LOS 53.g: Define and compare the spot curve, yield curve on coupon bonds, par curve, and forward curve.

CFA Level I Fixed Income
#1

Dear everyone,

I have a question that unable to solve the equation as written in Page 50, Level I, Book 5 in SchweserNotes CFA 2017 Exam Prep:

"Consider a 3-year annual-pay bond and spot rates for one, two, and three years of S1, S2, and S3. The following equation can be used to calculate the coupon rate necessary for the bond to be trading at par.

[PMT / (1+S1)] + [PMT / (1+S2)^2] + [PMT + 100 / (1+S3)^3] = 100

With spot rates of 1%, 2%, and 3%, a 3-year annual par bond will have a payment that will satisfy:

[PMT / 1.01] + [PMT / (1.02)^2] + [PMT + 100 / (1.03)^3] = 100, so the payment is 2.96 and the par bond coupon rate is 2.96%."

I only know how to calculate the one-year par rate, but I don’t know how to calculate the two-year and three-year par rate. How to use BA II Plus calculator to calculate this equation in order to get the answer is 2.96 for payment?

Appreciate if anyone could guide me and and show me the steps to calculate the par rates for the PMT.

Thanks a lot.

#2

In general, you can’t do it on your calculator.

If you have only one period, it’s trivial (you’re solving a linear equation), and if you have two periods it’s almost trivial (you’re solving a quadratic equation). For three and four periods you can, theoretically, solve for the payments (using, respectively, Ferrari’s and Cardano’s formulae for solving, respectively, cubic and quartic equations), but it’s well beyond what you learned in algebra, and would be a pain in the neck on a calculator. With five or more periods, you’re out of luck: Galois proved that quintic and higher order equations cannot be solved in general using integer powers and roots.

In general, you solve these problems numerically, using something like Goal Seek or Solver in Excel.

Fortunately, you don’t have to do anything of the sort on the CFA exams. I’d suggest putting it out of your mind immediately.

I wrote an article on the spot, par, and forward curves that may be of some help: https://www.financialexamhelp123.com/par-curve-spot-curve-and-forward-curve/

Full disclosure: as of April 25, 2016, there is a charge to read the articles on my website. The article above is one of the free sample articles I have available so that prospective customers can see what they’re getting.

#3

Dear S2000magician,

Thanks for your suggestions. I saw you managed to help out the algebra of the following question. Perhaps you could teach me how to calculate the bold wording part as highlighted below:

100 = [PMT / (1.05263)^1] + [(PMT + 100) / (1.05616)^2]

100 = [PMT / (1.05263)^1] + [PMT / (1.05616)^2] + [100 / (1.05616)^2]

100 = 0.95(PMT) + 0.89648(PMT) + 89.648

10.352 = 1.84648(PMT)

PMT = 5.606

I tried to press “y^x” in the calculator to calculate 1.05263^1, it equals to 1.05263 instead of 0.95; and 1.05616^2 = 1.115474 instead of 0.894648.

Thanks for your kindness and considerations. Thanks very much!

#4

I did so in the other thread.

Merry Christmas!

#5

Hi, I got stuck on the same page on the same problem. Were you able to solve it manually without the calculator? I want to know the working math behind it, it’d be great if I can get some help.