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**Financial Engineering**

Lecture 9

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**Continuous Compounding**

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

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**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% = (1+.06) (1+.06)2 (1+.06)3

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**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% = e.06 e.06x2 e.06x3

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**Yields YTM Example zero coupon 3 year bond with YTM = 6% and**

par value = 1,000 Price = 1000 / (1 +.06)3 =

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**Yields YTM Example zero coupon 3 year bond with YTM = 6% and**

par value = 1,000

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**Term Structure & Spots Rates**

8.04 6.00 4.84

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**Pure Term Structure Maturity (years) YTM 1 3.0% 5 3.5% 10 3.8% 15 4.1%**

% % % % % % 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.”

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**Forward rates f3-1 Rn = spot rates fn = forward rates Rates year**

Rn = spot rates fn = forward rates

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Spot/Forward rates R2 R3 year f2 f3 f3-2

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

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**Spot/Forward rates Forward Rate Computations**

(1+ Rn)n = (1+R1)(1+f2)(1+f3)....(1+fn)

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**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%

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**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 YTM 2 yr 1000 x ( )2 = 3 yr 1000 x ( )3 = IRR of ( FV= & PV= ) = 11%

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**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

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**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%

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**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 / ( )5 = .6843

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**Spot/Forward rates Bond Value = C1 + C2 (1+r) (1+r)2**

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

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

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**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) = 1 / (1.0929)(1+f2) f1 = 9.29% f2 = 9.58% PROOF

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**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%

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

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**Forward rates & Prices 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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**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%

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**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:

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Spot/Forward rates coupons paying bonds to derive rates

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

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**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

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**Short Sale Example Purchase of shares**

April: Purchase 500 shares for $120 -$60,000 May: Receive dividend 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 $ ,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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**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

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**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.

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**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?

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**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 $ Repay loan at $ $0.45

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**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 $ Receive 68 x e.5x $4.55

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**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

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**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?

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**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

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**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?

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**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

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**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

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**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

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**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%?

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**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%?

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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.

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**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

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