Breaking down the annuity/amortization questions

Question

I'm teaching an intro math class, and the students are having a rough time with calculator use when punching in the values for annuity or amortization problems.

The standard annuity formula is something like $$Payments=P \frac {R(1+R)^N}{(1+R)^N-1}$$

Tomorrow night, I'm going to try out a little more structure and add some steps/check points to the calculation.

Using $$R=\frac{r}{n}=\frac{APR}{\text{number of times compounded}}$$ and $$N=nt=(\text{number of times compounded per year})(\text{number of years})$$

I define the D value to be $$D=1-(1+R)^{-N}$$

So the amortization formula then becomes, $$Payments=\frac{P R}{D}$$

So my question: Does anyone know if there is a name and standard variable for this D Value quantity, so I can at least attempt to align with standard texts on the subject?

Answer 1

I learned to do this using an excel spreadsheet,

$\begin{matrix} \text {period} & \text {discount rate} & \text {cash flow} & \text {discounted cash flow}\\ n&(1+r)^{-n}& CF_n& CF_n\cdot(1+r)^{-n}\\ 1 & 0.995025 & 100 & 99.5025 \\ 2& 0.990075&100&99.0075\\ \vdots \end{matrix}$

$100$ as the monthly payment was chosen arbitrarily

$r = \frac {6\%}{12}$ above

And for a $30$ year mortgage.

The sum of the discounted cash flows $\frac {1-(1+r)^{-360}}{r} = \text {Ballance} $

or the monthly $\text {Payments} = \frac {r}{1-(1+r)^{-360}}\cdot\text{Loan amount}$

I think it is tougher to make a calculator input error this way. But you might not have excel so readily available in the classroom.

One big advantage of this approach, is that when you move to steady cash flows to something more complicated (such as bond pricing, internal rate or return, yield to maturity, duration, etc.), you don't need to change your calculation methodology. While the formula is a one trick pony.

Answer 2

$$Payments=P \frac {R(1+R)^N}{(1+R)^N-1}$$

Often it is the case that you divide the numerator and denominator by $(1+R)^N$ to obtain as slightly simpler formulae: $$P \frac {R}{1-(1+R)^{-N}}$$

You can then think of this as the repayment ratio where $PR$ is the interest payable on a fixed-rate interest only loan, and the denominator modifies this to accommodate capital repayments with a fixed repayment structure.

I used to work in a large financial institution and i think i must've been only one of a handful of people who knew this calculation. No-one ever referred to the denominator by a specific name though. The whole calculation (usually done with free online calculators) was just called 'capital repayment amount'.

I recommend you move this thread over to a finance forum as they are likely to know the industry terminology.