1 Interest Rate Markets. 2 Measuring Interest Rates The compounding frequency used for an interest rate is the unit of measurement.

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Interest Rate Markets

2

Measuring Interest Rates

• The compounding frequency used for an interest rate is the unit of measurement.

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Continuous Compounding

• In the limit as we compound more and more frequently we obtain continuously compounded interest rates

• Rs100 grows to Rs100eRT when invested at a continuously compounded rate R for time T

• Rs100 received at time T discounts to Rs100e-RT at time zero when the continuously compounded discount rate is R

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Conversion Formulas

Define

Rc : continuously compounded rate

Rm: same rate with compounding m times per year

R m

R

m

R m e

cm

mR mc

ln

/

1

1

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Example

1.What is the equivalent rate for continuous compounding for an interest rate which is quoted as 10% per annum semi annual compounding.

2.A bank quotes you an interest rate of 14% per annum with quarterly compounding. What is the equivalent rate with (a) Continuous compounding (b) annual compounding.

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Solution

• 1. 2ln(1+0.1/2)=0.09758• 2.(a) 4ln (1+0.14/4)=13.76

(b)[ (1+0.14/4)^4]-1=14.75

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Some More Questions

Q1. An Interest rate is quoted as 5% p.a. with semi annual compounding. Calculate the rate with:

(a)Annual compounding

(b)Monthly compounding

(c)Continuous compounding

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Solution

• (a) [(1+.05/2)^2]-1=5.0625%• (b) 12[(1.050625)^1/12]-1=4.9485%• (c) 2ln[1+.025]=4.9385%

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Types of Rates

• Bank Rates• Deposit Rates• Prime Lending Rates• Home Loan and Consumer Loan Rates.

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Types of Rates

Types of interest rates important for understanding Futures and Options are

• Treasury rates• LIBOR rates• Repo rates

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Treasury Rates

• Interest rates applicable to borrowing by a government in its own currency.

• It is usually assumed that there is no chance that a government will default on an obligation denominated in its own currency.

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LIBOR Rates

• Two types LIBOR and LIBID• Refers to rates at which International banks

lend or borrow (accept deposit) from each other.

• The duration is usually 1 month, 3 month, 6 month, and 12 months.

• If bank quotes bid rates at 6% pa and offer rates at 6.5%pa it is willing to accept deposits from other banks at 6% and lend to another bank at 6.5%.

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Repo Rates• A form of short term borrowing for dealers in

government securities. The dealer sells the government securities to investors, usually on an overnight basis, and buys them back the following day.

For the party selling the security (and agreeing to repurchase it in the future) it is a repo; for the party on the other end of the transaction (buying the security and agreeing to sell in the future) it is a reverse repurchase agreement

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Zero Rates ( Zero Coupon Rate)

• A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T

• The n year zero rate is the rate of interest earned on an investment that starts today and lasts for n years.

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Example

• Suppose the five year treasury zero rate with continuous compounding is quoted as 5% per annum. It means that Rs 100 if invested at this risk free rate for five years would grow to 100e0.05*5 =128.40.

• Most interest rates are not zero rates. Eg. A five year govt bond with 5% coupon has some returns realized before five years.

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Example

Maturity(years)

Zero Rate(% cont comp)

0.5 5.0

1.0 5.8

1.5 6.4

2.0 6.8

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Bond Pricing

• To calculate the cash price of a bond we discount each cash flow at the appropriate zero rate

• In our example, the theoretical price of a two-year bond providing a 6% coupon semiannually is

3 3 3

103 98 39

0 05 0 5 0 058 1 0 0 064 1 5

0 068 2 0

e e e

e

. . . . . .

. . .

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Bond Yield• The bond yield is the discount rate that

makes the present value of the cash flows on the bond equal to the market price of the bond

• Suppose that the market price of the bond in our example equals its theoretical price of 98.39

• The bond yield is given by solving

to get y=0.0676 or 6.76%.3 3 3 103 98 390 5 1 0 1 5 2 0e e e ey y y y . . . . .

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Par Yield• The par yield for a certain maturity is the

coupon rate that causes the bond price to equal its face value.

• In our example we solve: (c/m)A +100d=100

g)compoundin s.a.(with 876get to

1002

100

222

0.2068.0

5.1064.00.1058.05.005.0

.c=

ec

ec

ec

ec

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Sample Data

Bond Time to Annual Bond

Principal Maturity Coupon Price

(Rupees) (years) (Rupees) (Rupees)

100 0.25 0 97.5

100 0.50 0 94.9

100 1.00 0 90.0

100 1.50 8 96.0

100 2.00 12 101.6

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The Bootstrap Method

• An amount 2.5 can be earned on 97.5 during 3 months.

• The 3-month rate is 4 times 2.5/97.5 or 10.256% with quarterly compounding

• This is 10.127% with continuous compounding

• Similarly the 6 month and 1 year rates are 10.469% and 10.536% with continuous compounding

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The Bootstrap Method continued

• To calculate the 1.5 year rate we solve

to get R = 0.10681 or 10.681%

• Similarly the two-year rate is 10.808%

9610444 5.10.110536.05.010469.0 Reee

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Zero Curve Calculated from the Data

9

10

11

12

0 0.5 1 1.5 2 2.5

Zero Rate (%)

Maturity (yrs)

10.127

10.469 10.536

10.681

10.808

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Question• The following table gives the prices of

bonds:

• Calculate zero rates for 6,12,18 & 24 months. Also calculate fwd rates for 6to 12, 12 to 18, & 18to 24 months.

Principal Time to maturity

Annual Coupon

Bond Price

100 0.5 0 98

100 1.0 0 95

100 1.5 6.2 101

100 2.0 8.0 104

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Answer

Zero Rates Fwd Rates

4.0405 4.0405

5.1293 6.1281

5.4429 6.0700

5.8085 6.9054

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Forward Rates

The forward rate is the future zero rate implied by today’s term structure of interest rates

27Calculation of Forward Rates

Zero Rate for Forward Rate

an n -year Investment for n th Year

Year (n ) (% per annum) (% per annum)

1 10.0

2 10.5 11.0

3 10.8 11.4

4 11.0 11.6

5 11.1 11.5

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Formula for Forward Rates

• Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded.

• The forward rate for the period between times T1 and T2 is

R T R T

T T2 2 1 1

2 1

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Upward vs Downward SlopingYield Curve (ZCYC)

• For an upward sloping yield curve:

Fwd Rate > Zero Rate > Par Yield

• For a downward sloping yield curve

Par Yield > Zero Rate > Fwd Rate

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QuestionQ1. The 6 month,12 month, 18 month and

24 month zero rates are 4%, 4.5%, 4.75% and 5% with semi annual compounding. What are rates with continuous compounding? What is the forward rate for 6 month period beginning in 18 months?

Q2. What is the value of an FRA which promises to pay you 6% compd sa on a principal of $1million for 6 months period starting in 18 months

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Solution

Month Interest Rate (pa with sa compdg)%

Interest Rate(pa with continuous compdg) %

Fwd Rate (4.94*2-4.69*1.5)/2-1.5=(9.88-7.04)/0.5=5.68%

6 4 3.96

12 4.5 4.45

18 4.75 4.69 =5.76% pa with sa compdg

24 5 4.94

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Answer for FRA

• L(Rk-Rm)(T2-T1)e^(-R2T2)• Where Rk=6%• Rm=5.76%• T2=2• T1=1.5• R2=4.94%

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Question

• What is the 2 year par yield when zero rates are as calculated by you? What is the yield on a 2 year bond that pays a coupon equal to par yield?

Hint: c=[(100-100d)m]/A

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Forward Rate Agreement

• A forward rate agreement (FRA) is an agreement that a certain rate will apply to a certain principal during a certain future time period

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Bharti Instrument’s Interest Rate Risk

• As an example, Bharti instruments has taken out a three-year, floating-rate loan in the amount of US$10 million (annual interest payments).

• The loan is priced at US Dollars LIBOR +1.5%. The LIBOR base will be reset each year on an agreed upon date.

• Although LIBOR is floating the spread of 1.5% is a fixed component of the loan.

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Bharti Instrument’s Interest Rate Risk

• Bharti will not know the actual cost of the loan until the loan has been completely repaid. Although the treasury managers may forecast what LIBOR will be for the life of the loan, they will not know with certainty until all payments have been calculated.

• This uncertainty is not only an interest rate risk but also an actual cash flow risk associated with the interest payment.

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Management of Interest Rate Risk through FRA’s

• A forward rate agreement (FRA) is an interbank-traded contract to buy or sell interest rate payments on a notional principal.

• These contracts are settled in cash.

• The buyer of an FRA obtains the right to lock in an interest rate for a desired term that begins at a future date.

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Management of Interest Rate Risk through FRA’s

• The contract specifies that the seller of the FRA will pay the buyer the increased interest expense on a nominal sum (the notional principal) of money if interest rates rise above the agreed rate, but the buyer will pay the seller the differential interest expense if interest rates fall below the agreed rate.

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FRA for Bharti

• Bharti can lock in the first interest payment (due at the end of year 1), by buying an FRA that locks in a total interest payment of 6.5%.

• If LIBOR rises above 5% by the end of year 1, Bharti would receive a cash payment from the FRA seller that would reduce the interest rate to 5%.

• If LIBOR were to fall during the year below 5% ( not expected by treasury managers at Bharti), Bharti would make a cash payment to the seller of the FRA effectively raising its LIBOR payment to 5% and the total loan payment to 6.5%.

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Example

• A Corporate treasurer plans to raise Rs 10 Crore six months from now for three months. The current 3 month rate is 8% pa. The treasurer wants to hedge himself against a rise in interest rates using a 6/9 FRA that a bank is quoting at 8.1% pa. What will be effective cost for the firm if the three month MIBOR rate six months later is

(i) 8.5%(ii) 7.5%

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Meaning of 6X9 & 3X6

• The phrase “6 vs. 9” refers to a 3-month interest rate observed 6 months from the present, for a security with a maturity date 9 months from the present.

• Similarly “3 vs 6” refers to a 3 month interest rate observed 3 months from the present for a security with a maturity date 6 months from the present.

• What does 3 vs 12 FRA mean?

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Solution( When interest rate rises)

• The treasurer buys the 6/9 FRA at 8.1 percent per annum.

• If the three month MIBOR rate six months later is 8.5% the bank compensates the firm for the difference in interest rates.

= (.085-.081)*10 Crores*90/360= Rs100,000Since the actual payment is after maturity

period three months discounted value of Rs 100,000 @ 8.5% changes hands i.e. Rs 97919 (which when invested @8.5% = Rs100,000)

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Solution

• If Firm borrows Rs 10 Crores at 8.5% for three months.

=(0.085)*10Crores*90/360.=Rs 2,125,000

Net cost = 2,125,000-100000=2,025,000.

=(2,025,000)/10Cr* 360/90=8.1% pa.

Hence company has locked the cost at 8.1%.

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Solution ( When the interest rate falls to 7.5%)

• If the three month MIBOR six month later is 7.5%, the firm compensates the bank for the difference in interest rates.

• The difference is (8.1-7.5)%.

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Forward Rate Agreementcontinued

• An FRA is equivalent to an agreement where interest at a predetermined rate, RK is exchanged for interest at the market rate

• An FRA can be valued by assuming that the forward interest rate is certain to be realized

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FRA’s Revised

• A forward rate agreement (FRA) is an interbank-traded contract to buy or sell interest rate payments on a notional principal.

• These contracts are settled in cash.• The buyer of an FRA obtains the right to

lock in an interest rate for a desired term that begins at a future date.

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FRA’s Revised

• The contract specifies that the seller of the FRA will pay the buyer the increased interest expense on a nominal sum (the notional principal) of money if interest rates rise above the agreed rate, but the buyer will pay the seller the differential interest expense if interest rates fall below the agreed rate.

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Forward Rate Agreements

• For an FRA which agrees that a financial institution will earn an interest rate RK for the period of time between T1 and T2 on a principal L

RK : The rate of interest agreed to in FRA.

RF: The forward LIBOR interest rate between times T1and T2

RM:The actual LIBOR interest rate observed between time T1and T2.

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Forward Rate Agreements

• Assuming discrete compounding (i.e. T2-T1 =0.25 for s.a. compounding etc).

• Normally a company X would earn RM from a LIBOR loan (investment).

• The FRA means it will earn RK

• The extra interest earned due to FRA will be RK-RM.

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Forward Rate Agreements• The cash flow to company X will be: L(RK-RM)(T2-T1).• To the company Y borrowing money from X

the cash flow will be: L(RM-RK)(T2-T1).Thus we can also say that FRA is an agreement

where company X receive interest on principal between T1 and T2 at the fixed rate RK and pay interest on the principal at the realized market rate of RM .

Similarly company Y will pay interest on principal between T1 and T2 at the fixed rate of RK and receive an interest of RM

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Payoffs for X and Y from FRAs

• Usually FRAs are settled at time T1 rather than T2. If settled at T1 they are discounted from T2 to T1 at the market rate RM.

Payoff for X:

{L(RK-RM)(T2-T1)}/(1+RM)(T2-T1)

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Valuing FRAs

• Assuming discrete compounding such that rates of interest are measured with a compounding frequency equal to their maturity the cash flows at two time periods will be:

Time T1= -L

Time T2=+L[1+RK(T2-T1)]

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Valuing FRAs

• Remember K is delivery price in a forward contract &

F0 is forward price that would apply to the contract today

• The value of a long forward contract, ƒ, is ƒ = (F0 – K )e–rT

• Similarly, the value of a short forward contract is

(K – F0 )e–rT

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Valuing FRAs

• Compare the two FRAs. • The first promises that the forward rate

RF will be earned on a principal of L between times T1 and T2. The second promises that RK will be earned between the same two dates.

• The contracts are the same except for the payments received at time T2.

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Valuing FRAs• The excess of second contract over the

first is therefore the present value of the difference between these interest payments or,

L(RK-RF)(T2-T1)e-R2

T2 where R2 is the

continuously compounded riskless zero rate for maturity T2.

• As the value of FRA promising RF is zero the value of FRA promising RK is

V= L(RK-RF)(T2-T1)e-R2

T2

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Example

• Suppose that the three month LIBOR rate is 5% and the 6 month LIBOR is 5.5% with continuous compounding. Consider an FRA where we will receive a rate of 7% measured with quarterly compounding on a principal of Rs1million between the end of month 3 and the end the month 6. What is the value of the FRA? The risk less zero rate is 5.5%for six months.

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Solution

• The forward rate is 6% with continuous compounding or 6.0452% with quarterly compounding. (refer slide no. 24).

• The value of the FRA is

1000000*(0.07-.060452)*0.25*e-0.055*0.5 =Rs2322

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Theories of the Term Structure

• Expectations Theory: forward rates equal expected future zero rates.

• Market Segmentation: short, medium and long rates determined independently of each other.

Conjectures that there need be no relationship between short medium and long term interest rates.

• Liquidity Preference Theory: forward rates higher than expected future zero rates. Asserts that the investors prefer liquidity and invest their funds for short term whereas borrowers prefer fixed interest rates for a long period of time.

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Conclusions• If the interest rates offered by Banks and FIs

follows the first theory long term interest rates would equal the average of expected future short term interest rates.

• If not given incentives deposits would only be short term and borrowings would be long term causing excessive interest rate risks.

• Thus most FIs and banks raise long term interest rates relative to expected future short term interest rates.

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Conclusions

• This strategy reduces the demand for long term fixed rate borrowings and encourages investors to deposit for long term.

• The last theory leads to a situation in which forward rates are greater than expected future zero rates.

• It is also consistent with the empirical result that yield curves tend to be upward sloping more often they are downward sloping.

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• Duration of a bond that provides cash flow c i at time t i is

where B is its price and y is its yield (continuously compounded)

• This leads to

tceBi

i

ni

yti

1

yDB

B

Duration

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How?

ΔB=dB/dyΔy for small change in y.

Using the bond pricing relationship

B= Σ cie-yti substituting dB/dy we get

ΔB= -BDΔy.

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Example

• Calculate the duration for a 3 year 10% coupon bond with a face value of 100. The yield on the bond is 12% per annum with continuous compounding. The coupons are paid every six months.

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Calculation of Duration

Time CF PV Wt. T * Wt

0.5 5 4.709 0.050 0.025

1.0 5 4.435 0.047 0.047

1.5 5 4.176 0.044 0.066

2.0 5 3.933 0.042 0.083

2.5 5 3.704 0.039 0.098

3.0 105 73.256 0.778 2.333

Total 130 94.213 1.000 2.653

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Spreadsheet Solution

• Duration Example.xls

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Understanding Duration

• For the bond considered the B=94.213 and D=2.653. Therefore

ΔB= -94.213 * 2.653Δy orΔB=-249.95 Δy thus when the yield

increases by 0.1% the bond price goes down to 93.963.

The same can be checked by using the conventional formula for price of the bond.

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Duration Continued• When the yield y is expressed with

compounding m times per year

• The expression

is referred to as the “modified duration”

my

yBDB

1

D

y m1

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Example

• What is the modified duration of the bond with price 94.213 and duration 2.653. The yield with semiannual compounding is 12.3673%.

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Solution

• D=(2.653)/(1.123673/2) =2.499.

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Situation

• Portfolio A consists of a 1 year zero coupon bond with a face value of Rs 2000 and a 10 year zero coupon bond of Rs 6000. Portfolio B consists of a 5.95 year zero coupon bond with a face value of Rs 5000. The current yield on all bonds is 10% perannum.

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Question

1.What is the duration of both portfolios?

2. What is the percentage change in value of each portfolio if yield goes up by 0.1% per annum and 5% for both?

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Answer 1

Duration of portfolio A= weighted average of durations.

(Dur. of B1* Price of B1+ Dur. of B2*Pr. Of B2)/(Price of A + Price of B

Duration of Second portfolio is 5.95 years.

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Answer 2

• Discount the price of each bond by .101 instead of 0.1 and compare with values in previous slide.

• Percentage decrease in value is .59%• Repeat the process for 5% values.