Question

A saver goes to a bond trader on 1 January and purchases a bond of face value $100,000 with time to maturity 2 years and which pays a coupon of $10,000 on 31 December on each of the two years. The current market interest rate (the average yield on the collective portfolio and thus the opportunity cost of holding the bond) is 5% per year.

a) Write the equation for the price the saver must pay for the bond in terms of its face value, the coupon payments and its yield.

b) Calculate, showing and briefly explaining your algebraic workings, the price she must pay for the bond and explain its relationship to the "par value" or the face value.

c) Calculate, showing and briefly explaining your algebraic workings, the remaining duration of the bond.

Answer 1

(a) Price of the bond:

We are given the Face value of the bond, time to maturity and yield to maturity (opportunity cost).

The time to maturity of the bond is 2 years. The saver receives coupon at the end of the first year & second year. And additionally he will receive the face value of the bond at the end of the 2nd year along with the coupon payment.

So to know the price to be paid today, we need to calculate the Present Value (PV) of all future cash flows to be received.

The Future cash flow (CF) should be discounted by the yield to maturity / Opportunity cost.

Year 1 CF: Coupon payment (PMT) ==> Has to be discounted by YTM for 1 year

Year 2 CF: Coupon payment (PMT) + Face vaue (FV) ==> Has to be discounted by YTM for 2 years

This can be represented by the below equation.

............................Eqn (1)

(b). Now let us calculate the price

PMT = 10,000, YTM = 5%, FV = 100,000.

Substituting these values in Eqn(1), we get

PV = 9523.8095 + 99773.2426

PV = 109,297.0521

The price to be paid is greater than the Face value that will be received after 2 years. This means the bond is trading at premium.

This is because we receive a coupon rate of 10% (PMT/FV = 10,000/100,000) which is greater than the yield of 5%

(c) Duration of the bond:

We can calculate the remaining duration of the bond using Macaulay Duration. It is the weighted average of the number of years until each of the bond's promised cash flows is to be paid.

The weights will be calculated by taking PV of each CF as a percentage as bond's price (PV)

We know

CF at yr 1 = 10,000 & PV of CF1 = 9523.8095

CF2 at yr 2 = 110,000 & PV of CF2 = 99773.2426

We can calculate the weights by dividing PV_CF of each year by total PV

W1 = 9523.8095 / 109,297.0521 = 0.0871

W2 = 99773.2426 / 109,297.0521 = 0.9129

Macaulay Duration = W1* Year 1 + W2 * Year 2

Remaining Duration = 0.0871 * 1 + 0.9129 * 2 = 1.9129 years

We can calculate Modified Duration to find the approximate percentage change in bond's price for a 1% change in bond's YTM

= 1.9129/1.05 = 1.822

Approximate % change in bond price = -Modified duration * Change in YTM

= -1.822 * 1% = -1.822%, which means for a 1% increase in YTM, the bond price will decrease by 1.8128%