New Video: Internal Rate of Return - Infinity is Really Big

Lecture 5 on Financial Mathematics for Actuarial Science

Financial Math for Actuaries, Lecture 5: Internal Rate of Return (IRR), a.k.a. Yield Rate

I have been working on developing video resources for Bethel University‘s course on Financial Mathematics for Actuarial Science for the past couple of years.

Mostly this has consisted of making many very detailed problem-solving and conceptual videos on financial mathematics. The problems solved have typically been taken either from the 6th or 7th edition of “Mathematics of Investment & Credit”, by Samuel Broverman, or the 2nd Edition of “The Theory of Interest“, by Stephen Kellison, or from various FM sample exams online.

But I have also created some longer “summary lectures”. My work on this has been more sporadic because of busy-ness in other parts of my life and job. These lectures take quite a bit of time to put together, in part because I want to make conceptually helpful Mathematica demonstrations to show in them.

Lecture 5, made on March 1, 2019, was no exception. I am particularly excited about the Mathematica content I was able to make for this lecture, embedded below. In particular, I show a (hopefully) very enlightening way to visualize how the internal rate of return (IRR), also know as a yield rate, of a set of financial transactions “balances” the present values of the cash inflows and outflows. The discounted values of these cashflows are represented visually both as heights of rectangles and as points on an axis. This balancing must occur in such a way that the sum of these heights must equal zero (with negative “heights” representing negative cashflows). The Mathematica content starts about 35 minutes into the video.

Financial Math for Actuaries, Lecture 5: Internal Rate of Return (IRR), a.k.a. Yield Rate

A Few Mathematical Details on the Internal Rate of Return

In this video, we assume that there are cashflows (positive, negative, or zero) at times $0=t_{0}<t_{1}<t_{2}<\ldots<t_{n}$ (where oftentimes, $t_{k}=k$ for all $k=0,1,2,\ldots,n$ ). We let $C_{k}$ be the net cashflow at time $t_{k}.$

For an arbitrary interest rate $i,$ the function $P(i)=\displaystyle\sum_{k=0}^{n}C_{k}(1+i)^{-t_{k}}=\displaystyle\sum_{k=0}^{n}C_{k}v_{i}^{t_{k}},$ where $v_{i}=(1+i)^{-1},$ is the (net) present value of the cashflow.

By definition, an internal rate of return (IRR) of this cashflow is a value of $i$ so that $P(i)=0;$ in other words, it is a root or a zero of this function. Of course, this function could have many roots. Therefore, the IRR is typically not unique, according to this definition.

In the most ideal case, there would be a small positive real root, and this would be the one to use in typical financial settings. In the end, the idea is that if you are comparing the relative merit of two cashflows, you would pick the one with the higher IRR, assuming they are both real numbers that are close to zero.

It is emphasized in the video through examples that there can be complex roots and also positive real roots that are both small. It is therefore less clear in those cases how the IRR should be interpreted.

Lecture 5 on Financial Mathematics for Actuarial Science

I have been working on developing video resources for Bethel University‘s course on Financial Mathematics for Actuarial Science for the past couple of years.

A Few Mathematical Details on the Internal Rate of Return

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