Management 408: Financial Markets Spring 2009 Professor Mark Grinblatt

Management 408: Financial MarketsFall 2009Professor Torous

Final ReviewPerpetuityA security that pays a fixed amount C per period forever starting next period.

Present Value:

Growing PerpetuityA growing perpetuity makes a payment of C next period. After that the payment grows at a rate g.

Present value:

AnnuityAn annuity pays a fixed cash flow C every period until some final Time T.

Present Value:

Different Compounding FrequenciesInterest 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 assetsThe 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 MathematicsPortfolio 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 FrontierThe standard deviation is on the x-axis.The expected return is on the y-axis.

Mean Variance FrontierThe 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 PortfolioThe 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 PortfolioThe tangency portfolio is the combination of risky assets that has the highest Sharpe ratio (excess return over standard deviation).

CAPMThe 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 RiskThe 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 HypothesisInformation in past stock pricesAll Public InformationAll Available Information including inside or private informationSince 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 ALL available info[3] Weak-form ALL available infoEfficient Market HypothesisIf the market is weak-form efficient:Technical analysis or charting becomes ineffective. You wont 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 PricingThe 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 bonds 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 bonds price.

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

BootstrappingSuppose 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).

19Forward RatesOne-period Forward Rate one period from now:

In general,

MacAuley DurationA measure of the average effective maturity of a bonds cash flows

Given the MacAuley Duration you can calculate the %-change in the bond price for a given change in the yield to maturity (YTM)

or

We call D* the modified duration. Do not confuse D with D*.

21Duration and Different Compounding FrequenciesSemiannual Compounding of YTMD* = D/(1+YTM/2)Monthly Compounding of YTMD* = D/(1+YTM/12)Continuous Compounding of YTMD* = 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 ContractA forward/futures contract specifies:The underlying assetThe date on which it is to be boughtThe 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 differences24Forward ContractA 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)Tt0TLong Forward0ST - FTForward 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, DividendsF0 = S0 + costs benefitsCosts and benefits are measured at date T (FV)

26OptionsWith options, one pays money to have a choice in the futureA 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 dateEuropean options can be exercised only on exercise date

Options - Long Positions

Options - Short Positions

Put-Call ParityCall-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

Long Put

Portfolio

Portfolio of a Long Call and a Zero Bond with Face Value = Strike PriceUnderlyingPortfolioZero BondLong CallPortfolio of a Long Call and a Zero Bond with Face Value = Strike PriceBinomial Option PricingSimple up-down case illustrates fundamental issues in option pricingTwo periods, two possible outcomes onlyShows how option price can be derived from no-arbitrage-profits conditionBinomial Option Pricing, Cont.S = current stock priceu = 1+fraction of change in stock price if price goes upd = 1+fraction of change in stock price if price goes downr = risk-free interest rate Binomial Option Pricing, Cont.C = current price of call optionCu= value of call next period if price is upCd= value of call next period if price is downK = strike price of optionH = hedge ratio, number of shares purchased per call soldHedging by writing callsInvestor writes one call and buys H shares of underlying stockIf price goes up, will be worth uHS-CuIf price goes down, worth dHS-CdFor what H are these two the same?This is the Hedge-Ratio:

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

Black-Scholes Option PricingFischer Black and Myron Scholes derived continuous time analogue of binomial formula, continuous trading, for European options onlyBlack-Scholes continuous arbitrage is not really possible, transactions costs, a theoretical exerciseCall T the time to exercise, 2 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 FormulaLimiting case of binomial formulaPeriods get shorter, more frequent

Interpreting the formulaN(d): is number of shares in tracking portfolioSame as delta is risk free borrowing