distributed? Individual stock returns may not be normally
distributed. The normal return distribution allows
for any outcome, including a negative stock price. An alternative assumption: the continuously
compounded annual rate of return (r) is normally
distributed.
[Chapter 5, BKM] [Quant_Review.pdf]
Hsiulang Chen Portfolio Analysis 2 Are assets returns normally
distributed? If re is the effective annual rate, then re=er1
and re will be lognormal. Because er cannot
be negative, the smallest possible value for
re is –100%.
x
~ ~ N ( µ , σ 2 ) ⇒ ~ = e ~ is lognormal.
Note : x
z
~ ) = e µ + 1 σ 2 and Var ( ~ ) = e 2 µ +σ 2 ( eσ 2 − 1)
2
Its E ( z
z Hsiulang Chen Portfolio Analysis 3 Further Discussion on Lognormal
Suppose a stock’s price at the beginning of the year is
P0=$10. With continuous compounding, if r turns out to be
.23, then the endofyear price will be P1=P0 (1+re)= P0e.23
=$12.59 representing an observed effective annual rate of
return of re=(P1P0)/P0=er1=.259. That is why BlackScholes Option Pricing Model assumes that the price of
the asset underlying the option is lognormally distributed.
Recall: ln(P̃1/P0 )=ln(1+r̃e)=ln(er̃)=r̃ is normally distributed!
If a stock’s continuously compounded return is
normally distributed, then the future stock price is
necessary lognormally distributed.
Hsiulang Chen Portfolio Analysis 4 Why are continuously compounded
rates extensively used in Finance?
Since information flows into capital markets
continuously, rates of return are best modeled
as evolving and compounding continuously.
Pricing models for derivative assets often are
based on continuously rebalanced portfolios.
The need to model stock prices as trading
continuously calls for continuously
compounded rates of return.
Hsiulang Chen Portfolio Analysis 5 Are assets returns normally
distributed? Call re (t) the effective rate over an investment
period of length t. With rt normally distributed, the
effective holding return over short time period t (a
fraction of a year) may be taken as approximately
normally distributed.
re(t) = ert1 ≈ rt as t─>0 Suppose continuously compounded annual rate
of return (r) is normally distributed with µ and σ2.
The monthly continuously compounded return on
the stock has:
µ(monthly)= µ(annual)/12;
σ2(monthly )= σ2(annual)/12
Hsiulang Chen Portfolio Analysis 6 Will the distribution of returns of a large
portfolio resemble a normal distribution? NORMINV in Excel
Hsiulang Chen Portfolio Analysis 7 Measuring Mean Returns Arithmetic mean T rA = Geometric mean ∑r
t =1 t T T rG = [∏ (1 + rt )] 1/ T −1 t =1 When returns are normally distributed, rG = rA0.5σ2
Hsiulang Chen Portfolio Analysis 8 When do we use rG and rA? Consider a stock that will either double in
value with probability of 0.5, or halve in
value with probability 0.5. t2 t1
+100% What is rG?
Hsiulang Chen Now
50% What is rA?
Portfolio Analysis 9 When do we use rG and rA? Which one describes the average past
performance per year? Why? Which one approximates the expected
return per year in the future? Why? Hsiulang Chen Portfolio Analysis 10 Measuring Risk An ideal risk measure can be used to describe
the likelihood and magnitudes of “surprises”
(deviations from the mean).
A typical measure is the standard deviation
(SD) of returns, the square root of the variance.
When the returns are normally distributed, the
probability of having a return that is within one,
two, three SD of the mean is approximately
0.68, 0.95, and 0.997, respectively. Hsiulang Chen Portfolio Analysis 11 Asymmetric Return Distribution Consider the two probability distributions
for returns of a portfolio. [BKM, p. 132 (8th)
or 136 (9th)] [Quant_Review.pdf]
A Pr(r) Pr(r)
B small losses more likely;
extreme gains less likely E(rA) small gains more likely;
extreme losses less likely E(rB) By construction, both have identical expected values and variances.
Which one is more likely preferred by a riskaverse investor?
Hsiulang Chen Portfolio Analysis 12 The asymmetry of the distribution is called skewness, which
we measure by the third central moment, given by
n
M3 = ∑ Pr( s )[ r ( s ) − E ( r )] 3 s =1 It gives greater weight to larger deviations, so the “long
tail” of the distribution dominates the measure. Thus the
skewness of the distribution will be positive for a
distribution skewed to the right such as A and negative for
a distribution skewed to the left such as B. The σ overestimates risk when the distribution is
positively skewed while the σ underestimates risk when
the distribution is negatively skewed. Why? The use of options or nonlinear trading rules in the
portfolio may cause portfolio returns be nonnormally
distributed because such strategies reduce variance
asymmetrically.
Hsiulang Chen
Portfolio Analysis 13 Figure 5.5A Normal and Skewed
Distributions 514 FatTail Risk
n M4 = ∑ Pr( s )[r ( s ) − E ( r )]4 s =1 Kurtosis A measure of the
thickness of the tails. High frequency of
extreme returns
leads to fat tails! SD underestimates
the likelihood of
extreme events. Portfolio Analysis
Hsiulang Chen 15 Other risk measures: Tracking error = σ(∆r) ≡ σ(rrB) where rB is the
benchmark return. It measures how closely the
investment tracks the benchmark. Probability of shortfall = Probability (r<r*). It gives the
probability that undesirable event might occur but gives
no hint as to how sever it might be. Expected shortfall = E[rr*] over the range where rr*<0.
It represents the magnitude of shortfall times the
probability of occurring. A major problem is that it treats
a large probability of a small shortfall as equivalent to a
small probability of a large shortfall. However, the last two measures about downside risk
cannot be built up so easily from individual securities to
portfolios. Q: Using usdata.xls to construct Slide 18 & 20.
Hsiulang Chen Portfolio Analysis 16 Usdata.xls (19262009)
Rates of Return 19262009
World Portfolios World Equity
Return in US
Dollars Year US Markets World Bond
Return in US
Dollars Small
Stocks Large
Stock LongTerm
TBonds TBills Real Tbill
Inflation Rates Diversified 1926 25.24 8.10 8.91 12.21 4.54 3.19 1.12 4.36 12.20 1927 23.15 9.62 35.48 35.99 8.11 3.13 2.26 5.51 21.10 1928 28.62 2.44 51.31 39.29 0.93 3.54 1.16 4.76 24.29 1929 12.56 3.45 48.35 7.66 4.41 4.74 0.59 4.13 14.63 1930 22.6 6.04 48.36 25.90 6.22 2.43 6.40 9.43 19.11 1931 39.94 12.32 53.17 45.56 5.31 1.09 9.32 11.48 32.20 1932 1.46 18.26 8.58 9.14 11.89 0.95 10.27 12.50 6.01 1933 70.81 29.26 153.18 54.56 1.03 0.30 0.76 0.46 66.35 1934 0.15 3.87 34.78 2.32 10.15 0.18 1.52 1.32 10.08 1935 22.44 1.41 72.87 45.67 4.98 0.14 2.99 2.77 27.29 1936 18.84 0.49 77.01 33.55 6.52 0.18 1.45 1.25 26.78 1937 17.7 0.96 55.05 36.03 0.43 0.29 2.86 2.50 19.73 1938 6.21 0.65 15.46 29.42 5.25 0.04 2.78 2.82 7.77 1939 5.6 5.11 6.79 1.06 5.90 0.01 0.00 0.01 2.39 1940 7.97 11.32 14.15 9.65 6.54 0.02 0.71 0.72 3.12 1941 13.26 5.61 13.42 11.20 0.99 0.04 9.93 9.00 4.24 1942 0.56 3.69 39.66 20.80 5.39 0.28 9.03 8.03 9.27 Hsiulang Chen Portfolio Analysis 17 Figure 5.6 Histograms of Rates of
Return for 19262009 Hsiulang Chen Portfolio Analysis 18 Inferred Normal Distribution Hsiulang Chen Portfolio Analysis 19 Wealth Indexes of Investment Hsiulang Chen Portfolio Analysis 20 Investment horizon: Is it relevant? It is commonly assumed that investors
with longer horizons should allocate a
larger fraction of their savings to risky
assets than investors with shorter
horizons. [Chapter 7.5, BKM] Hsiulang Chen Portfolio Analysis 21 Consider a oneyear investment versus a fiveyear investment. Suppose that a risky asset yield an annual rate
of return with E(rP)=15% & σP=30%, and returns
are independent over time. What is the return standard deviation of the fiveyear investment if your primary concern is on
the averaged annual returns? What is the return standard deviation of the fiveyear investment if your primary concern is on
the total returns?
Hsiulang Chen Portfolio Analysis 22 Hsiulang Chen Portfolio Analysis 23 The Relevant Metrics Paul Samuelson argues that the notion of
time diversification is specious, because
the relevant metrics is terminal wealth, not
annualized return. As the investment horizon increases, the
dispersion of terminal wealth diverges from
the expected terminal wealth. Thus, the
total return becomes more uncertain the
longer the investment horizon.
Hsiulang Chen Portfolio Analysis 24 Hsiulang Chen Portfolio Analysis 25 Consider a risky investment of $100 that
has a 50 percent chance of a onethird
gain and a 50 percent chance of a onefifth
loss under two cases.
Case1: Returns are independent from one
period to the next.
Case 2: The risky asset has a 60 percent
chance of reversing direction and only a 40
percent chance of repeating its prior return.
Hsiulang Chen Portfolio Analysis 26 Case 1: Random Returns
Initial
Wealth Distribution of Wealth after
One Period Two Periods Three Periods
1/8 x 237.04
¼ x177.78 1/8 x 142.22
½ x133.33
1/8 x 142.22
¼ x106.67 1/8 x 85.33
100
1/8 x 142.22
¼ x106.67 1/8 x 85.33
½ x 80.00
1/8 x 85.33
¼ x 64.00 1/8 x 51.20
Expected Wealth?
Expected Utility if U= 1/W?
Hsiulang Chen Portfolio Analysis 27 Case 2: MeanReverting Returns
Initial
Wealth Distribution of Wealth after
One Period Two Periods Three Periods
0.08x237.04
0.2x177.78 0.12x142.22
½ x133.33
0.18x142.22
0.3x106.67 0.12x 85.33
100
0.12x142.22
0.3x106.67 0.18x 85.33
½ x 80.00
0.12x 85.33
0.2x 64.00 0.08x 51.20
Expected Wealth?
Expected Utility if U= 1/W?
Hsiulang Chen Portfolio Analysis 28 Conclusion of Time Diversification
If investors are more riskaverse than log
wealth investors, time diversification might
work when investment returns meanrevert rather than follow a random walk.
Questions: Will the implication of time
diversification be changed if investors’
degree of risk aversion changes, the
gain/loss ratio changes, or the probability
of meanreverting changes?
Hsiulang Chen Portfolio Analysis 29 ...
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 Fall '11
 HengChen

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