Day 3: Pricing/valuation of forwards and futures Selected discussion from Chapter 9 (pp. 287 – 312) FIN 441 Fall 2011

Cost of carry (carry abitrage) model - Introduction What does “futures price” or “forward price” mean? What does “value” mean when discussing futures or forward contracts? Unlike stocks, bonds, options, etc., price and value are entirely different concepts for futures/forward contracts.

Cost of carry model: Initial value of futures or forward Price of futures or forward is merely the agreed-upon price at which the future delivery will be made. Value refers to how much is paid by buyer to enter into contract. At inception of futures or forward contract, the value is always “zero!”

Cost of carry model: Notation V t (0,T) = Value of forward contract created at time 0, as of time t, with expiration at time T. V t (T) = Value of corresponding futures contract. F(0,T) = price at time t of forward contract with expiration at time T. f t (T) = price at time t of corresponding futures contract.

Cost of carry model: Value of forward contract F(T,T) = S T –Forward price of forward created at time of expiration. –Forward price = spot price of asset (S). –Trivial case, but HAS to be true (or arbitrage). V T (0,T) = S T – F(0,T) –Value of forward contract at expiration. –Example: entered into “long” forward to buy asset for $500 in 1 month. One month later, spot price of asset is $550. How much (opportunity) profit did you earn on the forward contract? V t (0,T) = S t – F(0,T)(1+r) -(T-t) –Value of forward contract prior to expiration. –Use the same example, but suppose it’s 10 days into contract and spot price is at $540 and annual interest rate is 10% (assume 365 days per year). –Value = $42.60 (SHOW IT!!!!)

Cost of carry model: Forward price (relative to spot) Based on last equation and fact that forward contract has zero value at creation, –V 0 (0,T) = S 0 – F(0,T)(1+r) -T = 0 Solving last equation for F(0,T), –F(0,T) = S 0 (1+r) T –Forward price is the spot price compounded to the contract’s expiration. –If not true, then an arbitrage opportunity exists. Example –What must the spot price of gold be if 2-year forward contracts for gold are priced at $850? Assume 2-year T-bills yield 1.5% per year? (see kitco.com)

Forward currency price Illustration of interest rate parity. –Fundamental relation between spot and forward prices and interest rates in 2 countries. F(0,T) = S 0 (1 +ρ) -T (1+r) T –ρ = foreign currency interest rate –r = domestic currency interest rate –S 0 = spot rate (in domestic currency/unit of foreign currency) –NOTE: can “reverse” the notation! Suppose you believe that you can earn higher interest rate in another county. –Example: What should be the forward price (in US$) of NZ$100,000 to be delivered in 1 year if 1-year US T-bills yield 3.25% per year, the NZ 1-year rate is 7.5%, and the spot rate is NZ$1.3333/US$1 (or US$0.75/NZ$1)?

An example of a forward currency strategy Buy NZ$133,333 for US$100,000. Enters into forward contract to buy US$ with NZ$ in 1 year. After 1 year, treasurer holds NZ$143,333 (i.e., 133,333 x 1.075). Convert back to US$. What forward rate at time of original investment would guarantee that treasurer could convert NZ dollars into $103,250 (i.e., 100,000 x )? F(0,T) = NZ$1.3882/US$1 (or US$0.7204/NZ$1) = (1.075)/(1.0325) = 0.75(1.0325)/(1.075). The answer to the question posed previously is US$72,035 (see forward rate above). Moral: even though I can earn higher interest rate in another currency, the forward rate is more expensive (US$0.72 vs. spot of US$0.75). See example on page 299 for similar example.

Cost of carry model: Value of futures contract f T (T) = S T –Futures price at expiration of contract. –Same result as with forward price. v t (T) = f t (T) – f t-1 (T) before contract is marked-to-market. –Suppose I bought November crude oil at $99.00 at market open on Oct 1, but at 1 PM, November crude is trading at $ –The contract is worth negative 50 cents per barrel to me. v t (T) = 0 as soon as the contract is marked-to-market. –Suppose settlement price for November crude is $ –My account is credited with the $0.30 per barrel gain, so the futures contract itself has zero value once this happens.

Cost of carry model: Futures price relative to spot f t (T) = S 0 (1+r) T This fact is true immediately after each daily settlement (i.e., after marking to market). Thus, futures price = forward price.

Cost of carry for underlying that generates cash flows Standard example: single-stock futures or stock index futures. –Cash flow on underlying = dividends. Assume dividend (D T ) on stock paid at expiration of futures contract. f 0 (T) = S 0 (1+r) T – D T –Futures price is equal to the compounded spot price of stock minus the amount of the dividend. –If dividends are paid at multiple times between time 0 and time T, D T represents the future value of all expected dividend payments.

Cost of carry with cash flows on underlying (cont’d) f 0 (T) = (S 0 – D 0 )(1+r) T –D 0 is present value of expected dividends. f 0 (T) = S 0 e (r – δ)T –If dividends are paid continuously (like on large index), use continuous compounding with δ as the continuous dividend yield. Stock futures pricing examples. Value of forward contract on underlying with cash flows (see page 297).

Additional issues in futures/forward pricing Storage costs –Denoted by “s” (page 300). –Futures price equals compounded spot price plus storage costs. Risk premium –Under certainty or risk neutrality, price of an asset (today) equals expected price of asset at future date minus storage costs and incurred interest costs. –If investors are risk averse AND there is uncertainty about future asset prices, today’s price reflects a risk premium, so the spot price also includes a discount to reflect risk premium. Final definition of “cost of carry” (page 302): –Denoted by θ. –Combination of storage costs and “net interest.” –Cost of carry applies to most assets traded on futures exchanges.

Future/forward pricing equilibrium Futures price equals spot price plus cost of carry. –f 0 (T) = S 0 + θ Explain what happens if this equality does not hold. Pricing Implications: –Positive cost of carry (true for most commodities) Contango (futures price > spot price) –When can cost of carry be negative? –Can futures price < spot price? Convenience yield = Premium earned by those who hold inventories of a commodity that is in short supply. Backwardation (inverted market) More common for assets with negligible storage costs (financials). Occasional occurrence in commodity markets (see first point).

Futures prices and risk premia We’ve considered the idea of risk premiums for investing in “spot” assets. Are there risk premiums for investing in futures contracts? –No risk premium hypothesis f 0 (T) = E(S T ) Futures price is an unbiased expectation of future spot price. –Risk premium hypothesis f 0 (T) < E(S T ) = f 0 (T) + E(φ) = E(f T (T)) Futures prices are (downward) biased expectations of future spot prices. Suggested by economists arguing that spot and futures markets are dominated by those who are “naturally long” in the underlying (i.e., farmers who own wheat, corn, etc.). Risk premium could be negative if hedgers are predominantly buyers of futures contracts.

Next 3 classes Hedging with futures (Chapter 11) –Price risk –Short vs. long hedge –Basis and basis risk –Hedge ratio –Liquidating a hedge –Hedging a “spot” transaction vs. hedging an “ongoing” transaction. Introduction to class project