Question

Part a.

Explain why bond price and interest rates are negatively related. What is the role of coupon rate and term to maturity in this relationship?

                                                                                                                     

Part b.

Consider a 3-year bond with 14 percent semi-annual coupon payments and currently priced to yield 12 per cent per annum.

1. Calculate duration of the bond.
2. What is the percentage of price change if interest rate increases to 12.15% per annum?

Answer 1

Part a.) Price of a bond is equal to the present value of future cash flows discounted. Simply speaking it refers to the sum of the present values of all likely coupon payments plus the present value of the par value at maturity discounted by yield of simillar bonds with exact same features. It given by:

This equation shows that

• There is an inverse relationship between price and yield.
• This relationship is convex to the origin i.e.
  • When r falls , price of the bond rises at higher rate.
  • When r rises, price of the bond falls at lower rate.

Since it is mathametically dificult to deal with the curve we a draw a tangint on it. Pice volatility is the slop of the tangint.

Step: 1. Relationship between price of a bond and Interest rates.

Price of the bond and the interest rate are inversely related to each other because yield of bond is strongly related to prevailing interest rates in the market. When the interest rates in the market rises or falls the investor would now require higher or lower yield on the bond therefore fall or rise in the price of the bond.

For Example: If a zero-coupon bond is trading at $950 and has a par value of $1,000 (paid at maturity in one year), the bond's rate of return at the present time is 5.26%. If current interest rates were to rise, where newly issued bonds were offering a yield of 10%, then the zero-coupon bond yielding 5.26% would be much less attractive. The effect of such rise in interest rates would be on preious bond would be that price from $950 (at 5.26% ytm) will fall to approximate value of $909.09 (at 10% ytm).

Step: 2. Relationship between Coupon rates and term to maturity on the bond.

Coupon rates: Coupon rates have strong impact on the price of the bond i.e. if coupon rate on the bond are higher than the prevailing market interest rates price of the bond will rise and if coupon rate on bond are lower than the prevailing market interest rates price of the bond will fall.

For example a Bond A with 14% coupon rate maturing in 5 years have the same yield of 12% as the Bond B with 11% coupon rate with the same maturity. Bond A will higher price than the Bond B because the varriable factor of coupon rate is higher than the yield of 12%.

Term to Maturity: Bond and the term the term to maturity have the inverse relationship between each other i.e. the time taken to repay the value on the bond will effect more negatively for higher delay. Bonds with higer term to maturity also have the risk of rising inflation in between which will increase yield therefore decrease in the price of the bond.

For example we have two ZCB with same face value of $100 and same YTM of 10% but the term to maturity on Bond A is 5 years and on Bond B is 7 years. Bond A price ($ 62.07) would be higher than the Bond B ($ 51.32).

Part b.)

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