LIBOR Case

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Interest Rates

Chapter 4

Types of Rates

Treasury rate

LIBOR

Fed funds rate

Repo rate

Treasury Rates

Rates on instruments issued by a government in its own currency

LIBOR

LIBOR is the rate of interest at which a AA bank can borrow money on an unsecured basis from another bank

For 10 currencies and maturities ranging from 1 day to 12 months it is calculated daily by the British Bankers Association from submissions from a number of major banks

There have been some suggestions that banks manipulated LIBOR during certain periods. Why would they do this?

The U.S. Fed Funds Rate

Unsecured interbank overnight rate of interest

Allows banks to adjust the cash (i.e., reserves) on deposit with the Federal Reserve at the end of each day

The effective fed funds rate is the average rate on brokered transactions

The central bank may intervene with its own transactions to raise or lower the rate

Similar arrangements in other countries

Repo Rates

Repurchase agreement is an agreement where a financial institution that owns securities agrees to sell them today for X and buy them bank in the future for a slightly higher price, Y

The financial institution obtains a loan.

The rate of interest is calculated from the difference between X and Y and is known as the repo rate

LIBOR swaps

Most common swap is where LIBOR is exchanged for a fixed rate (discussed in Chapter 7)

The swap rate where the 3 month LIBOR is exchanged for fixed has the same risk as a series of continually refreshed 3 month loans to AA-rated banks

OIS rate

An overnight indexed swap is swap where a fixed rate for a period (e.g. 3 months) is exchanged for the geometric average of overnight rates.

For maturities up to one year there is a single exchange

For maturities beyond one year there are periodic exchanges, e.g. every quarter

The OIS rate is a continually refreshed overnight rate

The Risk-Free Rate

The Treasury rate is considered to be artificially low because

Banks are not required to keep capital for Treasury instruments

Treasury instruments are given favorable tax treatment in the US

OIS rates are now used as a proxy for risk-free rates

Measuring Interest Rates

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

The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers

Impact of Compounding

When we compound m times per year at rate R an amount A grows to A(1+R/m)m in one year

Compounding frequency	Value of $100 in one year at 10%
Annual (m=1)	110.00
Semiannual (m=2)	110.25
Quarterly (m=4)	110.38
Monthly (m=12)	110.47
Weekly (m=52)	110.51
Daily (m=365)	110.52

Continuous Compounding (Pages 86-87)

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

$100 grows to $100eRT when invested at a continuously compounded rate R for time T

$100 received at time T discounts to $100e-RT at time zero when the continuously compounded discount rate is R

Conversion Formulas (Page 87)

Define

Rc : continuously compounded rate

Rm: same rate with compounding m times per year

Examples

10% with semiannual compounding is equivalent to 2ln(1.05)=9.758% with continuous compounding

8% with continuous compounding is equivalent to 4(e0.08/4 -1)=8.08% with quarterly compounding

Rates used in option pricing are usually expressed with continuous compounding

Zero Rates

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

Example (Table 4.2, page 88)

Maturity (years)	Zero rate (cont. comp.
0.5	5.0
1.0	5.8
1.5	6.4
2.0	6.8

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

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% (cont. comp.)

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

Par Yield continued

In general if m is the number of coupon payments per year, d is the present value of $1 received at maturity and A is the present value of an annuity of $1 on each coupon date

(in our example, m = 2, d = 0.87284, and A = 3.70027)

Data to Determine Treasury Zero Curve (Table 4.3, page 90)

Bond Principal	Time to Maturity (yrs)	Coupon per year ($)*	Bond price ($)
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

* Half the stated coupon is paid each year

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

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%

Zero Curve Calculated from the Data (Figure 4.1, page 91)

Zero Rate (%)

Maturity (yrs)

10.127

10.469

10.536

10.681

10.808

Application to OIS Rates

OIS rates out to 1 year are zero rates

OIS rates beyond one year are par yields,

Forward Rates

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

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

This formula is only approximately true when rates are not expressed with continuous compounding

Application of the Formula

Year (n)	Zero rate for n-year investment (% per annum)	Forward rate for nth year (% per annum)
1	3.0
2	4.0	5.0
3	4.6	5.8
4	5.0	6.2
5	5.5	6.5

Upward vs Downward Sloping Yield Curve

For an upward sloping yield curve:

Fwd Rate > Zero Rate > Par Yield

For a downward sloping yield curve

Par Yield > Zero Rate > Fwd Rate

Forward Rate Agreement

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

Forward Rate Agreement: Key Results

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

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

This means that the value of an FRA is the present value of the difference between the interest that would be paid at interest rate RF and the interest that would be paid at rate RK

FRA Example

A company has agreed that it will receive 4% on $100 million for 3 months starting in 3 years

The forward rate for the period between 3 and 3.25 years is 3%

The value of the contract to the company is +$250,000 discounted from time 3.25 years to time zero at the OIS rate

FRA Example Continued

Suppose rate proves to be 4.5% (with quarterly compounding

The payoff is –$125,000 at the 3.25 year point

Often the FRA is settled at tiem 3 years for the present value of the known cash flow at time 3.25 years.

Theories of the Term Structure Pages 97-98

Expectations Theory: forward rates equal expected future zero rates

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

Liquidity Preference Theory: forward rates higher than expected future zero rates

Liquidity Preference Theory

Suppose that the outlook for rates is flat and you have been offered the following choices

What would you choose as a depositor? What for your mortgage?

Maturity	Deposit rate	Mortgage rate
1 year	3%	6%
5 year	3%	6%

Liquidity Preference Theory cont

To match the maturities of borrowers and lenders a bank has to increase long rates above expected future short rates

In our example the bank might offer

Maturity	Deposit rate	Mortgage rate
1 year	3%	6%
5 year	4%	7%

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