Class Business

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Probability Models

Suppose price to play = $0.85 We can draw a model of net returns:

Two-state probability model– Two states– Two returns– Two probabilities

Expected Return

The expected return to playing this game once is

In general, the expected return of any two-state probability model is

– p1 and p2 are the probabilities of the two states

– r1 and r2 are the returns received in the two states

1 1[ ] 18% ( 6%) 6%

2 2E r

1 1 2 2[ ]E r p r p r

Expected Return: Example

Coin flipping game:– Cost: $1– If heads: $2– If tails: $1– Probability of heads: 0.75

What is expected return from playing?

[ ] 0.75(100%) .25(0%) 75%E r

Expected Return Suppose

– We don’t know the true probability model– But we can observe past data from the game

100% 100% 0% 100% 0% 0% 100%

• Then we could estimate the expected return • Find simple average: add-up all values and divide by the number of values you observe

• With many observations, this would be very close to expected return derived from the probability model

Probability Models

Realistic probability models are very complex and involve an infinite # of possible outcomes.

– Example: the normal distribution To get an estimate of the expected return, it is

usually easiest to just estimate simple mean from past data if available.

Simple probability models with only two possible outcomes, though unrealistic, help us understand finance theory.

Uncertainty

Game 1:– 10% return with 50% probability– 20% return with 50% probability

Game 2:– 0% return with 50% probability– 30% return with 50% probability

Which game do you prefer?

Uncertainty We need a measure of uncertainty. Both games have expected return of 15%. How about expected deviation from mean? Game 1 Deviations from mean:

– 10%-15%=-5% with 50% probability– 20%-15%=5% with 50% probability– Expected deviation from mean is zero.

Uncertainty

Game 2 Deviations from mean:– 0%-15%=-15% with 50% probability– 30%-15%=15% with 50% probability– Expected deviation from mean is zero.

The expected deviation from mean will always be zero for any probability model.

Need a more helpful measure

Uncertainty

How about expected squared deviation from mean?

Game 1 squared deviations– (-5%)2=0.0025– (5%)2= 0.0025– Expected squared deviation from mean is

0.0025.

Uncertainty Game 2 squared deviations

– (-15%)2=0.0225– (15%)2= 0.0225– Expected squared deviation from mean is 0.0225.

Expected squared deviations:– Game 1: 0.0025– Game 2: 0.0225

Uncertainty

VARIANCE: – Expected squared deviation from mean

STANDARD DEVIATION: – Square-root of the variance

Uncertainty: Example Coin flipping game:

– Cost: $1– If heads: $2– If tails: $1– Probability of heads: 0.75

What is variance of this game?

What is the standard deviation?

2 2[ ] 0.75(100% 75%) .25(0% 75%)

0.75(.0625) .25(0.5625) .1875

Var r

[ ] .1875 0.43Stdev r

Uncertainty Suppose

– We don’t know the true probability model– But we can observe past data from the game

The we could estimate the variance by– Estimating expected return (simple average)– Finding squared deviation for each outcome– Take simple average of squared deviations

We could estimate the standard deviation as– Square-root of estimated variance

Uncertainty Example: Suppose for coin flipping game we observe the following

outcomes:– 100%, 0%, 100%, 0%

Estimated expected return: 50% Deviations:

– 50%, -50%, 50%, -50% Squared Deviations:

– 0.25, 0.25, 0.25, 0.25 Estimated Variance: 0.25

– Std. Deviation: .50

From True probability model:– Expected return=75%– Variance = 0.1875

• Std. Deviation: .4330

Variance

We often use 2 to represent variance to represent standard deviation

Later in the course we will look at how risk is measured for portfolios that will include covariation as well as standard deviation

What does Standard Deviation Tell Us?

Helps us measure likelihood of extreme outcomes.

Prob(return < 1 standard deviation from mean) = 16%

Probability of Extreme Bad Events

Example: Your portfolio has an expected return of 10% with a standard deviation of 0.16 over the next year.

What is probability that realized return is <-22%?

Probability of Extreme Bad Events

1. How many standard deviations is outcome from mean?

.22 .102

.16

xz

.10-.22

2 Standard Deviations (.16)

Probability of Extreme Bad Events

2. Use excel function normsdist(z)This function gives probability of getting z

standard deviations from mean or less.

normsdist(-2) = 0.02275 = 2.275%

Data vs. Probability Model

Note that probability models are forward looking. They tell us about what we should expect in the future.

Estimates of means and variances from historical data are backward looking. They tell us about what happened in the past.

The hope is that the past will be indicative of the future.

The Historical RecordArith. Stan.

Series Mean% Dev.%

Lg. Stk 12.49 20.30

Sm. Stk 18.29 39.28

LT Gov 5.53 8.18

T-Bills 3.85 3.25

Inflation 3.15 4.40

Real Rates of Return Suppose at the beginning of the year, the cost of a pizza is $10.00.

You have $100 in cash. You could buy 10 pizzas, but instead, you invest the $100 in a long term gov. bond. The return on the bond is 5%. Inflation over the year is 3%.

The investment provides you a nominal income at year end of 100(1.05) = $105.

At year end, the cost of a pizza is 10.00(1.03)=$10.30. At year end, you could buy 10.19 pizzas (105/10.3)=10.19. Your real return is therefore only ____?%

Real Rates of Return

C = amount of cash at beginning of period P = price of a good at beginning of period rn = nominal rate of return, rr = real return i = inflation rate The real (gross) rate of return was found above by

solving the following equation

Since ir

PCiPrC

r nr

11

/)1(/)1(

1

irri nr with eapproximatcan wesmall is Because