# bond value using forward-rates-calculated spot rate

Hello,

I’m having hard time understanding this concept.

Schweser Book 5 - FIXED INCOME, DERIVATIVES, AND ALTERNATIVE INVESTMENTS, page 52, says that you can calculate 2 year spot rate the following way: (1+S_{2})^{2}=(1+S_{1})*(1+1y1y), which means that S_{2}=[(1+S_{1})*(1+1y1y)]^{1/2}-1

On page 55 there is this example:

*“The current 1-year rate, S _{1}, is 4%, the 1-year forward rate for lending from time = 1 to time = 2 is 1y1y = 5%, and the 1-year forward rate for lending from time = 2 to time = 3 is 2y1y = 6%. Value a 3-year annual-pay bond with a 5% coupon and a par value of $1,000.”*

According to Schweser, the answer is:

bond value = 50/(1+S_{1})+50/[(1+S_{1})*(1+1y1y)]+1,050/[(1+S_{1})*(1+1y1y)*(1+2y1y)]=

= 50/(1.04)+50/[(1.04)*(1.05)]+1,050/(1.04*1.05*1.06)=$1,000,98

however, when I calculate the 3-year spot rate using the forward rates: S_{3}=(1.04*1.05*1.06)^{1/3}-1=4,99968% and I use this 3-year spot rate to calculate the bond’s present value using the calculator, I get a very different number than when doing it using the individudal forward rates, why is that?

On financial calculator: N=3, I/Y=4,99968%, PMT=50, FV=1000, CPT, PV. PV= $1,000,08646

Thank you very much in advance for the explanation!

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Basically, you cannot use the 3-year Spot rate as YTM for coupon-paying bond. Spot rates are discount rates for one-time payments. Since this is a 3-year coupon paying bond, each coupon is discounted at each equivalent spot rate. If it were a 3-year zero coupon bond then you can use the 3-yr sspot rate as YTM.

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but why?

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The coupon payment one year from today is supposed to be discounted at the 1-year spot rate. That’s what a spot rate means: the discount rate for a single (spot) payment at that maturity.

The coupon payment two years from today is supposed to be discounted at the 2-year spot rate.

The coupon payment and par payment three years from today are supposed to be discounted at the 3-year spot rate.

If those three rates aren’t equal, then discounting all of the cash flows at the 3-year spot rate will not give you the correct price today.

Simplify the complicated side; don't complify the simplicated side.

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why do we calculate the implied spot/forward rates then?

Because you discount the each period’s payment at the appropriate spot rate. For example, the present value of the second period’s cash flow is

50/[(1+S

_{1})*(1+1y1y)]and since

(1+S

_{2})^{2 = }(1+S_{1})*(1+1y1y)this is equivalent to 50/[(1+S

_{2})^{2}In other words, you’re simply calculating the PV of each cash flow at its appropriate spot rate and then summing them.

If the yield curve was flat (i.e. S

_{1}=S_{2}=S_{3}), you could use the same rate for all cash flows (this is effectively what you do when you price a bond using the Yield To Maturity). However, since the curve isn’t flat, you use different rates for each cash flow.You keep using that word. I do not think it means what you think it means.Studying With

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