Brandon Groeger April 6, 2010

I. Stocks a. What is a stock? b. Return c. Risk d. Risk vs. Return e. Valuing a Stock II. Bonds a. What is a bond? b. Pricing a bond III. Financial Derivatives a. What are derivatives? b. How are derivatives valued? IV. Discussion

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

 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. Average Return =

DateOpenMnth RtrnIndexTotal Rtrn 4/1/ % % 5/1/ % % 6/1/ % % 7/1/ % % 8/3/ % % 9/1/ % % 10/1/ % % 11/2/ % % 12/1/ % % 1/4/ % % 2/1/ % % 3/1/ % % 4/1/ Average Monthly Return4.32% Expected Value of Monthly Return4.58%

 1a. Geometric Mean = [(1+1)(1+-.5)(1+-.5)(1+1)]^(1/4) - 1 = 1 – 1 = 0  1b. Arithmetic mean = [ ] / 4 = 1/4 = 25%  1c. $100 * (1 + 0)^4 = $100 = true value  1d. $100 * (1 +.25)^4 = $244.14

 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

 2a.  APR = 4% * 12 = 48%  APY = (1.04)^ = 60.1%  2b.  18%/12 = 1.5% monthly  (1.015)^ = 19.56%

 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

 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%

 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)

 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.

 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

 β = Cov(Rm,R) / (σ R ) 2 = Cor(Rm,R) * σ Rm / σ R  σ R is the standard deviation of the asset’s returns  σ Rm 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.

 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.

 Two assets are almost always correlated due to market risk.   Equivalently,  2 p = W 1 2  W 2 2  W 1 W 2  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.

 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 * *.08 = 6.5%  V(Rp) =.5 2 * * *.5 *.5 *.25 *.1 *.06 =  σ Rp = V(Rp) ^.5 = 6.44%

 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.

 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  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 = * $5.24 = $76.08  Chevron is actually trading around $77.55

 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.

 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.

 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

 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) / (1 +.08/2)^10 = $108.11

 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.

 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.

 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.

 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 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.

 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?