drake drake university fin 288 interest rates futures fin 288 futures options and swaps

DrakeDRAKE UNIVERSITY

Fin 288

Interest Rates Futures

Fin 288Futures Options and Swaps

DrakeDrake University

Fin 288Interest Rate Future Contracts

Traded on the CBOT30 Year Treasury Bond &30 Yr Mini10, 5, & 2 year Treasury note futures30 Day Fed Funds5 & 10 year SwapGerman Debt

Traded on CMEEurodollar Futures


Fin 288

A quick look at contract Specifications

Treasury Bonds and Notes-Range of delivery dates

Fed Funds FuturesPrice

SwapsDelivery

MuniUnderlying Asset


Fin 288Treasury Securities

Since a majority of the interest rate instruments we will use are related to treasury securities, we need to discuss some basics relating to the pricing of Treasury securities.


Fin 288Some Pricing Issues

Day Count ConventionsUsed to determine the interest earned between two points in timeUseful in calculating accrued interestSpecified as X/Y

X = the number of days between the two datesY = The total number of days in the reference period


Fin 288Day Count Conventions

Day Count Convention Market Used

US Treasury Bonds

Corporate and MunicipalBonds

US T-Bills & money Market Instruments

period)(in ActualActual

36030

360Actual


Fin 288

Price Quotes for Treasury Bills

Let Yd = annualized yield, D = Dollar Discount F= Face Value, t = number of days until maturityPrice = F -D

360

tFxYD

t

360

F

DY

d

d


Fin 288Price Quotes on T- Bills

Note: Return was based on face value invested, not the actual amount invested.360 day convention makes it difficult to compare to notes and bonds.CD equivalent yield makes the measure comparable to other money market instruments

d

d

tY- 360

360Y yieldeqivalent CD


Fin 288Accrued Interest

When purchasing a bond between coupon payments the purchaser must compensate the owner for for interest earned, but not received, since the last coupon payment

PeriodCoupon in Days

period AIin Days

2

Coupon $ Annual AI


Fin 288Price Quotations

QuotationsThe quoted and cash price are not the same due to interest that accrues on the bond. In general:

DateCoupon Last

SinceInterest Accrued

Price

Quoted

Price

Cash


Fin 288Example

Assume that today is March 5, 2002 and that the bond matures on July 10, 2004 Assume we have an 11% coupon bond with a face value of $100. The quoted price is 90-05 (or 90 5/32 or 90.15625)Bonds with a total face value of $100,000 would sell for $90,156.25.


Fin 288Example continued

Coupons on treasuries are semiannual. Assume that the next coupon date would be July 10, 2000 or 54 days from March 5.The number of days between interest payments is 181 so using the actual/actual method we have accrued interest of

(54/181)(5.50) = $1.64

The cash price is then $91.79625 = $90.15625 + $1.64


Fin 288Conversion Factors

Since there are a range of bonds that can be delivered, the quoted futures price is adjusted by a conversion factor.

DateCoupon

Last Since

interest Accured

Factor

Conversion

Price

Futures

Quoted

Received

Cash


Fin 288Price based upon 6% YTM

The conversion factor is based off an assumption of a flat yield curve of 6% (that interest rates for all maturities equals 6%). By comparing the value of the bond to the face value, the CBOT produces a table of conversion factors.


Fin 288

Conversion Factor Continued

The maturity of the bond is rounded down to the nearest three months.If the bond lasts for a period divisible by 6 months the first coupon payment is assumed to be paid in six months. (A bond with 10 years and 2 months would be assumed to have 10 years left to maturity)


Fin 288Conversion Factor continued

If the bond does not round to an exact six months the first coupon is assumed to be paid in three months and accrued interest is subtracted. A bond with 14 years and 4 months to maturity would be treated as if it had 14 years and three months left to maturity


Fin 288Example 1

14% coupon bond with 20 years and two months to maturityAssuming a 100 face value the value of the bond would equal the price valued at 6%:

The conversion factor is then 1.92459/100 = 1.92459

192.459 )03.(1

100

)03.(1

17 V

40

40

1ttBond


Fin 288Example 2

What if the bond had 18 years and four months left to maturity? The bond would be considered to have 18 years and three months left to maturity with the first payment due in three months.Finding the value of the bond three months from today

329.187 )03.(1

100

)03.(1

17 V 36

36

1ttBond


Fin 288Example 2 continued

Assume the rate for three months is (1+r)2 = 1.03 r = .014889

Using this rate it is easy to find the PV of the bond

187.329/1.014889 = 184.581There is one half of a coupon in accrued interest

so we need to subtract 7/2=3.50184.581 - 3.50 = 181.081

resulting in a conversion factor of 181.081/100 = 1.81081


Fin 288Price Quote on T-Bills

Quotes on T- Bills utilize the actual /360 day count convention. The quoted price of the treasury bill is an annualized rate of return expressed as a percentage of the face value.


Fin 288T- Bills continued

The quote price is given by (360/n)(100-Y)

where Y is the cash price of the bill with n days until maturity

90 day T- Bill Y = 98(360/90)(100-98) =8.00


Fin 288Rate of Return

The quote is not the same as the rate of return earned by the treasury bill.The rate of interest needs to be converted to a quarterly compounding annual rate.

2/98(365/90) = .0828


Fin 288Quoted Price

The price quote on a Treasury bill is then given by 100 - Corresponding Treasury bill price quote

(quoted price = 8 so futures quote =92)

Given Z = the quoted futures priceY = the corresponding price paid for delivery

of $100 of 90 day treasury bills thenZ = 100-4(100-Y) or Y = 100-0.25(100-

Z)Z = 100-4(100-98) = 92


Fin 288Cheapest to Deliver Bond

There are a large number of bonds that could be delivered on the CBOT for a given futures contract. The party holding a short position gets to decide which bond to deliver and therefore has incentive to deliver the cheapest.


Fin 288Cheapest to Deliver

Upon delivery the short position receives

The cost of purchasing a bond isQuoted bond price + accrued interest

By minimizing the difference between the cost and the amount received, the party effectively delivers the cheapest bond:

DateCoupon

Last Since

interest Accured

Factor

Conversion

Price

Futures

Quoted

Received

Cash


Fin 288Cheapest to deliver

The bond for which

is minimized is the one that is cheapest to deliver.

Factor

Conversion

Price

Futures

Quoted

Price

Bond

Quoted

Coupon

Last Since

Interest

Accrued

Factor

Conversion

Price

Futures

Quoted

Coupon

Last Since

Interest

Accrued

Price

Bond

Quoted


Fin 288

Example: Cheapest to Deliver

Consider 3 bonds all of which could be delivered Quoted Conversion

Bond Price Factor 1 99.5 1.0382 99.5-(93.25(1.0382))

=2.69 2 143.5 1.5188 143.5-

(93.25(1.5188))=1.87 3 119.75 1.2615 119.75-

(93.25(1.2615))=2.12


Fin 288

Impact of yield changes on CTD

As yield increases bonds with a low coupons and longer maturities become relatively cheaper to deliver. As rates increase all bond prices decrease, but the price decrease for the longer maturity bonds is greaterAs yields decrease high coupon, short maturity bonds become relatively cheaper to deliver.


Fin 288Wild Card Play

Trading at the CBOT closes at 2p.m. however treasury bonds continue to trade until 4:00pm and a party with a short position has until 8pm to file a notice of intention to deliver. Since the price is calculated on the closing price in the CBOT the party with a short position sometimes has the opportunity to profit from price movements after the closing of the CBOT.If the Bond Prices decrease after 2 pm it improves the short position.


Fin 288Eurodollar Futures

Eurodollar – dollar deposited in a foreign bank outside of the US. Eurodollar interest rate is the interest earned on Eurodollars deposited by one bank with another bank. London Interbank Offer Rate (LIBOR) – Rate at which banks loan to each other in the London Interbank Market.


Fin 288Simple Hedge Example

Assume you know that you will owe at rate equal to the LIBOR + 100 basis points in three months on a notional amount of $100 Million. The interest expenses will be set at the LIBOR rate in three months.Current three month LIBOR is 7%, Eurodollar futures contract is selling at 92.90.


Fin 288Simple Hedge Example

100 - 92.90 = 7.10The futures contract is paying 7.10%

Assume the interest rate may either increase to 8% or decrease to 6%


Fin 288A Short Hedge

Agree to sell 10 Eurodollar future contracts (each with an underlying value of $1 Million).We want to look at two results the spot market and the futures market. Assume you close out the futures position and that the futures price will converge to the spot at the end of the three months.


Fin 288Rates increase to 8%

Spot position:Need to pay 8% + 1% = 9% on $10 Million

$10 Million(.09/4) = $225,000Futures Position:

Fut Price = $92 interest rates increased by .9%

Close out futures position: profit = ($10 million)(.009/4) = $22,500


Fin 288Rates Increase to 8%

Net interest paid$225,000 - $22,500 = $202,500

$10 million(.0810/4) = $202,500


Fin 288Rates decrease to 6%

Spot position:Need to pay 6% + 1% = 7% on $10 Million $10 Million(.07/4) = $175,000

Futures Position:Fut Price = $94 interest rates decreased by

1.1%Close out futures position:

loss = ($10 million)(.011/4) = $27,500


Fin 288Rates Decrease to 8%

Net interest paid$175,000 + $27,500 = $202,500

$10 million(.0810/4) = $202,500


Fin 288Results of Hedge

Either way the final interest rate expense was equal to 8.10 % or 100 basis points above the initial futures rate of 7.10%Should the position be hedged?It locks in the interest rate, but if rates had declined you were better off without the hedge.


Fin 288Simple Example 2

On January 2 the treasurer of Ajax Enterprises knows that the firm will need to borrow in June to cover seasonal variation in sales. She anticipates borrowing $1million.The contractual rate on the loan will be the LIBOR rate plus 1%The current 3 month LIBOR rate is 3.75% and the Eurodollar futures contract is 4.25%


Fin 288

Simple Example 2 Continued

To hedge the position assume the treasurer sells one June futures contract. Assume interest rates increase to 5.5% on June 13.Assume that the expiration of the contract is June 13, the same day that the loan will be taken out. The futures price will be

100-5.50 = 94.50


Fin 288Rates increase to 5.5%

Spot position:Need to pay 5.5%+1%= 6.5% on $1 Million $1 Million(.065/4) = $16,250

Futures Position:Fut Price = $94.50 interest rates

increased by 1.25%Close out futures position:

profit = ($1million)(.0125/4) = $3,125


Fin 288Rates Increase to 5.5%


$1 million(.0525/4) = $13,125which is the interest rate implied by the

Eurodollar futures contract 4.25% +1% = 5.25%


Fin 288Assumptions

The hedge worked because of three assumptions:

The underlying exposure is to the three month LIBOR which is the same as the loanThe end of the exposure matches the delivery date exactlyThe margin account did not change since the rate changed on the last day of trading.


Fin 288Basis Risk revisited

The basis is a hedging situation is defined as the Spot price of the asset to be hedged minus the futures price of the contract used. When the asset that is being hedged is the same as the asset underlying the futures contract the basis should be zero at the expiration of the contract.

Basis = Spot - Futures


Fin 288Basis Risk

On what types of contracts would you expect the basis to be negative? Positive? Why?(-) Low interest rates assets such as currencies or gold or silver (investment type assets with little or zero convenience yield. F = S(1+r)T

(+) Commodities and investments with high interest rates (high convenience yield)F = S(1+r+u)T Implies it is more likely that

F < S(1+r+u)T


Fin 288Mismatch of Maturities 1

Assume that the maturity of the contract does not match the timing of the underlying commitment.Assume that the loan is anticipated to be needed on June 1 instead of June 13.


Fin 288Simple Example Redone

On January 2 the treasurer of Ajax Enterprises knows that the firm will need to borrow in June to cover seasonal variation in sales. She anticipates borrowing $1million.The contractual rate on the loan will be the LIBOR rate plus 1%The current 3 month LIBOR rate is 3.75% and the Eurodollar futures contract is 4.25%


Fin 288

Simple Example 2 Continued

To hedge the position assume the treasurer sells one June futures contract. Assume interest rates increase to 5.5% on June 1.Assume that the futures price has decreased to 94.75 (before it had decreased to 94.50) implying a 5.25% rate (a 25 bp basis)


Fin 288Rates increase to 5.5%

Spot position:Need to pay 5.5%+1%= 6.5% on $1 Million $1 Million(.065/4) = $16,250

Futures Position:Fut Price = $94.75 interest rates

increased by 1.00%Close out futures position:

profit = ($1million)(.0100/4) = $2,500


Fin 288Rates Increase to 5.5%


$1 million(.055/4) = $13,750which is more than the interest rate implied

by the Eurodollar futures contract 4.25% +1% = 5.25%


Fin 288Minimizing Basis Risk

Given that the actual timing of the loan may also be uncertain the standard practice is to use a futures contract slightly longer than the anticipated spot position. The futures price is often more volatile during the delivery month also increasing the uncertainty of the hedge Also the short hedger could be forced to accept delivery instead of closing out.


Fin 288Mismatch in Maturities 2

Assume that instead of our original problem the treasurer is faced with a stream of expected borrowing.Anticipated borrowing at 3 month LIBOR

Date AmountMach 1 $15 MillionJune 1 $45 MillionSeptember 1 $20 millionDecember 1 $10 Million


Fin 288Strip Hedge

To hedge this risk, it to hedge each position individually.On January 1 the firm should:

enter into 15 short March contractsenter into 45 short June contractsenter into 20 short Sept contractsenter into 10 short December contracts


Fin 288Strip Hedge continued

On each borrowing date the respective hedge should be closed out. The effectiveness of the hedge will depend upon the basis at the time each contract is closed out.


Fin 288Rolling Hedge

Another possibility is to Roll the Hedge:January 2 enter into 90 short March contractsMarch 1 enter into 90 long March contracts

enter into 75 short June contractsJune 1 enter into 75 long June contracts

enter into 30 short Sept contractsSept 1 enter into 30 long Sept contracts

enter into 10 short Dec contractsDec 1 enter into 10 long Dec contracts


Fin 288Rolling the Hedge

Again the effectiveness of the hedge will depend upon the basis at each point in time that the contracts are rolled over.This opens the from to risk from the resulting rollover basis.


Fin 288Example

Now assume that the treasury has decided to borrow it the commercial paper market instead of from a financial institution.There is not a commercial paper futures contract so it must be decided what contract to use to hedge the possible interest rate change in the commercial paper market.Assume that the treasure wants to borrow $36 million in June with a one month commercial paper issue.


Fin 288Number of contracts part 1

You must choose what underlying contract best matches the 30 day commercial paper return.90 Day T-Bill. 90 day LIBOR Eurodollar, 10 year treasury bond. Assume 90 day LIBOR Eurodollar has the highest correlation so it is chosen.Assume now that the treasurer for Ajax has ran the regression and that the beta is .75



We also need to consider the asset underlying the three month LIBOR futures contract and one month commercial paper rate have different maturities.A 1 basis point movement in $1,000,000 of borrowing is $1,000,000(.0001)(30/360) = $8.33A one basis point change in $1,000,000 of the future contract is equal to:

$1,000,000(.0001)(90/360) = $25



The change in the three month contract is three times the size of the change in the one month this would imply a hedge ratio of 1/3 IF the assets underlying both positions was the same.Both sources of basis risk need to be considered.


Fin 288Number of Contracts

The treasurer will need to enter into:

$36(.75)(.33) = $9 millionOf short futures contracts


Fin 288The Cross Hedge

On January 23 month LIBOR = 3.75%June Eurodollar Future price is 95.75 implying 4.24% rateSpread between spot LIBOR rate and 1 month commercial paper rate is 60 basis pointsThis implies a 4.35% commercial paper rate.


Fin 288Expectations

Previously Ajax hoped to lock in a 4.25% 3 month LIBOR rate or an increase of 50 basis points form the current 3.75%Keeping the 50 basis point increase constant and using our hedge ratio of .75 the goal becomes locking in a .75 (50) = 37.5 basis point increase in the commercial paper rate.This implies a one month rate of 4.35% + 37.5BP = 4.725%


Fin 288Results Futures

Assume that on June 1 the 3 month LIBOR rate increases to 5.5% (as it did in our previous example), also assume that the futures contract price falls to 94.75.Closing out the Futures contract resulted in a profit of $2,500 per $1million. Since we have 9 $1 million contracts our profit is

9(2,500)=$22,500


Fin 288Results Spot

LIBOR increased by 1.75 % or 175 basis points, assuming our hedge ratio is correct this implies a .75(175) = 131.25 basis point increase in the one month commercial paper rate.So the new expected one month commercial paper rate is 4.35+1.3125 = 5.6625%However assume that the relationship was not perfect ant the actual one month rate is 5.75%


Fin 288Results

Given the 5.75% commercial paper rate the cost of borrowing has increased by

$36,000,00(.0575-.0435)(30/360) = $42,000

Subtracting our profit of 22,500 in futures market the net increase in borrowing cost is:

$42,000 - $22,500 = $19,200

This is equivalent to an increase of:36,000,000(X)(30.360) = $19,500 X = 65 BP


Fin 288Results

Using the 65 BP increase Ajax ended up paying 5% for its borrowing.The treasurer was attempting to lock in 4.725% or 27.5BP less than what she ended up paying.The 27.5 BP difference is the result of basis risk.


Fin 288Basis Risk

Source 1June 1 spot LIBOR was 5.5% the LIBOR rate implied by the futures contract was 5.25% a 25 BP differenceGiven the hedge ratio of .75 this should be a 25(.75) = 18.75 BP difference for commercial paper

Source 2Expected 1 month commercial paper rate is 5.6625%, actual is 5.75% a 8.75 BP difference


Fin 288Basis Risk

The result of the two sources of risk:

18.75 + 8.75 = 27.5 basis points


Fin 288Duration: The Big Picture

Calculation: Given the PV relationships, we need to weight the Cash Flows based on the time until they are received. In other words we are looking for a weighted maturity of the cash flows where the weight is a combination of timing and magnitude of the cash flows


Fin 288Calculating Duration

One way to measure the sensitivity of the price to a change in discount rate would be finding the price elasticity of the bond (the % change in price for a % change in the discount rate)


Fin 288

Duration MathematicsMacaulay Duration

Macaulay Duration is the price elasticity of the bond (the % change in price for a percentage change in yield).Formally this would be:

P

r)(1

price Original

yield Original

Yieldin Change

Pricein Change

yield originalyieldin change

price originalpricein change

DMAC

r

P


Fin 288Duration Mathematics

Taking the first derivative of the bond value equation with respect to the yield will produce the approximate price change for a small change in yield.


Fin 288Duration Mathematics

1n1n432 r)(1

(-n)MV

r)(1

(-n)CP

r)(1

(-3)CP

r)(1

(-2)CP

r)(1

(-1)CP

r

P

nn32 r)(1

MV

r)(1

CP

r)(1

CP

r)(1

CP

r)(1

CPP

nn32 r)(1

nMV

r)(1

nCP

r)(1

3CP

r)(1

2CP

r)(1

1CP

r1

1

r

P

The approximate price change for a small change in r


Fin 288

Duration MathematicsMacaulay Duration

P

r)(1

price Original

yield Original

Yieldin Change

Pricein Change

yield originalyieldin change

price originalpricein change

DMAC

r

P

nn32 r)(1

nMV

r)(1

nCP

r)(1

3CP

r)(1

2CP

r)(1

1CP

r1

1

r

P

substitute

P

r)(1

r)(1

nMV

r)(1

nCP

r)(1

3CP

r)(1

2CP

r)(1

1CP

r1

1nn32

MACD


Fin 288

Macaulay Duration of a bond

N

1tt

t

N

1tt

t

r)(1CF

r)(1)t(CF

MACD

P

1

r)(1

nMV

r)(1

nCP

r)(1

3CP

r)(1

2CP

r)(1

1CPnn32

MACD


Fin 288Duration Example

10% 30 year coupon bond, current rates =12%, semi annual payments

periods 3895.17

)06.1(1000$

)06.1(50

)06.1()1000($60

)06.1()50($

60

160

60

160

tt

tt

MAC

t

D


Fin 288Example continued

Since the bond makes semi annual coupon payments, the duration of 17.3895 periods must be divided by 2 to find the number of years.17.3895 / 2 = 8.69475 yearsAnother interpretation of duration is shown here: Duration indicates the average time taken by the bond, on a discounted basis, to pay back the original investment.


Fin 288

Using Duration to estimate price changes

P

r)(1DMAC

r

Pr)(1

rDMAC

P

PRearrange

% Change in Price

Estimate the % price change for a 1 basis point increase in the yearly yield

000820257.06.1

0001.69475.8

r)(1

rDMAC

P

P

Multiply by original price for the price change

-0.000820257(838.8357)=-.688061


Fin 288Using Duration Continued

Using our 10% semiannual coupon bond, with 30 years to maturity and YTM = 12%Original Price of the bond = 838.3857If YTM = 12.01% the price is 837.6986

This implies a price change of -0.6871Our duration estimate was -0.6881 a difference of .0010


Fin 288Note:

Previously yield increased from 12% a year to 12.01%.We used the Duration represented in years, 8.69475We could have also used duration represented in semiannual periods, 17.3895. The change in yield needs to be adjusted to .0001/2 = .00005 however, the original yield (1+r) stays at 1.06.

000820257.06.1

00005.3895.17

r)(1

rDMAC

P

P

The estimated price change is then the same as before: -0.000820257(838.8357)=-.688061


Fin 288Modified Duration

r)(1

DurationMacaulay D

Duration

ModifiedMOD

r)r1(

D

r)(1

rD MAC

MAC

P

P

Substitute DMOD

The % Change in price was given above as:

rDMODP

P


Fin 288

Modified vs Macaulay Duration

202533.806.1

69475.8DMODIFIED

00082025.06.1

0001.69475.8

r)(1

rDMAC

P

P

00082025.0

)0001(.202533.8r Drr)(1

DMODIFIED

MAC

P

P


Fin 288Duration - Continuous Time

Using continuous compounding the bond value formula becomes

And the Duration equation becomes

n

t

rttecV

1

V

etc

D

n

t

rtt

1


Fin 288

Change in Bond Price – Continuous Time

The estimated percentage change in the price of the bond is then given by letting value (V) = price (P):

By rearranging the actual price change is then

rDP

P

PrDP )(


Fin 288Duration Hedging

You can also estimate the hedge ratio using duration.

We know that the change in price can be estimated using duration. Assume that we have a bond portfolio with duration equal to DP

P=-PDPyLikewise the change in the asset underlying a futures contract should be estimated by

F=-FDFy



You can combine the two to produce a position with a duration of zero.

The optimal number of contracts is

Must assume a bond to be delivered

F

P

FD

PDN *


Fin 288Tailing the Hedge

Adjustments to the margin account will also impact the hedge and need to be made.The idea is to make the PV of the hedge equal the underlying exposure to adjust for any interest and reinvestment in the margin account.For N contracts this becomes Ne-rT contracts where r is the risk free rate and T is the time to maturity.



You can also estimate the hedge ratio using duration.

We know that the change in price can be estimated using duration. Assume that we have a bond portfolio with duration equal to DP

P=-PDPyLikewise the change in the asset underlying a futures contract should be estimated by

F=-FDFy

drake drake university fin 288 interest rates futures fin 288 futures options and swaps

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