# forward rates

Based on the following forward rates, compute the expected yield on a 5-year bond, in 1 year. Spot Rates 1 yr 4.0 2 yr 4.6 3 yr 4.9 4 yr 5.1 5 yr 5.3 6 yr 5.4 7 yr 5.5 a. 2.48% b. 5.63% c. 5.40% d. 5.68% - Dinesh S

C?

I thought D (1.04) * x^5 = 1.054^6 r1 r6 x = 1.0568 D

D?

cpk, you are correct… (1 + s6)^12 = (1 + s1)^2 * (1 + 5f1/2)^10 Annual Rate = 5.68 = D - Dinesh S

Given everything was annual, I do not know if we should approach as semi-annual. Answers are pretty close, anyways. Is it convention, that unless stated “Annual” go and use the semi-annual convention? A doubt, that I’d like to clarify. CP

i see my mistake. thanks!

Yes, It’s always Semi-Annual unless otherwise specified. Even zero coupon bonds are to be valuated semi-annually.

Huh? Exp(r*5) *Exp(0.04*1) = Exp(0.054*6) Exp(r*5) = 1.32843 r = 0.0568

Joey, That is what I had done as well. while Dinesh had solved it with the time * 2, but both answers were the same.

That’s because the more you compound it, the closer you get to continuous compounding.

i’m still having a problem understanding this. isn’t the formula (1.054)^6 / (1.04)^1 ? please advise.

That is correct. CP

CP, That gets me to 1.3182. What do I need to do further to get to the 5.68% answer? Fixed Income is my weakest link.

JoeyDVivre Wrote: ------------------------------------------------------- > That’s because the more you compound it, the > closer you get to continuous compounding. Is this because as the [Limit of (Compounting Frequency) --> 0] the effective yield will increase to a maximum value of [e^(Annual Yield) - 1]??? - Dinesh S

That 1.3182 = (1+r)^5 so find r now. (1+r6)^6 = (1+r1) (1+5f1)^5 and you are trying to find 5f1

I just wanted to clarify one thing. Does it mean that in case we have to find out the present value of a bond, we can either use the 5 year forward rate to discount every coupon payment or we can discount the coupon payments using the respective spot years. Does this make sense?

got it now, thanks for your help and patience buddy. time to review fixed income again tonight.

A quick question: is there a ever a scenario where you would not use (1+r1) on the right side?

it might be 1+r2 if they asked for some other forward rate. In this case, they asked for a compute the expected yield on a 5-year bond, in 1 year. so 1 year, 5 year and 6 year were involved in this transaction. If they had asked for a expected yield on a 4 year bond in 2 years 2 yr 4.6 3 yr 4.9 4 yr 5.1 5 yr 5.3 6 yr 5.4 (1+r2)^2 * (1+4f2)^4 = (1+r6)^6 or 1.046^2 * (1+r)^4 = 1.054^6 solve for r = 5.8%