Management 408: Financial Markets Fall 2009 Professor Torous Final Review

Perpetuity A security that pays a fixed amount C per period forever starting next period. Present Value:

Growing Perpetuity A growing perpetuity makes a payment of C next period. After that the payment grows at a rate g. Present value:

Annuity An annuity pays a fixed cash flow C every period until some final Time T. Present Value:

Different Compounding Frequencies Interest payments can occur more than once per year. If the interest rate is quoted as an APR (annualized percentage rate), then the present value can be computed as m is the compounding frequency (number of interest payments per year) r is the interest rate (APR) t is the number of years In the limit with continuous compounding

Portfolio Mathematics with two assets The expected return of a portfolio is the portfolio weighted average of expected returns: The variance of a portfolio is: Converting from the correlation to the covariance:

Portfolio Mathematics Portfolio weights have to sum up to 1. Do not mix up standard deviation and variance. Note: standard deviation is in the same units as the random variable (e.g. return in %) Do not mix up covariance and correlation. The value of the correlation is between -1 and +1 Do not calculate the variance of a sum as the sum of the variances. Take the covariance term into account

Mean Variance Frontier The standard deviation is on the x-axis. The expected return is on the y-axis.

Mean Variance Frontier The portfolios on the mean variance frontier offer the lowest standard deviation for a certain expected return. The mean variance frontier of risky assets is a hyperbola in mean standard deviation space. If you add a risk free asset, it becomes a straight line connecting the risk free rate with the tangency portfolio. This line is called the Capital Market Line.

Minimum Variance Portfolio The minimum variance portfolio is the combination of risky assets that provides the lowest variance. In the two asset case (Assets A and B), the Minimum Variance Portfolio weights are given by

Tangency Portfolio The tangency portfolio is the combination of risky assets that has the highest Sharpe ratio (excess return over standard deviation).

CAPM The beta coefficient measures the amount of market risk the asset is exposed to. The CAPM equation says that the risk premium of any asset is proportional to the risk premium of the market portfolio. The beta of a portfolio is the weighted sum of the betas of single assets. Note:

Systematic vs. Idiosyncratic Risk The variance of asset returns can be decomposed as The first component represents systematic (i.e. market) risk The second component is idiosyncratic risk

Efficient Market Hypothesis Since we are more interested in how efficient is the capital market, we define the following 3 forms of market efficiency hypothesis: “A market is efficient if it reflects ALL available information” [1] Strong-form ALL available info [2] Semi-strong form [3] Weak-form All Available Information including inside or private information All Public Information Information in past stock prices

Efficient Market Hypothesis If the market is weak-form efficient: Technical analysis or charting becomes ineffective. You won’t be able to gain abnormal returns based on it. If the market is semi-strong-form efficient: No analysis will help you attain abnormal returns as long as the analysis is based on publicly available information. If the market is strong-form efficient: Any effort to seek out insider information to beat the market are ineffective because the price has already reflected the insider information. Under this form of the hypothesis, the professional investor truly has a zero market value because no form of search or processing of information will consistently produce abnormal returns.

Bond Pricing The cash flows from a bond consist of coupon payments until maturity plus the final payment of par value (received at maturity). Therefore, Bond value = Present value of coupons + Present value of par value If we call the maturity date T and call the discount rate r, the bond value can be written as With the annuity formula, this can be rewritten as

Yield to Maturity (YTM) The YTM is the discount rate that makes the present value of a bond’s payments equal to its price. This rate is often viewed as a measure of the average rate of return that will be earned on a bond if it is bought now and held until maturity. To calculate the yield to maturity, we solve the bond price equation for the interest rate (YTM) given the bond’s price.

Premium, Par, and Discount bonds Here YTM means the yield to maturity over the coupon payment period.

Bootstrapping Suppose we have prices of 3 different coupon bonds with maturities 1, 2, and 3 years. Let F1, F2, and F3 and C1, C2, and C3 denote the face value and the coupon of each bond. Then bond prices satisfy We have three equations and want to find the one, two and three-year yields Y(1), Y(2), and Y(3). Bootstrapping works in an iterative fashion: Find Y(1) from the first bond pricing equation. Use Y(1) in the second equation and solve for Y(2). Use Y(1) and Y(2) in the third equation to get Y(3).

Forward Rates One-period Forward Rate one period from now: In general,

MacAuley Duration A measure of the average effective maturity of a bond’s cash flows Given the MacAuley Duration you can calculate the %-change in the bond price for a given change in the yield to maturity (ΔYTM) or We call D* the modified duration. Do not confuse D with D*.

Duration and Different Compounding Frequencies Semiannual Compounding of YTM D* = D/(1+YTM/2) Monthly Compounding of YTM D* = D/(1+YTM/12) Continuous Compounding of YTM D* = D (i.e. no adjustment)

Duration - Properties For Zero Coupon Bonds: Duration = Maturity Duration of a bond portfolio is the portfolio weighted average of durations of the individual bonds. Coupon bond duration is less than maturity. Higher coupon bonds have lower duration.

Forward Contract A forward/futures contract specifies: The underlying asset The date on which it is to be bought The price at which it will be bought No money changes hands when you enter the transaction (unlike options) For each long position, someone takes a short position For understanding pricing, ignore forward /futures differences

Forward Contract A forward contract involves two parties: The party who agrees to buy the underlying holds a long position. The party who sells the underlying holds a short position. At the contract date t=0 no money changes hands. At the settlement date t=T the party long the contract pays FT and gets the raw materials. Instead of physical delivery of the underlying many forward contracts can specify cash settlement. Cash Flow from a long position: Forward price: FT = S0 (1+rf)T t T Long Forward ST - FT

Forward Parity Formula F0 = S0 + [S0(1+rf)T – S0] - benefits spot price + cost of holding spot Cost of holding spot is the cost of financing the position: S0(1+rf)T – S0 In general, there may be other costs: Financing costs (rf), Transportation costs, Storage costs, Lost interest There may also be benefits: Interest earned, Dividends F0 = S0 + costs – benefits Costs and benefits are measured at date T (FV)

Options With options, one pays money to have a choice in the future A call option gives its holder the right to purchase an asset for a specified price, called the exercise, or strike price, on or before some specified expiration date. A put option gives its holder the right to sell an asset for a specified exercise or strike price on or before some expiration date. American options – can be exercised any time until exercise date European options – can be exercised only on exercise date

Options - Long Positions

Options - Short Positions

Put-Call Parity Call-plus-bond portfolio must cost the same as stock-plus-put portfolio. Each call costs C. The riskless zero-coupon bond costs X/(1 + rf)T. Therefore, the call-plus-bond portfolio costs C+ X/(1 + rf)T to establish. The stock costs S0 to purchase now (at time zero), while the put costs P. Hence, we have established In words: Put option price – call option price = present value of strike price – price of stock

Portfolio of a Long Put and the Underlying Portfolio of a Long Call and a Zero Bond with Face Value = Strike Price Portfolio of a Long Call and a Zero Bond with Face Value = Strike Price Portfolio Portfolio Zero Bond Underlying Long Call Long Put

Binomial Option Pricing Simple up-down case illustrates fundamental issues in option pricing Two periods, two possible outcomes only Shows how option price can be derived from no-arbitrage-profits condition

Binomial Option Pricing, Cont. S = current stock price u = 1+fraction of change in stock price if price goes up d = 1+fraction of change in stock price if price goes down r = risk-free interest rate

Binomial Option Pricing, Cont. C = current price of call option Cu= value of call next period if price is up Cd= value of call next period if price is down K = strike price of option H = hedge ratio, number of shares purchased per call sold

Hedging by writing calls Investor writes one call and buys H shares of underlying stock If price goes up, will be worth uHS-Cu If price goes down, worth dHS-Cd For what H are these two the same? This is the Hedge-Ratio:

Binomial Option Pricing Formula One invested HS-C to achieve riskless return, hence the return must equal (1+r)(HS-C) (1+r)(HS-C)=uHS-Cu=dHS-Cd Subst for H, then solve for C

Black-Scholes Option Pricing Fischer Black and Myron Scholes derived continuous time analogue of binomial formula, continuous trading, for European options only Black-Scholes continuous arbitrage is not really possible, transactions costs, a theoretical exercise Call T the time to exercise, σ2 the variance of one-period price change (as fraction) and N(x) the standard cumulative normal distribution function (sigmoid curve, integral of normal bell-shaped curve) =normdist(x,0,1,1) Excel (x, mean,standard_dev, 0 for density, 1 for cum.)

Black-Scholes Formula Limiting case of binomial formula Periods get shorter, more frequent Interpreting the formula N(d): is number of shares in tracking portfolio Same as delta Δ is risk free borrowing