bond value using forward-rates-calculated spot rate

CFA Level I
#1

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+S2)2=(1+S1)*(1+1y1y), which means that S2=[(1+S1)*(1+1y1y)]1/2-1

On page 55 there is this example:

“The current 1-year rate, S1, 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+S1)+50/[(1+S1)*(1+1y1y)]+1,050/[(1+S1)*(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: S3=(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!

#2
#3

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.

#4

but why?

#5

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.

#6

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

#7

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+S1)*(1+1y1y)]

and since

(1+S2)2 = (1+S1)*(1+1y1y)

this is equivalent to 50/[(1+S2)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. S1=S2=S3), 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.

#8

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