I’ve just begun reading up on bonds for Level I and had a few simple questions regarding that the texts didn’t provide enough detail on. 1. “Discount bonds, such as zero coupon bonds, face higher interest rate risk.” I understand the logic behind this, but could someone provide the mathematical reasoning to help be better understand it. 2. “For a given change in interest rates, price sensitivity is lower when the level of interest rates in the market is high; price sensitivity is higher when the level of interest rates is low.” I’m having a tough time understanding this and all explanations would be very much appreciated. Thanks
- time value of money is the reason. of the total payments due to you as the investor, with coupon bonds, since a portion is returned sooner, a change in interest rate has less impact on payments you will receive in the early years (or no impact on payments you have already recieved!) But when the whole return (interest plus principal) will be received at the end, an increase in interest rates will discount the total payment to be received much more, in relative terms. 2. just look at a graph of a bond’s price/yield relationship. Duration is linear, but the price/yield curve is actually convex. As interest rates increase, price decreases, but at a declining rate of change. When rates are high to begin with, you are receiving high coupon payments to reflect the higher interest rates. So, again, think of the total compensation the investor will receive over the life of the bond (Coupon plus Principal at the end). Since you are getting a relatively greater proportion of your total compenstion earlier in the life of the bond, a change in interest rates after issuance has less impact on the value of what you will receive. I hope this makes some sense. You are asking a thoughtful question that indicates you are trying to learn this, not just memorize for exam purposes. keep it up and you will pass!
i like how cfastudent answered both of your questions, especially, the second one. Here is mathematical reasoning for higher duration for discount bonds. Bond price = sum(Fi/(1+r)^i: i = 1,T) d Bond price/ dr = sum(-Fi*i/(1+r)^(i+1)) = -(1/(1+r))*sum(Fi*i/(1+r)^i) duration = -(d Bond price/dr)/Bond price = (1/(1+r))*sum(i*(Fi*i/(1+r)^i)/sum(Fi/(1+r)^i)) if we denote weights wi = (Fi/(1+r)^i)/sum(Fi/(1+r)^i) -> sum(wi) = 1 duration = (1/(1+r))*sum(i*wi) <= (1/(1+r))*T*1 -> highest for pure discount bond.