Mathematics in Finance

Brandon GroegerApril 6, 2010

Outline

I. Stocksa. What is a stock?b. Returnc. Riskd. Risk vs. Returne. Valuing a Stock

II. Bondsa. What is a bond?b. Pricing a bond

III. Financial Derivativesa. What are derivatives?b. How are derivatives valued?

IV. Discussion

What is a stock?

Stock represents a share of ownership in a company.

Companies issue stock to raise money to run their business.

Investors buy stock with expectation of future income from the stock. This is why stock has value.

Stocks can be publicly or privately traded.

Primerica Initial Public Offering (IPO)

Primerica, a financial services company had their IPO this past Thursday, April 1st.

The company sold 21.4 million shares for $15 a piece. The market price closed at around $19 per share.

The market price of a stock is determined like the price at an auction: it is a compromise between the buyers and the sellers.

Measuring Stock Growth

Stock returns can be measured daily, weekly, monthly, quarterly, or yearly.

The geometric mean is used to calculate average return over a period.

The arithmetic mean is used to calculate the expected value of return given past data.

nn

ii

nn

ii rr

/1

1

/1

1

)(1)1(

n

iirn 1

1

Average Return =

Monthly Returns for GoogleDate Open Mnth Rtrn Index Total Rtrn4/1/2009 343.78 14.91% 1.15 14.91%5/1/2009 395.03 6.00% 1.21802 21.80%6/1/2009 418.73 1.31% 1.23393 23.39%7/1/2009 424.2 5.79% 1.30531 30.53%8/3/2009 448.74 2.44% 1.33713 33.71%9/1/2009 459.68 7.25% 1.43406 43.41%10/1/2009 493 8.94% 1.56228 56.23%11/2/2009 537.08 9.51% 1.71077 71.08%12/1/2009 588.13 6.60% 1.8237 82.37%1/4/2010 626.95 -14.73% 1.55506 55.51%2/1/2010 534.6 -1.01% 1.53936 53.94%3/1/2010 529.2 7.96% 1.66196 66.20%4/1/2010 571.35

Average Monthly Return 4.32%Expected Value of Monthly Return 4.58%

Exercise 1

1a. Geometric Mean = [(1+1)(1+-.5)(1+-.5)(1+1)]^(1/4) - 1 = 1 – 1 = 0

1b. Arithmetic mean = [1 + -.5 + -.5 +1] / 4 = 1/4 = 25%

1c. $100 * (1 + 0)^4 = $100 = true value

1d. $100 * (1 + .25)^4 = $244.14

Converting Rates Between Periods

APR or annual percentage rate = periodic rate * number of periods in a year

APY or annual percentage yield = (1 + periodic rate) ^ number of periods in a year - 1

APR ≠ APY Example: 1% monthly rate has a 12%

APR and a 12.68% APY

Exercise 2

2a. APR = 4% * 12 = 48% APY = (1.04)^12 - 1 = 60.1%

2b. 18%/12 = 1.5% monthly (1.015)^12 - 1 = 19.56%

Risk

Risk is the probability of unfavorable conditions.

All investments have risk. There are many types of risk.

Specific risk Risk associated with a certain stock

Market risk Risk associated with the market as a whole

Measuring Risk Standard Deviation (σ) of

returns can be used to measure volatility which is risky.

Standard Deviation 10.78%

Arithmetic Mean 2.49%

Geometric Mean 1.93%

Standard Deviation

Assume a normal distribution. 68% confidence interval for monthly return: Mean ± σ =

(-8.29%, 13.27%) 95% confidence interval: Mean ± 2σ = (-19.07%, 24.05)

Risk vs. Return

Investors seeks to maximize return while minimizing risk.

Sharpe Ratio = (return – risk free rate) / standard deviation.

Can be computed for an individual asset or a portfolio.

The higher the Sharpe ratio the better.

Capital Asset Pricing Model (CAPM)

E(R) = Rf + β(Rm-Rf) E(R) = expected return for an asset Rf = risk free rate β = the sensitivity of an asset to

change in the market. It is a measure of risk

Rm = the expected market return Rm - Rf = the market risk premium

Calculating Beta

β = Cov(Rm,R) / (σR)2 = Cor(Rm,R) * σRm / σR

σR is the standard deviation of the asset’s returns

σRm is the standard deviation of the market’s returns

Cov(Rm,R) = covariance of Rm and R Cor(Rm,R) = correlation of Rm and R

In practice beta can also be calculated through linear regression.

Implications of CAPM

Higher beta stocks have higher risks which means that the market should demand higher returns.

Assumes that the market is efficient or equivalently perfectly competitive.

Portfolio Diversification

Two assets are almost always correlated due to market risk.

Equivalently, 2

p = W121

2 + W222

2 + 2W1W212 where

12 = 12*1*2

This statistical result implies that the variance for a two or more assets is not equal to the sum of there variances which implies the risk is not equal to the sum each assets risk.

In general two assets have less risk than one asset.

Diversification example

Suppose stock A has average returns of 5% with a standard deviation of 6% and stock B has average returns of 8% with a standard deviation of 10%. The correlation between stock A and B is .25. What is the expected return and standard deviation (risk) of a portfolio with 50% stock A and 50% stock B.

E(Rp) = .5 * .05 + .5 * .08 = 6.5% V(Rp) = .52 * .062 +.52 * .12 + 2 * .5 * .5 *.25 * .1 *

.06 = .00415 σRp = V(Rp) ^.5 = 6.44%

Diversification Graphs

Methods of Valuation for stock

Discounted Cash Flow Analysis uses the assumptions of time value of money and the expected earnings of a company over time to compute a value for the company.

Relative Valuation bases the value of one company on the value of other similar companies.

For a publicly traded company the market value of the company is the market stock price multiplied by the number or shares outstanding.

Relative Valuation Example

Royal Dutch Shell and Chevron Corporation are comparable companies.

Royal Dutch Shell is trading at $59.26 per share and has a P/E ratio of 14.52

Chevron Corporation has earnings per share of $5.24

What would you expect for the price of a share of Chevron Corporation?

P/E * EPS = 14.52 * $5.24 = $76.08 Chevron is actually trading around $77.55

What is a bond?

A Bond is “a debt investment in which an investor loans money to an entity (corporate or governmental) that borrows the funds for a defined period of time at a fixed interest rate.” (Investopedia)

Bonds can be categorized into coupon paying bonds and zero-coupon bonds.

Example: $1000, 10 year treasury note with 5% interest pays a $25 coupon twice a year.

Bond Terminology

Principal/Face Value/Nominal Value: The amount of money which the bond issuer pays interest on. Also the amount of money repaid to the bond holder at maturity.

Maturity: the date on which the bond issuer must repair the bond principal.

Settlement: the date a bond is bought or sold Coupon Rate/Interest Rate: the rate used to

determine the coupon Yield: the total annual rate of return on a bond,

calculated using the purchase price and the coupon amount.

Bond Pricing

Based on time value of money concept.

C = coupon payment n = number of payments i = interest rate, or required yield M = value at maturity, or par value 

Exercise 3

Price a 5 year bond with $100 face value, a semiannual coupon of 10% and a yield of 8%.

Price = 5 * [1 - [1 / (1+.08/2)^10]] / (.08 / 2) + 100 / (1 + .08/2)^10 = $108.11

Other Considerations for Bond Pricing The previous examples have assumed that the bond

is being priced on the issue date of the bond or a coupon pay date, but a bond may be bought or sold at any time.

Pricing a bond between coupon periods requires the following formula where v = the number of days between settlement date and next coupon date.

Some bonds use a convention of 30 days per month, other bonds use the actually number of days per month.

11

1 )1()1()1()1(

nv

n

itv rr

M

rr

CP

Types of Bonds that Affect Valuation

Municipal bonds pay coupons that are often exempt from state or local taxes. This makes them more valuable to residents but not to others.

Some bonds have floating interest rates that makes them impossible to accurately price.

Some bonds have call options which allow the bond issuer to buy back the bond before maturity.

Some bonds have put options which allow the bond holder to demand an early redemption.

What is a Financial Derivative?

A derivative is a financial instrument that derives it value from other financial instruments, events or conditions. (Wikipedia)

Derivatives are used to manipulate risk and return. Often to hedge, that is to say generate return regardless of market conditions.

Derivatives are bought and sold like any other financial asset, relying on market conditions to determine pricing.

Examples of Derivatives

Options A call gives the buyer the right to buy an asset at a certain

price in the future. A put gives the buyer the right to sell an asset at a certain

price in the future. Future

A contract between two parties to buy and sell a commodity at a certain price at a certain time in the future.

Swaps Two parties agree to exchange cash flows on their assets.

Collateralized Debt Obligations (CDO’s) Asset backed fixed income securities (often mortgages) are

bundled together and then split into “tranches” based on risk.

Valuing Derivatives

Valuing derivatives can be difficult because of the many factors that effect value and because of high levels of future uncertainty.

Stochastic Calculus is used frequently. For very complex derivatives Monte Carlo

simulation can be used to determine an approximate value.

Many people are critical of complex derivatives because they are so difficult to value.

Discussion

What else would you like to know about finance or the mathematics of finance?

Is it possible to find a “formula” for the stock market?

What regulations should financial markets have? Why?