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# Fixed Income -- Par curve algebra

It has been over 25 years since I tool an algebra course and I am stumped (and annoyed at the CFA Institute curriculum for not clarifying). In reading 53, section 4 dealing with par curves, the curriculum lays out equation 13 (100 = PMT/(1+Z1) + PMT/(1+Z2)^2 + ... + (PMT+100)/(1+ZN)^N) and says "the problem is to solve for PMT algebraically." Yeah, no kidding. But it doesn't show its work. It just cavalierly declares what the answer is. From what I can tell, there is no shortcut way to do this like with YTM on the 12C. You just have to brute-force your way through it. I am tempted to blow it off, given that there will probably be at most 1 question on it and I don't want to spend a whole day trying to figure it out. Can someone please show me the algebra? I'm sure it's easy and I'll feel silly for asking. But I am starting to get frazzled and it is affecting my ability to continue studying.

Thanks,

BW
Tagged:

• OK. I figured it out. I was right it was silly and easy. Use the distributive property. All the PMT/(1+Z) terms can be rewritten as PMT x (1/1+Z1) + PMT x (1/(1+Z2)^2) + ... PMT x (1/(1+ZN)^N + 100 x (1/(1+ZN)^N. All the 1/1+Z terms can be grouped so you get PMT ((1/1+Z1) + (1/(1+Z2)^2 + etc.) + 100 x (1/(1+ZN)^N = 100. The last term is the face value discounted for N periods. Solve all the 1/Z terms and the 100/(1+ZN)^N. So you get PMT = (100 - (100/(1+ZN)^N))/(the sum of all the 1/1+Z terms)

Sorry if I wasted anyone's time.
• It took me two hours when I was studying l1 to figure this out • Wouldn't you use the calculator's TVM functions to get to [PMT] --> [CPT]? No need to know how to do algebra.
"Slow is smooth, smooth is fast."

"Never half-ass two things. Whole-ass one thing."
• If the Z terms are not the same, the TVM functions won't help you since they only allow for 1 discount rate.  You need to set up the equation in terms of each period's discount rate.

• haha. yeah, the CFAI is great at letting you figure out how a formula operates. I watched a number of youtube videos. It was really helpful learning how to use the calculator for SD calcs and others! I remember seeing geometric mean the first time and not knowing how to calculate with my calculator. Seems silly, but we have all been there in one formula or another.
• Prash, you can't use the TVM functions of an HP12C for problems with more than one discount rate.