Lecture 9. Continuous Compounding Warning: Answers in book will be slightly different than...

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.

e

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


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


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


Spot/Forward rates


Answer FV of principal @ YTM

IRR of ( FV= 1336.16 & PV= -1197.10) = 11.62%

Spot/Forward rates

16.133610003

10.119710002309660.

208995.

eyr

eyr

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

A:


8903.

1Price

11162.

e

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:


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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C

1

1 d1

fe

21

12d

ff ee

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



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$

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50.045.0

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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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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.

rTe

KFalue

)(V 0

Long Contract Value

rTe

FKalue

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Short Contract Value



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

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22.2$

)55.6980.71(Value

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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.

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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.

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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

Lecture 9. Continuous Compounding Warning: Answers in book will be slightly different than...

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