UW AFM 472

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Probability distribution  Return distribution  Terminologies AFM472 INVESTMENTS Characterizing Returns with Distributions BKMPR chapter 4 1/20

Transcript of 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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Probability distribution   Return distribution   Terminologies

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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Probability distribution   Return distribution   Terminologies

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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Probability distribution Return distribution Terminologies

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Skewness and Kurtosis

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Sample statistics(arithmetic) mean:

µ  =  1

∑i =1

r i 

variance:

σ 2 =

  1

∑i =1

(r i −µ )2

skewness (lack of symmetry)

skew  =1N ∑

i =1(r i −µ )3

σ 3

kurtosis (peakedness)

kurt  =1N ∑

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

Probability distribution Return distribution Terminologies

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Example

Month Return (%)1 -52 53 154 105 0

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Probability distribution Return distribution Terminologies

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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∏

i =1(1 + r i )

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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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Preparation for the Next Class

BMKPR ch. 5

What are the building blocks for optimal portfolio selection?

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