Bonds and Force of Interest?

Question

Studying for FM/2 and ran into this problem dealing with bonds;A 1,000 par value 3 year bond with annual coupons of 50 for the first year, 70 for the second year, and 90 for the third year is bought to yield a force of interest$$\\delta_t=\\frac{2t-1}{2(t^2-t+1)}, t\\ge0.$$Calculate the price of the bond.So, $\\delta_1=\\frac{1}{2}, \\delta_2=\\frac{1}{2},$ and $\\delta_3=\\frac{5}{14}.$ And now, where do I go from here?

Answer 1

I don't know where this term "force of interest comes from".

If the bond is priced at Par then

1000 = Present value of Coupons plus principal. Let d= discount factor = (1/(1+ i)) where i= yield to maturity.

then 1000 = 50d +70d² +90d³+1000d³.=P

We solve for d and, therefore, i iteratively. The coupon, on average is 7%. That is a good place to start. If i = .07 the d = .93458 and P =997.63. With a YTM of 7% the bond would trade at a discount to par as the cash flows are back-ended.

Price the bond again at 7.01%. In this case the bond price would fall to 997.37. so the bond price sensitivity to a 1 basis point change in yield is 997.37-997.63. or about .-.26.

we want the yield that values the cash flows at PAR (1,000).. We need the PV to rise by 2.37 points. we could take a better guess now. if a one basis point change gets us .26, 9 will get us about 2.34 (if the DV01 is constant which it is not). So lets try a yield of 6.91%

When i= 6.91 d= .93536 and the PV of the cash flows is 1000.025 . This is pretty close and we can keep repeating the process until we are as close to 1000 as we would like.

I hope this helps.

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