Now that Charlie’s father has promised to give £100,000 to his son in ten
years time, he has to save the money up. This he plans to do with a stream of
equal saving account payments made at the end of each year for ten years. By
the end of this period he will then be able to hand over the money to Charlie on
the day he has promised, and Charlie can hand the money to the bank to pay off
the loan.
Regular payments formula
Charlie’s father manages to find an account that allows him to do this, an
account that pays an annual interest rate of 8% for the whole ten year term. He
gives Charlie the task of working out the annual payments. With the compound
interest formula Charlie was concerned with one payment (the original principal)
but now he is concerned with ten payments made at different times. If regular
payments R are made at the end of each year in an environment where the
interest rate is i, the amount saved after n years can be calculated by the regular
payments formula.
Charlie knows that S = £100,000, n = 10 and i = 0.08 and calculates that R =
£6902.95.
Now that Charlie has his brand new Porsche, courtesy of the bank, he needs a
garage to put it in. He decides to take out a mortgage for £300,000 to buy a
house, a sum of money he will pay back in a stream of equal annual payments
over 25 years. He recognizes this as a problem in which the £300,000 is the
present value of a stream of payments to be made and he calculates his annual
payments with ease. His father is impressed and makes further use of Charlie’s
prowess. He has just been given a retirement lump sum of £150,000 and wants
to purchase an annuity. ‘That’s OK,’ says Charlie, ‘we can use the same formula,
as the mathematics is the same. Instead of the mortgage company advancing me
money that I repay in regular instalments, you are giving them the money and
they are making the regular payments to you.’
By the way, the answer to Henry Dudeney’s brainteaser is £130, made up of
the £51 Norman gave the customer and the £79 he paid for the bike.