Par yield from Spot Rates.

RaamZen

New Member
Hello

This is in regards to explanation on Hull 04.13 question on the par yield calculation from sets of upward sloping spot rates.

Suppose that the 6-month, 12-month, 18-month, and 24-month zero rates are 5.0%, 6.0%, 6.5%, and 7.0%, respectively. What is the 2-year par yield?

Answer was:
Using the notation in the text, m = 2, d = e^-0.07x2 = 0.8694. Also A = e^-0.05x0.5 + e^-0.065x1.5 + e^-0.07x2.0 = 3.6935 The formula in the text gives the par yield as (100 – (100 x 0.8694) x 2)/3.6935 = 7.072

How do you come to arrive to this formula.
I am bit confused.
Can you pls explain?

Thanks,
 

Dr. Jayanthi Sankaran

Well-Known Member
Hi @RaamZen,

The par yield is the coupon rate that causes the Bond price = par value. Usually, the bond is assumed to provide semi-annual coupons. If the coupon rate is c per annum, then c/2 is the coupon rate every six months.

Using the zero spot rates above, the value of the bond is equal to its par value of $100 when:

(c/2)*e^-0.05*0.5 + (c/2)*e^-.06*1.0 + (c/2)*e^-.065*1.5 + (100 + c/2)*e^-.07*2 = $100

(c/2)*0.975310 + (c/2)*0.941765 + (c/2)*0.907102 + (c/2)*0.869358 = $100 - $100*e^-.07*2 = $100 - $86.935824

(c/2)*[0.975310 + 0.941765 + 0.907102 + 0.869358] = $13.064176
(c/2)*[3.693535] = $13.064176
(c/2) = $13.064176/3.693535 = 3.537039
c = 3.537039*2.0 = 7.074077%

The formula in the text is as follows:

c = (100 - 100*d)*m/A
where d = present value of $1 received at the maturity of the bond, A = value of an annuity that pays one dollar on each coupon payment date and m is the number of coupon payments per year, then the par yield c must satisfy

100 = A*c/m + 100*d
In the above example, m = 2, d = e^-.07*2= 0.869358 and
A = e^-0.05*0.5 + e^-.06*1.0 + e^-0.065*1.5 + e^-0.07*2.0 = 3.693535
Hence c = (100 - 100*0.869358)*2/3.693535 = 7.074090%

Hope that helps:)
Thanks!
Jayanthi
 

David Harper CFA FRM

David Harper CFA FRM
Subscriber
HI @leenaabbasali I don't think you can because (i) the above assume continuous compound frequency and (ii) the spot rate curve is not flat. If, on the other hand, we were to assume a flat spot rate curve at 6.50% per annum with semi-annual compounding, then we could solve for the present value of $1.00 (which is what A represents; A is the sum of discount factors) with:
2*2 = 4 = N
6.5 / 2 = 3.25 = [I/Y]
1 = PMT (per the definition of A)
0 = FV
and CPT PV = 3.695 which turns out to be very near to the 3.6935
(I picked 6.50% because it's near to the 7.0%. The yield is a weighted average of spot rates but the final 7.0% has the most weight, so if I had to guess, I go to the next-highest spot rate). Thanks,
 
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