Par yield from Spot Rates.

[RaamZen]

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]

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

 

[RaamZen]

Thank you, Jayanthi.

 

[leenaabbasali]

A = e^-0.05*0.5 + e^-.06*1.0 + e^-0.065*1.5 + e^-0.07*2.0 = 3.693535

how could we find A , straight from BA II plus ?

 

[David Harper CFA FRM]

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,