Interest problems, exams, and calculators
The Society of Actuaries approves only a limited range of scientific calculators for use in their exams.
Only the following models of Texas Instruments calculators are approved:
- BA II Plus
- BA II Plus Professional
- TI-30X II (IIS solar or IIB battery)
- TI-30XS MultiView (or XB battery)
These calculators, while possessing basic scientific calculator features, do not have graphing capabilities, are not programmable, and cannot deal with symbolic algebraic expressions.
This means that the following relatively simple interest problem is, on the face of it, somewhat difficult to solve on such a calculator:
It is straightforward to find an equation for the interest because the interest earned by Georgia in the second half of the year is which we are given to be , so
which we can write as
If our calculator had graphing capabilities (which those approved by the Society of Actuaries for exam use do not) we could do a plot to get an estimate for the solution:
and then zoom in a couple of times:
to get a decent estimate .
Absent graphing or programming capabilities, what can we do to get an approximate, and sensible, solution to the equation (*)?
One powerful method, which takes advantage of the calculator’s memory function, is iteration, in which we start with a roughly approximate solution and then apply an appropriate function to get a better approximation, and then rinse and repeat, until we reach a desired degree of accuracy – say 4 decimal places.
A clue to using iteration is that we can re-write equation (*) as:
which tells us that our sought for value is a fixed point of the function .
Finding fixed points is easy for functions that have a small slope (less than 1 in size) close to the fixed point: that is because we can pick a guess close to the fixed point and then iterate the function which, due to the small slope, will converge relatively rapidly to the fixed point.
We are not completely in the dark as to what an approximate value of might be: this is an interest rate, and is unlikely to be greater then and may even be considerably less than that. So we can take as our initial guess for .
The function has derivative which, when , takes the value which is (considerably) less than 1 in size (= absolute value).
So here’s how, after a few iterations, we can get a solution to equation (*) accurate to 4 decimal places:
- place the initial guess in memory, say in variable A;
- place the constant 0.0551429 in memory, say in variable B;
- To update the value A, calculate B/(1+A/2)^11 and store this value in variable A.
- Repeat until 4 digit accuracy is obtained – in this example, about 5 iterations.
So there you have it – an illustration of how the mathematics of iteration to a fixed point and use of calculator memory can efficiently solve otherwise annoying equations arising in interest problems.
The real world
Of course outside of exams, in what’s often referred to as “the real world”, calculators do have graphing and symbolic algebra capabilities, and are programmable. What’s more, apps on your phone or computation websites such as Wolfram Alpha, will rapidly solve such equations:
Note: we had to replace “i” by something else – in this case “x” – because Wolfram Alpha interpreted i as the complex number whose square is -1.