Risk and ReturnPrimer

Expectations

Expected value (μ) is weighted sum of possible outcomes

E(X) = μ = p1X1 + p2X2 + …. psXs E(X) – Expected value of XXi – Outcome of X in state ipi – Probability of state is – Number of possible statesProbabilities have to sum to 1

p1 + p2 + …..+ ps = 1

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Horse Race

There are three horse racing in the Finance Derby. Your horse is “Love of NPV”. If your horse has a 30% chance of coming in first, and a 40% chance of coming in second. How much do you expect your horse to win? 1st pays $1,500 2nd pays $750 3rd pays $250

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Horse Race

There are three horse racing in the Finance Derby. Your horse is “Love of NPV”. If your horse has a 30% chance of coming in first, and a 40% chance of coming in second. How much do you expect your horse to win? 1st pays $1,500, 2nd pays $750, 3rd pays $250

Chance of coming in 3rd: 1-0.3-0.4 = 0.3 0.3*1,500 + 0.4*750 + 0.3*250 = $825

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What is risk?

Uncertainty

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Measuring Risk There is no universally agreed-upon

measureHowever, variance and standard deviation are both

widely accepted measures of total risk

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Statistics Review: Variance Variance (σ2) measures the dispersion of

possible outcomes around μ Standard deviation (σ) is the square root of

variance Higher variance (std dev), implies a higher

dispersion of possible outcomesMore uncertainty

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Different Variances

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Variance Calculation Variance = σ2 = Σpi * (Xi – μ)2: Use this one

Alternative formulas you may have seen σ2 = Σ(Xi – μ)2 / N σ2 = Σ(Xi – μ)2 / (N-1)

All give similar answers with large samplesBUT each give very different answers with small

samples

Ex. s=3σ2 = p1 * (X1 – μ)2 + p2 * (X2 – μ)2 + p3 * (X3 – μ)2

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Risk Example

Economy is “Good” with 20% probability DJIA will return 20%

Economy is “Fair” with 30% probability DJIA will return 5%

Economy is “Bad” with 50% probability DJIA will return -9%

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Calculations

Expected Return =

Variance =

Standard Deviation =

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Calculations

Expected Return = p1X1 + p2X2 + p3X3 = 0.2*0.20+0.3*0.05+0.5*(-0.09) = 0.01

Variance =

Standard Deviation =

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Calculations

Expected Return = 0.01

Variance = p1(X1- μX)2+p2(X2-μX)2+p3(X3-μX)2

=0.2*(0.20-0.01)2 + 0.3*(0.05-0.01)2 + 0.5*(-0.09-0.01)2

= 0.0127 =127 (%)2

Standard Deviation =

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Calculations

Expected Return = 0.01

Variance = 0.0127 =127 (%)2

Standard Deviation = √ σ2 √0.0127 = 0.113 = 11.3%

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Historical Data

In practice we do not know all of the possible states of the world, so we use historical data to form expectationsIdea: Look at what has happened in the past and

we can calculate the mean and variance What is each states probability of occurring?

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Risk Example 2

Sample Mean = 0.2*0.20+0.2*0.15+0.2*(-0.05)+0.2*0.05+0.2*0.10 = 0.09 = 9%

Sample Variance = = 0.2*(0.20-0.09)2 + 0.2*(0.15-0.09)2 + 0.2*(-0.05-0.09)2 + 0.2*(0.05-0.09)2 + 0.2*(0.10-0.09)2 = 74%2

Standard Deviation = √0.0074 = 0.086 = 8.6%

1996 1997 1998 1999 2000

20% 15% -5% 5% 10%

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Risk

A risky asset is one in which the rate of return is uncertain.

Risk is measured by ________________

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Risk

A risky asset is one in which the rate of return in uncertain.

Risk is measured by standard deviation. higher σ → more uncertainty

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General Securities

T-bills are a very safe investment No default risk, short maturity Risk free asset

Stocks are much riskier Bond’s riskiness is between T-bills and Stocks

Why Do We Demand a Higher Return Investors seem to dislike risk (ex. insurance)

Risk Averse If the expected return on T-Bills (risk-free), is

10%, and the expected return for Ford is 10%, which would you buy?The 10% offered by T-Bills is guaranteed while

this is not the case for FordA guaranteed 10% dominates a possible 10%

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Return Breakdown

A risky asset’s return has two components:Risk free rate + Risk premium

Risk free rate: The return one can earn from investing in T-Bills

Risk Premium: The return over and above the risk free rateCompensation for bearing risk

Average Risk Premiums (1926-2005)

Small company stocks : 17.4% – 3.8% = 13.6%

Large company stocks : 12.3% – 3.8% = 8.5%

Long-term corporate bonds : 6.2% – 3.8% = 2.4%

The more risk the larger the risk premium

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The Risk-Return Tradeoff

2%

4%

6%

8%

10%

12%

14%

16%

18%

0% 5% 10% 15% 20% 25% 30% 35%

Annual Return Standard Deviation

Ann

ual R

etur

n A

vera

ge

T-Bonds

T-Bills

Large-Company Stocks

Small-Company Stocks

Highest Risk & Return: Small Cap Stocks, Large Cap Stocks, L.T. Corp bonds, L.T. Gov Bonds, U.S. T-Bills

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Quick Quiz

Which of the investments discussed has had the highest average return and risk premium?

Which of the investments discussed has had the highest standard deviation?

Why we care?

This is the very basics of investing General knowledge that “finance” people

possess

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