lecture 9. continuous compounding warning: answers in book will be slightly different than...
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Financial EngineeringLecture 9
Continuous Compounding
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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
YieldsYTM
Examplezero coupon 3 year bond with YTM = 6% andpar value = 1,000Price = 1000 / (1 +.06)3 = 839.62
YieldsYTM
Examplezero coupon 3 year bond with YTM = 6% andpar value = 1,000
27.835
1000Price
306.
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Term Structure & Spots Rates
2 3 10
8.04
6.00
4.84
Pure Term StructureMaturity (years) YTM
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
0 1 2 3
Rates
f3-1
Rn = spot rates
fn = forward rates
year
Spot/Forward rates
R2
R3
f3
f3-2
f2
0 1 2 3 year
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
ExampleWhat is the 3rd year forward rate?2 year zero treasury YTM = 8.995%3 year zero treasury YTM = 9.660%
Spot/Forward rates
ExampleWhat 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%
Spot/Forward rates
example (using previous example )f3 = 11%Q: What is the 2 year forward price on a 1 yr bond?A: 1 / (1+.11) = .9009
Forward rates & Prices
ExampleTwo 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%
Spot/Forward rates
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 / (1 + 7.88)5 = .6843
Forward rates & Prices
coupons paying bonds to derive rates
Spot/Forward 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)
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 1value bonds 8% = 975 10%= 1010
Step 2 975 = 80d1 + 1080 d2 -------> solve for d11010 =100d1 + 1100d2 -------> insert d1 & solve for d2
Spot/Forward rates
example continuedStep 3 solve algebraic equationsd1 = [975-(1080)d2] / 80insert d1 & solve = d2 = .8350insert 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
ExampleWhat is the 3rd year forward rate?2 year zero treasury YTM = 8.995%3 year zero treasury YTM = 9.660%
Spot/Forward rates
ExampleWhat 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%
Spot/Forward rates
16.133610003
10.119710002309660.
208995.
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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:
Forward rates & Prices
8903.
1Price
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ExampleTwo 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%
Spot/Forward rates
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:
Forward rates & Prices
6640.
1Price
50819.
e
coupons paying bonds to derive rates
Spot/Forward rates
221 Value Bond
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21 Value Bond fff ee
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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 1value bonds 8% = 975 10%= 1010
Step 2 975 = 80d1 + 1080 d2 -------> solve for d11010 =100d1 + 1100d2 -------> insert d1 & solve for d2
Spot/Forward rates
example continuedStep 3 solve algebraic equationsd1 = [975-(1080)d2] / 80insert d1 & solve = d2 = .8350insert d2 and solve for d1 = d1 = .9150
Step 4
Insert d1 & d2 and Solve for f1 & f2.
f1 = 8.89% f2 = 9.15%
PROOF
Spot/Forward rates
1
1 .9150 fe
20889.
18350. fee
Short Sale ExamplePurchase of sharesApril: Purchase 500 shares for $120 -$60,000May: Receive dividend +500July: Sell 500 shares for $100 per share +50,000
Net profit = -$9,500
Short Sale of sharesApril: Borrow 500 shares and sell for $120 +60,000May: Pay dividend -$500July: Buy 500 shares for $100 per share -$50,000
Replace borrowed shares to close short position .
Net profit = + 9,500
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007
Futures Price Notation
S0: Spot price today
F0: Futures or forward price today
T: Time until delivery date
r: Risk-free interest rate for maturity T
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.
rTeSF 00
Futures Price Calculation
Example IBM stock is selling for $68 per share. The zero
coupon interest rate is 4.5%. What is the likely price of the 6 month futures contract?
55.69$
68
0
50.045.0
00
F
eF
eSF rT
Futures Price CalculationExample - 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 +70.00Repay loan at $69.55 -69.55
$0.45
Futures Price CalculationExample - 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 -65.00Receive 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
rTeISF )( 00
Futures Price Calculation
Example IBM stock is selling for $68 per share. The zero
coupon interest rate is 4.5%. It pays $.75 in dividends in 3 and 6 months. What is the likely price of the 6 month futures contract?
47.1$
75.75.50.045.25.045.
Iee
I
04.68$
)47.168(
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0
50.045.0
00
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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 $500. The zero coupon
interest rate is 4.5% and the index is known to produce a continuously compounded dividend yield of 2.0%. What is the likely price of the 6 month futures contract?
29.506$
500
0
50.)02.045(.0
)(00
F
eF
eSF Tqr
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.
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KFalue
)(V 0
Long Contract Value
rTe
FKalue
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Short Contract Value
Futures Price Calculation
Example IBM stock is selling for $71 per share. The zero
coupon interest rate is 4.5%. What is the likely value of the 6 month futures contract, if it only has 3 months remaining? Recall the original futures price was 69.55.
80.71$
71
0
25.045.0
00
F
eF
eSF rT
22.2$
)55.6980.71(Value
25.045.
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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.
rTeUSF 00
Futures price given constant yield storage
cost
Futures price given set price storage cost
TureSF )(00
u =continuously compounded cost of storage, listed as a percentage of the asset pricerTe
UCost Storaget
Futures Prices and StorageExample The spot price of copper is $3.60 per pound. The 6 month cost to store
copper is $0.10 per pound. What is the price of a 6 month futures contract on copper given a risk free interest rate of 3.5%?
76.3$
)098.60.3( 50.035.
00
e
eUSF rT098.
.1050.035.
eU
Futures Prices and StorageExample The spot price of copper is $3.60 per pound. The annual cost to store
copper is quoted as a continuously compounded yield of 0.5%. What is the price of a 6 month futures contract on copper given a risk free interest rate of 3.5%?
67.3$
60.3 50.)005.035(.
)(00
e
eSF Tur
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.
TyureSF )(00
Fundamentals of Futures and Options Markets, 6th Edition, Copyright © John C. Hull 2007 5.45
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
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