At retirement nine years from now, a client will have the option of receiving a lump sum of $400,000 or 20 annual payments of $40,000, with the first payment made at retirement. What is the rate the client would need to earn on a retirement investment fund to be indifferent between the two choices?
I understand how to solve this problem on the calculator but I was wondering if it was possible to do long hand (I try to learn all long hand equations too, to get a better understanding of the material)? I figured on dividing 400,000 by the PV annuity due factor then comparing the payment to the 40,000 payment and figuring out rate. The math for this gets a little complicated:
|{1-[(1+i)^-n]} / I | * (1+i). Simplifying led me to (1-i)^-20 * (1+i^2) – the math here seems a bit complicated. Is it better to just understand how this works on the calculator as opposed to knowing the underlying principles behind it?
Also, while I solved this problem in [BEG] mode (because the payments start at the beginning of each period) the book said you could also solve in [END] mode by inputting 360,000 (400,000-40000), and [N]=19. What is the reasoning behind subtracting 40000 from 400000?
The problem with solving for interest rate (or # of payments) on annuities is that it involves solving for the roots of a complicated higher-order polynomial. An unless you’re a world-clas math jock, that simply ain;t going to happen. In contrast, the calculator does it in a flash.
There is a closed form solution for the interest rate and # payments on lump sums. But even then, why bother - the calculator does it far, far faster (and speed on the exam is at a premium). So learn the calculator.
As for solving the probelm in END mode, if you focus on the actual NET cash inflows and outflows, it should become clear. At time zero (the date of retirement) you “recieve” the 40,000,lump sum, but “give up” the 4,000. So the net cash flow is 36,000.
Thanks for that - I kind of figured that there was no simple solution (without the calculator) but I wanted to make sure. Naturally, I am learning how to use the calculator (and have done very well thus far, if I do say so myself) but I also want to learn the concepts and formulas behind the problems. I guess that isn’t possible for every single one.
And even if you’re a world-class math jock, there are general formulae only for situations up to four payments; if you have five or more, there is no general formula for the solution, so it’s going to involve trial and error, which is what the calculator’s doing (albeit very quickly).