Chapter 5. Determination of Forward and Futures Prices. Joel R. Barber. Department of Finance. Florida International University.

Transcription

1 Chapter 5 Determination of Forward and Futures Prices Joel R. Barber Department of Finance Florida International University Miami, FL 33199

2 I. Primer on Continuous Compounding Why? Thetraditioninoptionpricingistodiscount cash flow using continuous compounding Simplifies interest rate problems For example, What is future value of $100 in 73 days at 6% compounded weekly? Review of discrete compounding Define R as the annual rate of interest compounded m times per year. What is the future value of $1 at 12% compounded annually for 10 years? After one year, we have $1 plus $.12 1in interest: 1.12

3 After two years, we have $1.12 plus $ in interest: 1.12 (1 +.12) = = Aftern years, we have: {z } n times =1.12 N In general, the future value of $A at rate R compounded n times is FV = A(1 + R) n Now suppose we can earn 12% compounded monthly for one year. This means that each month we earn 12% 12 =1% So over one year, the future value of $1 is compounded twelve times: FV =

4 Over two years, FV = In general, the future value of $A at rate R compounded m times per year for n years is µ FV = A 1+ R nm m As m increases the future value increases but is bounded by FV = Ae RT Wewill define r as the annual rate compounded continuously The future value is determined by FV = Ae rt (1)

5 Example Determine the future value of $1 at 12% for 2 years for various compounding frequencies. Annual (m =1): FV = Semi-annual (m =2): Quarterly (m =4): Monthly (m = 12): = FV = = FV = = FV = =

6 Daily (m =365): FV = (1+.12/365) = Continuously (m is very large): FV = e 2.12 = roughly the same as daily Converting from discrete to continuous We typically observe a discrete rate. Howdoweconvertadiscreteratetoacontinuous rate with the same future value. Set one-year future value under discrete equal to future value under continuous and solve for r e r =(1+ R m )m

7 Conversion formula: µ r =ln 1+ m R m (2) In words, take the natural logarithm of the future value of $1 under discrete compounding Example An account pays 12% compounded monthly What is the effective annual rate of interest? = = 12.68% Whatistheccrate? ln(1.1268) = = %

8 What is the future value of $100 in 17 months? FV = 100e (17/12) = or using discrete compounding Example FV = 100(1.01) 17 = What is future value in 73 days at 6% compounded weekly? Step 1 µ r = ln = Step 2 FV = 100 exp = ½ ¾ 365

9 II. Determination of Forward Prices Nature of Underlying Investment asset Consumption asset Commodity or perishable good Non-deliverable quantity future value of an economic variable future temperature outcome of presidential election terrorist activity

10 Investment Assets Held by significant numbers of people purely for investment purposes. Examples: gold, silver, stocks, bonds In other words, investment assets are held for income and potential capital gains. Consumption Assets Held primarily for consumption Examples: copper, oil Distinction not always clear cut For example, gold is also a consumption asset because it is used in manufacture.

11 Is your house an investment or consumption asset? Cost of Carry Market Cost of carry is the cost of holding an investment asset These cost include interest and storage. In this class we will focus on interest cost. In a cost of carry market, the forward price should equal the spot price plus the cost of holding the asset until the delivery date. Why? Because the alternative to buying the forward contract is to buy the spot asset and hold it to the delivery date.

12 For an investment asset, the two strategies should have the same forward price. For a consumption asset, we need to adjust for the convenience yield, which measures the benefit of currently owning the asset. Notice that price discovery does not occur in cost of carry market. The forward price does not tell you anything about investor s expectations Because the forward price is determined by spot price and spot interest rate. For example, the forward price of gold is always higher than the spot price.

13 Price Discovery Cost of carry model will not work for a perishable asset non-deliverable quantity Normal Backwardation Contango Notation F forward or futures price S spot price T delivery date r cc risk-free rate of interest for maturity T

14 Overview of cost of carry: F = S + cost of carry income from underlying Gold example Spot price equals $400 per oz. Forward matures in one year Current interest rate equals 10% compounded annually Theoretical forward price: F = = 440 Suppose market forward price equals $450 per oz. Switching argument: Instead of buying forward you should buy spot and hold to maturity

15 You would save $10 oz Arbitrage argument: Sell forward (since it is over priced). Hedge by buying spot asset. Finance spot purchase by borrowing at riskfree rate. You would make an arbitrage profit of $10 per oz. Since position is perfectly hedged, you are exposedtonorisk. Since you borrowed the spot price, you have invested no money. So you have invested no money, assumed norisk,amadeacertain$10profit. Not a bad deal!

16 Cost of carry model for investment asset that pays no income Theoretical forward price: F = Se rt (5.1) If F > Se rt, we could earn an arbitrage profit by Selling forward Borrowing S dollars Buying spot asset for S dollars This is an example of a forward carry If F<Se rt, we can earn an arbitrage profit with a reverse carry: Buy forward Shortspotasset

17 Invest proceeds at risk-free rate You will not be tested on reverse carry Cost of carry model for investment asset that pays a discrete income stream. Define I as the present value of the income stream paid by underlying asset over the life of the contract. Assumeincomestreamisrisk-free. So I equals present value of cash flows discounted at the rate r. Theoretical forward price: Why? Suppose F>(S I)e rt F =(S I)e rt (5.2)

18 Arbitrage profit is available using forward carry strategy. To hedge future delivery we don t need the income stream. So we sell off income stream. Therefore, we need to finance the difference between the spot price and the present value of the income stream. When an Investment Asset Provides a Known Yield Define q as cc rate that underlying asset pays. Theoretical forward price: F = Se (r q)t (5.3) Why? Carrycostisoffset by income yield q.

19 Stock Index Futures Can be viewed as an investment asset paying a dividend yield. The futures price and spot price relationship is therefore F = Se (r q)t where q is the dividend yield on the portfolio represented by the index. Index Arbitrage When F > Se (r q)t an arbitrageur buys the stocks underlying the index and sells futures When F < Se (r q)t an arbitrageur buys futures and shorts or sells the stocks underlying the index Index arbitrage involves simultaneous trades in futures and many different stocks

20 Very often a computer is used to generate the trades Occasionally (e.g., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F and S does not hold Spot Exchange rate Spot exchange is the rate at which one currency can converted to another For example: S =.5 $/SF means 1 Swiss Franc can be exchange for.5 dollars. Often times, we talk about buying or selling a foreign currency. This can be confusing, because what is really happening is that one currency is being exchanged for another.

21 The currency in the dominator (downstairs) is the one being bought or sold. The currency in the numerator (upstairs) is the one we pay or receive A spot quote of $/SF.5 means we can buy 1 SF for.5 dollars Or it means we can sell 1 SF for.5 dollars. If the spot rate increases to $/SF 1, then the Swiss Franc has appreciated. From the perspective of the Swiss, the exchange rate is SF/$ 2. This means that 1 dollar can be bought or sold for 2 SF. Now if the exchange rate decreases to SF/$ 1, the dollar has depreciated.

22 So the convention is the currency you buy or sell is in the denominator and the currency you pay or receive is in the numerator. Under this convention, a long makes money if the exchange rate increases and loses money if the exchange rate decreases. Compare to gold price of $400 /oz. you buy or sell the denominator. you pay or receive the numerator. Overview of Forwards on Currencies A forward contract to buy means you agree to accept delivery of the denominator currency and pay the numerator currency at the delivery date.

23 A forward contract to sell means you agree to deliver the denominator currency and receive the numerator currency at the delivery date. Example: forward contract for the delivery of SF 1000 in one year at a forward rate of $/SF.5. The buyer (or long) agrees to buy 1000 Swiss Franc in one year for $500. The seller (or short) agrees to sell 1000 Swiss Franc in one year for $500. If the spot exchange rate in one year is $/SF 1, the buyer makes a profit 1000(1-.5) = $500. On the other hand, the seller loses $500.

24 Determination of Forward Exchange Rate Alternative to buying forward: Convert dollars to foreign currency at spot rate Invest foreign currency at foreign risk-free rate to delivery date Cost of Carry Model Carry of cost rate r is reduced by foreign risk-free rate. Theoretical Forward Price: F = Se (r r f)t (5.9) Notice that we substitute r f for q. Arbitrage Argument: suppose F>Se (r r f)t Borrow Se r ft dollars at r.

25 Convert to e r ft units of foreign currency. Invest at r f to maturity T. e r ft units of foreign currency grows to 1 unit. Deliver 1 unit to forward contract and collect F dollars. Payoff dollar debt: Se r ft e rt = Se (r r f)t. Pocket difference: F Se (r r f)t > 0(by assumption) Example S =SF/$.5 T =1year r =5% r f = 10% (Swiss Bank Account)

26 Theoretical Price: F =.5e (.05.1)(1) = The forward price of SF is lower than spot price. Consider this strategy: Convert dollars to SF to earn higher return. Invest SF at 10% This strategy is risky because we don t know the future exchange rate. Consider this strategy: Convert dollars to SF to earn higher return. Invest SF at 10% for one year. Sell a one-year forward contract on SF.

27 In one year, we have locked in the rate at which we can convert SF back into dollars. This covered strategy is risk-free. What should the rate of return be? S dollars/sf 1SF e r ft SF Fe r ft dollars.5 dollars/sf 1 SF e.1 SF.4756e.1 dollars Under this strategy.5 dollars today becomes.4756e.1 dollars in one year. So the discrete annual return equals.4756e.1.5 1=5.124% Guess what the cc return is: ln( ) = 5.000%

28 Determination of Forward Price of Consumption Asset Because of convenience of owning asset, most market participants prefer to own the asset rather buy forward contract. Adjust the cost of carry r for convenience yield y. Theoretical Price: F = Se (r y)t Forward vs Futures Prices Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different:

29 A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price. A strong negative correlation implies the reverse. The reason for the difference is that futures are settled daily. So interest rate uncertainty will affect the net worth in a futures account at the delivery date. Valuing Forward Contracts on Investment Assets Let K equal the delivery price. At delivery date T,wepayK and receive the underlying asset. The payoff is S T K

30 What alternative strategy creates the same payoff? Buy spot asset and borrow Ke rt. The cost of alternative strategy should have the same value as the forward contract: or V FC = S Ke rt V FC =(F K) e rt (5.4) IfwesetV FC =0,thenK should equal forward price: K = Se rt Suppose S = $400/oz., T = 1 year, and r = ln(1.1) = 9.531%. What is forward price and delivery price of contract: K = F =400e = 440

31 and V FC =0 Suppose one month later S = $400/oz. Then value of forward contract with delivery price $440 is V FC = e /12 = The seller s position has improved. Suppose in one month S = $450/oz. Then V FC = e /12 = The buyer s position has improved. Suppose in one month S = $350/oz. Then V FC = e /12 =

32 The seller s position has improved.