Best books covering SOA Exam Fm

Question

Looking for book recommendations to self learn material for SOA exam FM. (This exam is mostly equations involving compound interest). Particularly something that covers the underlying math intuitively and well i.e. to give me the knowledge i would need to derive the formulas on my own. My background knowledge is somewhat limited I hope to fill in the gaps as I identify them. Thanks.

Answer 1

Mathematical Interest Theory by James Daniel and Leslie Federer Vaaler was the recommended book for interest theory. Derivatives markets by McDonald covered (you guessed it) derivatives markets, which was roughly 30% of the exam when I took it.

However, $\frac{1}{1-x}=\sum_{n=0}^\infty x^n$ is almost all of the math you need in order to do the FM exam. From this you can derive $$\sum_{n=r}^\infty x^n=x^r\sum_{n=0}^\infty x^n=\frac{x^r}{1-x},$$ $$\sum_{n=r}^R x^n=\sum_{n=r}^\infty x^n-\sum_{n=R+1}^\infty x^n=\frac{x^r-x^{R+1}}{1-x}.$$

For example, assume an annuity due with $N$ pay periods and an interest rate of $i$ compounded once per payment period. If each payment is $F$, the PV is $$\sum_{n=0}^{N-1}\frac{F}{(1+i)^n}=F\cdot \frac{1-v^N}{1-v},$$ where $v=\frac{1}{1+i}$, since you have payments of $F$ after $0,1,\ldots,N-1$ periods have elapsed.

You can also get annuity immediate with payments $F$, $N$ periods: $$PV=\sum_{n=1}^N \frac{F}{(1+i)^n}=Fv\cdot \frac{1-v^N}{1-v}.$$

You can also do future values for annuity due: $$FV=\sum_{n=0}^{N-1}F(1+i)^{N-n}=\sum_{n=1}^N F(1+i)^n=F(1+i)\cdot \frac{(1+i)^N-1}{i}.$$

And for annuity immediate: $$FV=\sum_{n=1}^N F(1+i)^{N-n}=\sum_{n=0}^{N-1}F(1+i)^n=F\cdot \frac{(1+i)^N-1}{i}.$$

We can also do perpetuities. We can do bonds (which have the terms above plus the final face value payment). We can do loans (similar to annuities).

We can have annuities that have non-level payments. For example, assume the payment changes by a factor of $g$ each time. That is, if the payment increases by $3\%$ each time, $g=1.03$. If it decreases by $5\%$, $g=.05$. Then for an annuity due, $$PV=\sum_{n=0}^{N-1}\frac{Fg^n}{(1+i)^n}=F\cdot \frac{1-(g/v)^N}{1-g/v}.$$

For arithmetically increasing payments (say, the first payment at time $0$ is $F$ and they increase by $H$ after that). Then the payment at time $F+Hn$, $n=0,\ldots,N-1$. We know $$\sum_{n=0}^{N-1}\frac{F}{(1+i)^n},$$ so we need to calculate $$H\sum_{n=0}^{N-1}\frac{n}{(1+i)^n}.$$ Let $$f(x)=\frac{1}{1-x}-\frac{x^N}{1-x}=\sum_{n=0}^\infty x^n-\sum_{n=N}^\infty x^n=\sum_{n=0}^{N-1}x^n.$$ Then $$x\frac{df}{dx}(x)=\sum_{n=0}^{N-1}n x^n =x\cdot \frac{d}{dx}\Bigl[\frac{1-x^N}{1-x}\Bigr]=x\Bigl[\frac{(1-x)(-Nx^{N-1})-(1-x^N)(-1)}{(1-x)^2}\Bigr].$$ Evaluate at $x=v=\frac{1}{1+i}$ and multiply by $H$.

Answer 2

Please reference the following link from the Society of Actuaries for the Spring 2024 Exam FM syllabus. In particular, the following texts are recommended:

• Broverman, S.A., Mathematics of Investment and Credit (Seventh Edition), 2017, ACTEX Publications, ISBN 978-1-63588-221-6
• Vaaler, L.J.F., Harper, S.K., and Daniel, J.W. Mathematical Interest Theory (Third Edition), 2019, The Mathematical Association of America, ISBN: 978-1-4704-4393-1
• Brown, R and Kopp, S, Financial Mathematics: Theory and Practice, 2012, Reprint: ACTEX Learning, Published by McGraw-Hill Ryerson: ISBN: 978-1-63588-694-8
• Francis, J. and Ruckman, C., Interest Theory – Financial Mathematics and Deterministic Valuation; (Third Edition), 2022, Actuarial Brew, ISBN 978-09981604-4-3
• Chan, Wai-Sum, and Tse, Yiu-Kuen, Financial Mathematics for Actuaries, Third Edition, 2022, World Scientific Publishing ISBN: 978-9811243271 (hard cover) or 978-9811245671 (paperback).

Not all of these books are required reading. Most of the content overlaps, although using a single text may not be adequately comprehensive. Differences will exist in presentation, pedagogy, and focus, so the choice of which text(s) to use will be highly dependent on the individual student.

I strongly recommend that you read the syllabus carefully and avail yourself of all of the resources it links to. While you could conceivably pass without doing so, in my experience, the more thorough your preparation, the less stressful the experience will be.