Question

why discount bond decreased linearly?

Answer 1

A discount bond is a bond that is issued for less than its par/face value ( at a discount).

The yield curve of a bond is a graph between the yield of the bond and maturity. Yield curve can have 3 different shapes, upward sloping (normal yield curve), inverted and flat.

How the bond yield and price reacts to change in interest rate is described by the duration and convexity of the bond. The duration of bond is the bond sensitivity to interet rate changes and convexity describes the interaction between bond price and yield when interest rate changes.

Now the impact of duration and convexity depends on the factors impacting the bond like coupon payments, call/put options, maturity etc.

Here the bond we have is a zero coupon discount bond which is issues at discount to par. Thus there are no coupon payments and only a face value of bond that will be paid kn maturity. Thus there is no volatile component impacting the bond yield and the o ly risk is the interest rate risk impacting only the value of the face value the bond at maturity. As time value of money is the major factor here which has to be considered by the investor while buying and by the issuer while issuing the bond to ensure that the discount on bond should be attractive enough to make up for the time value of money.

Thus as time value of money is the main concern here, the yield of the bond decreases (and price increases) when time decreases ( as we near maturity). Thus yield on discount bonds decrease linearly ( as time value is the main fac6driving the yield curve as the bond does not have components increasing convexity like optionality or even coupon payments or part payments).