Please help on the following question.A 5-year 8% coupon bond is trading for $1144.75 and a 5-year 4% coupon bond is trading for $968.83. coupons are paid annually and the face value is $1000 for both bonds. Can you compute the price of a 5-year zero coupon $1000 face value bond from the above information above? What is the 5-year zero coupon yield?Thanks

Hi ninogmiranda,I got 792.91 as price of the Zero coupon bond and the yield as 4.7% or 4.75% depending on semi-annual/annual compounding assumed.The workings are just simple arithmetic under cetain assumptions. These you can check in the sheet below.Manu

QuoteOriginally posted by: chiranjivHi ninogmiranda,I got 792.91 as price of the Zero coupon bond and the yield as 4.7% or 4.75% depending on semi-annual/annual compounding assumed.The workings are just simple arithmetic under cetain assumptions. These you can check in the sheet below.ManuI agree, provided you can assume that the yield to maturity (YTM) of the 8% coupon bond is the same as the yield to maturity of the 4% coupon bond is the same as the yield to matuirty of the zero coupon bond. It's not clear that this is a valid assumption. Given the spot rate curve, one backs out the bond yield to maturity (effectively, an average yield, given the cash flows, that reproduces the price of the bond). Assume the YTM of the 8% coupon bond is YTM8 and that of the 4% coupon bond is YTM4. Redo the algebra, do you come out with the same YTM (=YTM8 = YTM4)?

IS the ytm not clear once you specify the price, coupon and maturity? can't see how the algebra won't work.

QuoteOriginally posted by: marajeshIS the ytm not clear once you specify the price, coupon and maturity? can't see how the algebra won't work.Define the price of the bond with coupon r, YTM y and time to expiration T, face value F as P(r,y,T, F). Then P(r,y,T, F) = (rF/y)*{1 - (1/(1+y)^T)} + F/(1+y)^T. Specifically, P(r,YTM4,T, F) = (rF/YTM4)*{1 - (1/(1+YTM4)^T)} + F/(1+YTM4)^T and P(2r,YTM8,T, F) = (2rF/YTM8)*{1 - (1/(1+YTM8)^T)} + F/(1+YTM8)^T . 792.91 = 2*P(r,YTM4,T, F) - P(r,YTM8,T, F) = F/(1+YTM)^T, if YTM4 = YTM8 = YTM ! Given the spot rate curve, a series of cash flows can be priced, uniquely (implying a unique YTM). If that price is not equal to the market price, a spread is added to the spot rates to reproduce the market price. That spread will depend upon coupon maturity and market price. For different bonds, different spreads will result - consequently different YTMs will result. YTM is not the same as a spot rate!! Its difficult to write equations here so I can't be as specific as I 'd like.

What i was thinking was:Assuming the bond is of same credit/liquidity type characteristics, (say both are US Govt notes) then it is probably safe to assume that they share the same spot term structure. (the coupon strips are fungible in US bonds)When each cashflow is discounted using its own spot rate you get a price which is given in this problem. Given the market's need to think in terms of a single yield number it averages and summarises it as YTM (a number given the bond characteristics, does not give any more information than the price). Now a no arb condition for these bonds would imply they should have the same discount factors for cashflows ocurring at the same period, so while the YTMs can (and in general will be) different, the implied 10 year spot rate will not be. The algebraic manipulation essentially eliminates all cashflows except the last one, essentially turning the combined portfolio to a 10 year strip. So you should be able to infer the 10 year rate from that independent of what the YTMs of these bonds were.May be I am missing something which of course would not surprise me.

Agree with marajesh.Actually the assumption I am making is that the discount factors are the same..which, as marajesh pointed out, depend on credit/liquidity risk.As a matter of fact the YTM of the 8% coupon bond is 4.69%, the 4% coupon bond YTM is 4.71% and given the price I calculated for the zero coupon bond, its YTM is 4.75% (All on annual basis)These YTM's are a good example of the coupon effect when higher coupons imply lower YTMs and vice-versa. This is the main reason why discount factors extracted from the spot curve, rather than YTM are used for pricing bonds

Sell the 8% coupon bond, and buy 2 times the face value of the 4% coupon bond.Net amount paid = 2 x 968.33 - 1144.75 = 792.91.My net coupon (regardless of frequency) is zero.At maturity I receive 2000-1000 = 1000, meaning I've syntheticically replicated a 5yr zero coupon bond. The price is 792.91.As for the implied yield, this is solely dependent on how you want to quote it (eg 4.75% if annual compounding, 4.668% if quarterly compounding, 4.64% if continuous compounding).My only assumption is funding and investment rates being equal here.