Financial Engineering Lecture 9

Continuous Compounding Warning: Answers in book will be slightly different than calculator.

Bond Value Bond Value = C1 + C2 + C3 (1+r) (1+r)2 (1+r)3 Example $1,000 bond pays 8% per year for 3 years. What is the price at a YTM of 6% 1053.46 = 80 + 80 + 1080 (1+.06) (1+.06)2 (1+.06)3

Bond Value Bond Value = C1 + C2 + C3 er er2 er3 Example $1,000 bond pays 8% per year for 3 years. What is the price at a YTM of 6% 1048.39 = 80 + 80 + 1080 e.06 e.06x2 e.06x3

Yields YTM Example zero coupon 3 year bond with YTM = 6% and par value = 1,000 Price = 1000 / (1 +.06)3 = 839.62

Yields YTM Example zero coupon 3 year bond with YTM = 6% and par value = 1,000

Term Structure & Spots Rates 8.04 6.00 4.84 2 3 10

Pure Term Structure Maturity (years) YTM 1 3.0% 5 3.5% 10 3.8% 15 4.1% 1 3.0% 5 3.5% 10 3.8% 15 4.1% 20 4.3% 30 4.5% The “Pure Term Structure” or “Pure Yield Curve” are comprised of zero-coupon bonds These are often only found in the form of “US Treasury Strips.” http://online.wsj.com/mdc/public/page/2_3020-tstrips.html?mod=topnav_2_3000

Forward rates f3-1 Rn = spot rates fn = forward rates Rates year 0 1 2 3 Rn = spot rates fn = forward rates

Spot/Forward rates R2 R3 0 1 2 3 year f2 f3 f3-2

Spot/Forward rates example 1000 = 1000 (1+R3)3 (1+f1)(1+f2)(1+f3)

Spot/Forward rates Forward Rate Computations (1+ Rn)n = (1+R1)(1+f2)(1+f3)....(1+fn)

Spot/Forward rates Example What is the 3rd year forward rate? 2 year zero treasury YTM = 8.995% 3 year zero treasury YTM = 9.660%

Spot/Forward rates Example What is the 3rd year forward rate? 2 year zero treasury YTM = 8.995% 3 year zero treasury YTM = 9.660% Answer FV of principal @ YTM 2 yr 1000 x (1.08995)2 = 1187.99 3 yr 1000 x (1.09660)3 = 1318.70 IRR of ( FV= 1318.70 & PV= -1187.99) = 11%

Forward rates & Prices example (using previous example ) f3 = 11% Q: What is the 2 year forward price on a 1 yr bond? A: 1 / (1+.11) = .9009

Spot/Forward rates Example Two years from now, you intend to begin a project that will last for 5 years. What discount rate should be used when evaluating the project? 2 year spot rate = 5% 7 year spot rate = 7.05%

Forward rates & Prices Example (previous example) 2 yr spot = 5% 5 yr forward rate at year 2 = 7.88% Q: What is the price on a 2 year forward contract if the underlying asset is a 5year zero bond? A: 1 / (1 + 7.88)5 = .6843

Spot/Forward rates Bond Value = C1 + C2 (1+r) (1+r)2 coupons paying bonds to derive rates Bond Value = C1 + C2 (1+r) (1+r)2 Bond Value = C1 + C2 (1+R1) (1+f1)(1+f2) d1 = 1 d2 = 1 (1+R1) (1+f1)(1+f2)

Spot/Forward rates Example – How to create zero strips 8% 2 yr bond YTM = 9.43% 10% 2 yr bond YTM = 9.43% What is the forward rate? Step 1 value bonds 8% = 975 10%= 1010 Step 2 975 = 80d1 + 1080 d2 -------> solve for d1 1010 =100d1 + 1100d2 -------> insert d1 & solve for d2

Spot/Forward rates example continued Step 3 solve algebraic equations d1 = [975-(1080)d2] / 80 insert d1 & solve = d2 = .8350 insert d2 and solve for d1 = d1 = .9150 Step 4 Insert d1 & d2 and Solve for f1 & f2. .9150 = 1/(1+f1) .8350 = 1 / (1.0929)(1+f2) f1 = 9.29% f2 = 9.58% PROOF

Spot/Forward rates Example What is the 3rd year forward rate? 2 year zero treasury YTM = 8.995% 3 year zero treasury YTM = 9.660%

Spot/Forward rates Example What is the 3rd year forward rate? 2 year zero treasury YTM = 8.995% 3 year zero treasury YTM = 9.660% Answer FV of principal @ YTM IRR of ( FV= 1336.16 & PV= -1197.10) = 11.62%

Forward rates & Prices example (using previous example ) f3 = 11.62% Q: What is the 2 year forward price on a 1 yr bond? A:

Spot/Forward rates Example Two years from now, you intend to begin a project that will last for 5 years. What discount rate should be used when evaluating the project? 2 year spot rate = 5% 7 year spot rate = 7.05%

Forward rates & Prices Example (previous example) 2 yr spot = 5% 5 yr forward rate at year 2 = 8.19% Q: What is the price on a 2 year forward contract if the underlying asset is a 5year zero bond? A:

Spot/Forward rates coupons paying bonds to derive rates

Spot/Forward rates Example – How to create zero strips 8% 2 yr bond YTM = 9.43% 10% 2 yr bond YTM = 9.43% What is the forward rate? Step 1 value bonds 8% = 975 10%= 1010 Step 2 975 = 80d1 + 1080 d2 -------> solve for d1 1010 =100d1 + 1100d2 -------> insert d1 & solve for d2

Spot/Forward rates example continued Step 3 solve algebraic equations d1 = [975-(1080)d2] / 80 insert d1 & solve = d2 = .8350 insert d2 and solve for d1 = d1 = .9150 Step 4 Insert d1 & d2 and Solve for f1 & f2. f1 = 8.89% f2 = 9.15% PROOF

Short Sale Example Purchase of shares April: Purchase 500 shares for $120 -$60,000 May: Receive dividend +500 July: Sell 500 shares for $100 per share +50,000 Net profit = -$9,500 Short Sale of shares April: Borrow 500 shares and sell for $120 +60,000 May: Pay dividend -$500 July: Buy 500 shares for $100 per share -$50,000 Replace borrowed shares to close short position . Net profit = + 9,500

Futures Price Notation Spot price today F0: Futures or forward price today T: Time until delivery date r: Risk-free interest rate for maturity T Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007

Futures Price Calculation The price of a non interest bearing asset futures contract. The price is merely the future value of the spot price of the asset.

Futures Price Calculation Example IBM stock is selling for $68 per share. The zero coupon interest rate is 4.5%. What is the likely price of the 6 month futures contract?

Futures Price Calculation Example - continued If the actual price of the IBM futures contract is selling for $70, what is the arbitrage transactions? NOW Borrow $68 at 4.5% for 6 months Buy one share of stock Short a futures contract at $70 Month 6 Profit Sell stock for $70 +70.00 Repay loan at $69.55 -69.55 $0.45

Futures Price Calculation Example - continued If the actual price of the IBM futures contract is selling for $65, what is the arbitrage transactions? NOW Short 1 share at $68 Invest $68 for 6 months at 4.5% Long a futures contract at $65 Month 6 Profit Buy stock for $65 -65.00 Receive 68 x e.5x.045 69.55 $4.55

Futures Price Calculation The price of a non interest bearing asset futures contract. The price is merely the future value of the spot price of the asset, less dividends paid. I = present value of dividends

Futures Price Calculation Example IBM stock is selling for $68 per share. The zero coupon interest rate is 4.5%. It pays $.75 in dividends in 3 and 6 months. What is the likely price of the 6 month futures contract?

Futures Price Calculation If an asset provides a known % yield, instead of a specific cash yield, the formula can be modified to remove the yield. q = the known continuous compounded yield

Futures Price Calculation Example A stock index is selling for $500. The zero coupon interest rate is 4.5% and the index is known to produce a continuously compounded dividend yield of 2.0%. What is the likely price of the 6 month futures contract?

Futures Price Profit Calculation The profit (or value) from a properly priced futures contract can be calculated from the current spot price and the original price as follows, where K is the delivery price in the contract (this should have been the original futures price. Long Contract Value Short Contract Value

Futures Price Calculation Example IBM stock is selling for $71 per share. The zero coupon interest rate is 4.5%. What is the likely value of the 6 month futures contract, if it only has 3 months remaining? Recall the original futures price was 69.55.

Futures Prices and Storage Commodities require storage Storage costs money. Storage can be charged as either a constant yield or a set amount. The futures price of a commodity can be modified to incorporate both, as in a dividend yield. Futures price given constant yield storage cost Futures price given set price storage cost u =continuously compounded cost of storage, listed as a percentage of the asset price

Futures Prices and Storage Example The spot price of copper is $3.60 per pound. The 6 month cost to store copper is $0.10 per pound. What is the price of a 6 month futures contract on copper given a risk free interest rate of 3.5%?

Futures Prices and Storage Example The spot price of copper is $3.60 per pound. The annual cost to store copper is quoted as a continuously compounded yield of 0.5%. What is the price of a 6 month futures contract on copper given a risk free interest rate of 3.5%?

Convenience Yield Shortages in an asset may cause a lower than expected futures price. This lower price is the result of a reduction in the interest rate in the futures equation. The reduction is called the “convenience yield” or y.

The Cost of Carry (Page 117) The cost of carry, c, is the storage cost plus the interest costs less the income earned For an investment asset F0 = S0ecT For a consumption asset F0  S0ecT The convenience yield on the consumption asset, y, is defined so that F0 = S0 e(c–y )T c can be thought of as the difference between the borrowing rate and the income earned on the asset. C = r - q Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.45