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Lecture 9

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

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Bond Value = C 1 + C 2 + C 3 (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% = (1+.06)(1+.06) 2 (1+.06) 3

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Bond Value Bond Value = C 1 + C 2 + C 3 e r e r2 e r3 Example $1,000 bond pays 8% per year for 3 years. What is the price at a YTM of 6% = e. 06 e.06x2 e.06x3

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

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

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Maturity (years)YTM 13.0% 53.5% 103.8% 154.1% 204.3% 304.5% 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.” 020-tstrips.html?mod=topnav_2_3000

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Rates f 3-1 Rn = spot rates fn = forward rates year

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R2 R3 f3 f3-2 f year

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example 1000=1000 (1+R 3 ) 3 (1+f 1 )(1+f 2 )(1+f 3 )

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Forward Rate Computations (1+ Rn) n = (1+R1)(1+f 2 )(1+f 3 )....(1+f n )

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

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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 YTM 2 yr1000 x ( ) 2 = yr1000 x ( ) 3 = IRR of ( FV= & PV= ) = 11%

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

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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%

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Example (previous example) 2 yr spot = 5% 7 yr spot = 7.05% 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 / ( ) 5 =.6843

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coupons paying bonds to derive rates Bond Value = C 1 + C 2 (1+r)(1+r) 2 Bond Value = C 1 + C 2 (1+R 1 )(1+f 1 )(1+f 2 ) d1 = 1 d2 = 1 (1+R 1 )(1+f 1 )(1+f 2 )

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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% = %= 1010 Step = 80d d > solve for d =100d d > insert d1 & solve for d2

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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 f 1 & f = 1/(1+f 1 ).8350 = 1 / (1.0929)(1+f 2 ) f 1 = 9.29% f 2 = 9.58% PROOF

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

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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 YTM IRR of ( FV= & PV= ) = 11.62%

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

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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%

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Example (previous example) 2 yr spot = 5% 7 yr spot = 7.05% 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:

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coupons paying bonds to derive rates

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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% = %= 1010 Step = 80d d > solve for d =100d d > insert d1 & solve for d2

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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 f 1 & f 2. f 1 = 8.89% f 2 = 9.15% PROOF

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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

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Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 2007 Futures Price Notation S0:S0:Spot price today F0:F0:Futures or forward price today T:T:Time until delivery date r:r:Risk-free interest rate for maturity T

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

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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?

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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 6Profit Sell stock for $ Repay loan at $ $0.45

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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 6Profit Buy stock for $ Receive 68 x e.5x $4.55

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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

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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?

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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

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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?

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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 ValueShort Contract Value

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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

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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

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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%?

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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%?

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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.

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Fundamentals of Futures and Options Markets, 6 th Edition, Copyright © John C. Hull 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 F 0 = S 0 e cT For a consumption asset F 0 S 0 e cT The convenience yield on the consumption asset, y, is defined so that F 0 = S 0 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

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