Discounting and annuity explorer

You set four things: the interest rate you are discounting at, how many yearly payments the promise runs for, whether each payment lands at the start or the end of the year, and how fast the payments grow. It values a promise of £10,000 a year on that basis and prints the answer three ways: as a present value, as an annuity factor, and as the year-by-year column that adds up to the same total.

Everything runs in your browser. There is no account, no upload and nothing stored anywhere: close the tab and the numbers are gone. The annuity factor is the useful number to take away, because it is the value of £1 a year on your basis, so you can multiply it by any amount you like.

Discounting has one moving part. A pound you will be handed in a year is worth less than a pound today, because a pound today could be invested and be more than a pound by then. At an interest rate of 5% a year, the ratio between the two is 1 divided by 1.05, which is 0.9524. Actuaries call that v. A pound paid in 25 years is worth v multiplied by itself 25 times, which is 29.5p. That is the whole of it, and Figure 1 above is that number drawn for every year in between.

An annuity is a series of those payments, so its value is a series of those multiplications added up. Add one at the end of each of the next n years and the sum has a short form: (1 - v to the power n) divided by i. That total is the annuity factor. At 5% over 25 years it is 14.0939, so a promise of £10,000 a year is worth £140,939 today. If the payments come at the start of each year instead, every one of them lands a year earlier, so the factor is simply (1 + i) times bigger.

An increase on the payments needs one substitution rather than a new formula. A payment growing at g and discounted at i behaves exactly like a level payment discounted at the net rate j, where (1 + j) equals (1 + i) divided by (1 + g). Move the increase slider above zero and the tool prints j as its own step, because that substitution is the one people lose marks on.

Two settings break the textbook formula, and neither is an error. With no interest at all there is nothing to discount, so n payments of a pound are worth n pounds. With an increase exactly equal to the interest rate, the net rate is zero and the same thing happens. The working prints those two cases in words instead of printing a division by zero, which is the sort of detail that separates a model that has been checked from one that has not.

Payments are annual and land on the anniversary. The rate is an annual effective rate, held constant for the whole term. There is no tax, no expense loading, no investment strategy and no allowance for inflation beyond the increase you set yourself. The term stops at 60 years and the rate at 12%, which is enough to show the shape of the curve without illustrating a rate nobody is discussing.

The assumption that matters most is the one about the people. Every payment here is treated as certain. A real pension or annuity is contingent: each payment is made only if the person is alive to receive it, so each one is weighted by a survival probability read off a published mortality table, and any allowance for future improvement needs a named published projection model on top. This tool does none of that. It is not a valuation, not a quotation, not a transfer value and not anybody’s life expectancy. If you want to build the contingent version, you need a table to start from, and the Office for National Statistics publishes one for the UK.

The Institute and Faculty of Actuaries covers this material in Actuarial Mathematics for Modelling (CM1). Its curriculum page describes the subject as giving a knowledge of theories of interest rates and the mathematical techniques used to model and value cashflows which are either certain or are contingent on mortality, morbidity and/or survival. This page is the first half of that sentence: cashflows that are certain.

In a job it turns up everywhere. A scheme actuary discounts decades of benefit payments to see whether a fund is big enough. A pricing actuary values a promise before putting a premium on it. An investment actuary prices a bond by discounting its coupons. Our lessons take it further than a slider can:

• Valuing a promise in the pensions course walks through why a scheme has to discount at all, and it is free to read.
• Expected present value: pricing a promise adds the piece this tool leaves out, the probability the payment is made. Also free to read.
• Price a bond from scratch does the same arithmetic on coupons and a redemption payment.
• Excel, R, Python and SQL for actuaries has a lesson on building a discounted cashflow in Excel so that a reviewer can check it, which is how the calculation actually arrives at work.
• The IFoA exams explained sets out where the Core Principles subjects sit in the qualification.