Chapter 4: Interest Rates

Question 1

What is a zero rate and how is it calculated?

Answer 1

The n-year zero-coupon interest rate is the rate of interest earned on an investment that starts today and lasts for n years. All the interest and principal is realized at the end of n years and there are no intermediate payments.

If we have a bond with the following characteristics: principal = $100, T - t = 6 months, coupon = 0, and price = $98 then the zero rate is calculated as:

If the bond pays coupons the calculation of the zero rate is different, because some of the return on the bond is realized in the form of coupons prior to expiration.

Question 2

How is a bond priced?

Answer 2

The theoretical price of a bond can be calculated as the present value of all the cash flows that will be received by the owner of the bond. Sometimes bond traders use the same discount rate for all the cash flows underlying a bond, but a more accurate approach is to use a different zero rate for each cash flow.

Consider the example with a bond with a principal of $100 and providing coupons of 6% per annum semi-annually:

Question 3

What is a bond’s yield?

Answer 3

A bond’s yield is the single discount rate that, when applied to all cash flows, gives a bond price equal to its market price.

Question 4

What is the bootstrap method and how is it used?

Answer 4

The bootstrap method is a way to determine Treasury zero rates from Treasury bills and coupon- bearing bonds.

Assume we’re given a table with 4 bonds where the two with the shortest time to maturity doesn’t pay coupons, but the third and fourth bond dies, The zero rate for the two first bonds can easily be calculated (as per the previous flashcard), but to determine the zero rate for the third and fourth bond we need to use the bootstrap method.

So if we have a bond with the following characteristics: principal = $100, T - t = 18 months, annual coupon = $6.2 with semiannual payments and bond price = 101 it’s valued the following way:

Question 5

What is a forward rate and how is it calculated?

Answer 5

Forward interest rates are the future rates of interest implied by current zero rates for periods of time in the future. So if we have a forward rate of 3.5% for year 2 this means that 3.5% is the implied interest rate by the zero rates for the period of time between the end of the first year and the end of the second year. It can be calculated from the 1-year zero interest rate of 3% per annum and the 2-year zero interest rate of 4% per annum. It is the rate of interest for year 2 that, when combined with 3% per annum for year 1, gives 4% overall for the 2 years.

The forward rate can be calculated using the formula:

Question 6

What is a forward rate agreement (FRA) and how is it valued?

Answer 6

A forward rate agreement (FRA) is an over-the-counter transaction designed to fix the interest rate that will apply to either borrowing or lending a certain principal during a specified future period of time.
If the agreed fixed rate is greater than the actual LIBOR rate for the period, the borrower pays the lender the difference between the two applied to the principal. If the reverse is true, the lender pays the borrower the difference applied to the principal.

Notation:

R_K = The fixed rate of interest agreed to in the FRA
R_F = The forward LIBOR interest rate for the period between times T(1) and T(2), calculated today.
R_M = The actual LIBOR interest rate observed in the market at time T(1) for the period between times T(1) and T(2).
L = The principal underlying the contract.
R_2 = The continuously compounded riskless zero rate for a maturity T(2)

Valuation formulas are shown below: