1. Financial Engineering
Lecture 9

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

3. 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

4. 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

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

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

7. Term Structure & Spots Rates
8.04
6.00
4.84

8. 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.”

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

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

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

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

13. 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%

14. 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%

15. 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

16. 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%

17. 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

18. 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)

19. 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

20. 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

21. 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%

22. 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%

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

24. 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%

25. 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:

26. Spot/Forward rates
coupons paying bonds to derive rates

27. 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

28. 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

29. 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

30. 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

31. 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.

32. 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?

33. 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

34. 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

35. 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

36. 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?

37. 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

38. 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?

39. 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

40. 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

41. 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

42. 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%?

43. 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%?

44. 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.

45. 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
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