Question

A company offers a bond with exactly 100 years to maturity (now 2000 to 2100) and its face value $2,000 that makes coupon payments at the end of each year.

Issue date: 2000; Maturity date: 2100

Coupon rate: 8%

2000 price: $2,000

2005 price: $1500

2010 price: $2,020.5

Q1: What is the interest rate in 2005 in market?

Q2: Sam bought this bond in 2005(yr) right after it made its coupon payment (he did not collect the coupon). Bob bought another bond at the same time with the same coupon rate, issue date and face value but a maturity date of 2015. Bob did not collect the coupon in 2005 either. It is 2010 now, if they were to sell their bonds today who would make more money?

Answer 1

Solution to Q1:

In case of bonds, the interest rate in the market is used to discount the cash flows of the bond to its present value. This present value is the price of the bond. Therefore, logically, if the bond is trading below its face value, this implies that the market interest rate might have been higher than the coupon rate and when such higher rate is used to discount the cash flows, the price of the bond results in a value lower than the face value.

Here, in 2005 the bond is trading at $1500. This means that surely the interest rates in the markets are higher than the 8% coupon rates.

We can calculate the market rate using a financial calculator.

There are still 95 years to maturity, hence 95N, -1500 PV, 160PMT, 2000FV, CPT I/Y, we get 10.67%.

Solution to Q2:

We can solve it logically, just by knowing a theme which is relevant for bonds.

That is, Long term bonds are more volatile to changes in interest rates.

Therefore, the bond with 100 years of maturity still has 90 years left as on 2010 while the bond that Bob purchased is relatively short term 5 more years to maturity. Therefore the short term bond would not be affected much with the changes in interest rates.

Here we can see that the bond prices increased from the year 2005 to 2010 which implies that market interest rates must have fallen, and this effect must have had a much higher impact on the long term bond and its price in 2010 increased to $2020.50.

The price of the bond that Bob purchased must have also gone up but not that much as that of the long term bond.

Therefore, if bonds have to be sold today, Sam will earn more by selling the bond at a higher rate than Bob.