Fin415 Week 2 Slides

FIN415 Week 2

Risk Mathematics

A Random WalkEugene Fama 1965Fama’s Website

http://faculty.chicagogsb.edu/eugene.fama/research/index.htm

Efficient Market HypothesisThe Behavior of Stock-Market Prices (January 1965)

Random WalkRandom Walks in Stock Market Prices (September-October

1965)

Picture of a Random Walk

What are the Odds?Gambling in France during the 17th Century

Chevalier de MéréBlaise PascalPierre de Fermat

Pascal’s Trianglehttp://en.wikipedia.org/wiki/Pascal's_triangle

Interpreting Pascal’s Triangle The triangle is a “binary system.”

Each row represents 2 to a corresponding power. The first row is 2 to the 0 power. The second row is 21 and the

third row in 22. This represents the total number of possible outcomes to a 50-50 bet like tossing a coin.

If you toss the coin once, the second row tells you that you have a 1 in 2 chance of heads and a 1 in 2 chance of tails.

If you toss the coin 6 times, look at row 7. The numbers in row 7 total 64 i.e. 26, so there are 64 possible results from tossing a coin 6 times. The probability of getting 6 heads or 6 tails is 1/64. The probability of getting 5 heads or 5 tails is 6/64. The probability of getting 4 heads or 4 tails is 15/64. The probability of getting 3 heads and 3 tails is 20/64.

Pascal’s Triangle Fractions and Percentages

Six Flips of a Coin - Row 7 (26)

Fraction Percentage

1/64 0.015625

6/64 0.093750

15/64 0.234375

20/64 0.312500

15/64 0.234375

6/64 0.093750

1/62 0.015625

Total 64/64 Total 1.000000

DiceIndependent EventSix possible outcomesPossible Outcomes = 6 raised to the power of

number of throws.Possible outcomes from 10 throws = 610

Playing CardsCan be an Independent Event, if card is replaced

and shuffled between draws.52 possible outcomes (no wild cards)Possible Outcomes = 52 raised to the power of

number of draws.Possible outcomes from 10 draws = 5210

Playing Cards - DependentHow do things change what you do not replace

the card?1/(52X51X50X49) = Odds of picking 4 specific

cards. Calculate the odds of drawing 4 Aces in a row

from a well shuffled deck?

and/orIf two or more events must happens (A and

B and C) then multiply the probability of each event to calculate the probability of all three events.

If any of two or more events must happen (A or B or C) then add the probability of each event to calculate the probability of any one of the three events happening.

These rules require independent events.

Information and ProbabilitiesOften additional information can change the

probabilities. If additional information is received it must be

taken into account.If you were playing Black Jack and learned that

the deck had not been shuffled, how would that change your calculation?

Risk NeutralPotential Return X Probability = Expected ReturnWhat is the value of a lottery ticket with a $23

million jackpot and a 1/30 million probability of winning the lottery?

TimeTime is not free.In finance, we always have to consider the

movement through time. Apples must be compared to apples and oranges

to oranges. In finance apples turn into oranges over time.

Discrete Time vs. Continuous TimeDiscrete – Time is broken up into chunks. (Years,

months, days, hours, minutes seconds, plank time)

Continuous – Time flow continuously. It is unbroken.

Which do you believe?

Present Value (PV) - Discrete

The PV equation for discrete time is: 1/(1+r)t

Present Value (PV) - Continuous

The PV equation for continuous time is: 1/ert

What Do Buyers and Sellers Believe

If you are the potential buyer of a cash flow producing asset, are you more likely to believe in discrete time or continuous time?

What is you are the seller of such an asset?

NPVNet Present Value: This equation converts

several future cash flows to one present value amount. (Sums up all of the PV for each time t.)

Mean

Mean is the average.Add up all of the relevant numbers and divide by

how many numbers you added up. 3+8+7=18 18/3=6 Mean=6

Median

The median in the number in the center of numbers. Like the mean it is a measure of centrality, but not as susceptible to extremes.

2,4,5,7,20 Median =5

Mode

Mode is another measure of centrality. It is the most common number in a distribution.

2,3,5,2,7,3,4,2,34 Mode=2

Normal Curve

Also Bell Curve or Gaussian Curve.

Standard Deviation

Describes how tightly a distribution is dispersed about the mean.

Assuming a normal distribution, approximately 68% will be dispersed within 1 SD.

Approximately 95% will be within 2 SD.

Calculating SDThis example comes from Wikipedia.

http://en.wikipedia.org/wiki/Standard_deviation

Suppose we wished to find the standard deviation of the set of the numbers 3, 7, 7, and 19.

Step 1: find the arithmetic mean (or average) of 3, 7, 7, and 19, (3 + 7 + 7 + 19) / 4 = 9.

Step 2: find the deviation of each number from the mean,

3 − 9 = − 6; 7 − 9 = − 2; 7 − 9 = − 2; 19 − 9 = 10.

Calculating SD continuedStep 3: square each of the deviations (amplifying larger deviations

and making negative values positive), ( − 6)2 = 36

( − 2)2 = 4( − 2)2 = 4102 = 100.

Step 4: sum the obtained squares (as a first step to obtaining an average),36 + 4 + 4 + 100 = 144.

Step 5: divide the sum by the number of values, which here is 4 (giving an average), 144 / 4 = 36.

Step 6: take the non-negative square root of the quotient (converting squared units back to regular units),

So, the standard deviation of the set is 6. This example also shows that in general the standard deviation is different from the average deviation (which is 5 in this example).

Standard Deviation = RiskIt is argued that SD is a measure of risk.The more returns tend to vary from the mean, the

greater the risk associated with a particular asset. This is also called volatility. A US Treasury Bond is risk free, because its return

does not vary from the mean. If you buy a 10 year Treasury Bond at 4% you know precisely what the return will be.

Because the US has never defaulted on a bond and because the government can always print more money, there is no default risk.

SD=0

Standard Deviation = Risk

Corporate bonds and sovereign bonds issued by less financial stable nations have SDs higher than zero.

The interest rate may be fixed, but there is default risk.

Standard Deviation = Risk

Shares of stock are risky assets because their returns constantly change.

Shareholders are the residual claimants – they own the leftovers.

Profits = money left after everyone is paid.

SamplingWhen we cannot measure everything, we have to

measure a sample and assume that it looks like the whole.

What can we measure, which will look like the market as a whole?

Is more better, or is less better?What periods should we measure?How does distance in time effect a sample?

CAPM

Capital Asset Pricing Model ERA=RF + βA (RM - RF)Discount Rate=Risk Free Rate + Beta(Average

Market Rate-Risk Free Rate)

β=Relative RiskBeta is the relative risk of a particular asset as

compared to the average of all risky assets in the market.

“p” stands for portfolio. In this case we are talking about the “market portfolio.

CAPM and NPVCAPM will give you the discount rate you will use

with your NPV calculation to determine the present value of an asset based on its expected cash flows.

The higher the discount rate, the lower the value of the asset will be.

This is the inverse relationship between Risk and Value.

The “implied assumption”Calculating risk as the SD of historical data

assumes that the past is a statistical “sample” of the future.

Is their good reason to assume that the future will look like the PAST?