UW AFM 472
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Probability distribution Return distribution Terminologies
AFM472 INVESTMENTS
Characterizing Returns with Distributions
BKMPR chapter 4
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Outline
Probability distribution
Return distribution
Terminologies
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Probability distribution Return distribution Terminologies
Random Event
Flipping a coin: head or tail?
Forecasting tomorrow’s return: down or up?
Mathematical tools for random event: outcomes and their
likelihoodprobability distribution
e.g. Binomial distribution
X = 0, with probability p ;
1, with probability 1−p .
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Probability distribution Return distribution Terminologies
Normal distribution
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Events that may not be normal
Market crashes and positive surprises:
“Black Monday”: October 19, 1987, TSX down 11%On January 31, 2001, the Nasdaq composite index gained
more than 14% in one daythe Crash of 1929 and the Great Depression October 29: over4 days markets down 62%...
What are the probabilities of these events if returns are
normally distributed?
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P b bili di ib i R di ib i T i l i
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Probability distribution Return distribution Terminologies
Annual returns (HPR): historical experience
µ (%) σ (%)Canada (1957–2009) 11 17US (1871–2006) from Robert Shiller 10.5 17.7
Using µ = 10%, σ = 15% and 252 trading days in a year, what’sthe probability of, say, a (worse than) 11% drop in a day?
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P b bilit di t ib ti R t di t ib ti T i l i
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What Normal Fails to Capture...
Large movements (both up and down) in stock prices canhardly be captured by the normal distribution.
Historical stock returns exhibit fat tails.
If we make financial decisions based on normal distribution, wewill miss out on the large movements.
There are more positive returns than negative returns
Negative returns tend to be larger but less frequent
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Skewness and Kurtosis
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Probability distribution Return distribution Terminologies
Sample statistics(arithmetic) mean:
µ = 1
N
N
∑i =1
r i
variance:
σ 2 =
1
N
N
∑i =1
(r i −µ )2
skewness (lack of symmetry)
skew =1N ∑
N
i =1(r i −µ )3
σ 3
kurtosis (peakedness)
kurt =1N ∑
N
i =1(r i −µ )4
σ 4
( Note: Some of you will point out that we should use 1N −1 instead of 1
N in the above
equations. This is a consistency issue in statistics which we don’t discuss.) 10/20
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Example
Month Return (%)1 -52 53 154 105 0
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Value-at-Risk: Another measure for “Tail” risk
Value-at-risk or VaR is another measure of riskFor a pre-selected probability level, what is the minimum lossthat we can expect?What is the value at risk?It highlights the potential loss from extreme, negative
movement of the underlying variable, such as “large” or“catastrophic” risks (for example, “100-year flood”)
Example: if a portfolio of stocks has a one-day 5% VaR of $1million, there is a 0.05 probability that the portfolio will fall invalue by more than $1 million over a one day period
VaR is the standard for large financial institutions and is usedboth as a risk measure and as the basis for calculating capitalreserves
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y g
VaR
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VaR
The VaR value can be used to illustrates departures fromnormality over the left tail
VaR assuming normality: L*standard deviation, where L is thestandard normal critical value. For 5% use 1.65. For 1% use2.33
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Exercise in Class
In the past century, historical stock return has a mean of about10% and standard deviation of about 15%. If returns are normallydistributed, what is the 5% VaR?
You plotted the return distribution, and found that the empiricaldistribution gives you a 5% VaR of −10%. Is this consistent withthe negative skewness and fat tail property of return distribution?
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Geometric mean return, and continuously compoundedreturn
Arithmetic mean may be misleading in interpreting averagereturns over multiple periods
Case A: 0 then 100% returnsCase B: 50% and 50%
Geometric mean:
µ GEO =
N
∏i =1
(1 + r i )
1N
−1
Continuous mean (return):
µ CON = ln∏
N
i =1(1 + r i )
N
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Nominal and real returnsWhat is your real return after removing the effect of inflation?
1 + r nominal,t = (1 + r real,t )(1+ i t )Example:
If during a period, nominal return is 5.06%, inflation is 2%,then real return = ?
What if during last month the return was 5.06%; but inflationis only reported quarterly, and for the last quarter inflationwas 2%, now what’s your real return for the past month?
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Historical Experience of Canada: 1957–2009
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