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Mathematics in Finance

Brandon Groeger April 6, 2010. Mathematics in Finance. Stocks What is a stock? Return Risk Risk vs. Return Valuing a Stock Bonds What is a bond? Pricing a bond Financial Derivatives What are derivatives? How are derivatives valued? Discussion. Outline.

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Mathematics in Finance

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  1. Brandon Groeger April 6, 2010 Mathematics in Finance

  2. Stocks • What is a stock? • Return • Risk • Risk vs. Return • Valuing a Stock • Bonds • What is a bond? • Pricing a bond • Financial Derivatives • What are derivatives? • How are derivatives valued? • Discussion Outline

  3. 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. What is a stock?

  4. 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. Primerica Initial Public Offering (IPO)

  5. 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. Measuring Stock Growth Average Return =

  6. Monthly Returns for Google

  7. 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 Exercise 1

  8. 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

  9. 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%

  10. 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

  11. 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%

  12. 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) Standard Deviation

  13. 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.

  14. 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

  15. 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.

  16. 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.

  17. Two assets are almost always correlated due to market risk. Equivalently, 2p = W1212 + W2222 + 2W1W212 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. Portfolio Diversification

  18. 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 example

  19. Diversification Graphs

  20. 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. Methods of Valuation for stock

  21. 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

  22. 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. What is a bond?

  23. 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 Terminology

  24. 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  Bond Pricing

  25. 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 Exercise 3

  26. 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. Other Considerations for Bond Pricing

  27. 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. Types of Bonds that Affect Valuation

  28. 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. What is a Financial Derivative?

  29. 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. Examples of Derivatives

  30. 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. Valuing Derivatives

  31. 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? Discussion

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