Sinking funds are used to save up for a targeted future expense (car, boat, school fees). You start saving with a target in mind and you put money into your fund regularly. The easiest way to look at these is to assume that any inflation during the saving period is accounted for by assuming the interest rate used is that which is in excess of inflation.

First, let us look at what happens if you save frequently. Assume you have £100 per month to put aside and let’s keep it easy; the rate on offer is 0.25% per month, 3% a year. Then after a year of saving you have the first payment with 12 months interest and the twelfth with one month (and you’re about to make the next payment but haven’t done it). So the amount totals:

100 x (1.0025 +1.0025^{2} +…. +1.0025^{12}) = 100 x 1.0025 x (1 + j + j^{2} +… + j^{11})

= 100.25 x (1 – j^{12})/(1-j) = 100. 25 x (j^{12}-1)/(j-1)

= 100.25 x 0.030416/0.0025 = 1219.68

This uses that (1 + j + j^{2} +… + j^{100}+…) = 1/(1-j) which is unknown to most junior students. Test the idea for j = ½ and you have in binary 1.1111111111… which IS 10_{2}, or two.

Note that if you were to pay compound interest of 10% (per time period) for a very long time on a loan of £100, the total would be 100 x (1.1 + 1.1^{2} + 1.1^{3} +.....) = 100 / 0.1 = £1000 which leads to the idea of Present Value, on another page in this series.

The general formula is, replacing j-1 with i,

A= P* [(1-j^{-n}) / i] = P * [1-(1+i)^{-n} / i]

for periodic payment P

Turning it round, P = Ai / (1-(1+i)^{-n}) = Ai / (1-j^{-n}) gives the periodic payment for the target amount A in an interest climate of i% per period. Be careful to use the right value for interest.

So, if you want £100k in 20 years time and you think you can find 2% above inflation, you would invest …. £100x0.02/(1-1.02^{20}) k = 4.11567k = £4115.67 per year. This is if 2% is only calculated annually. If the calculation and payment is done on a monthly basis and on a 365-day year, then I have £100k x .02/12 / (1 – (1 +2/1200)^{240})) = £339.22 each month and about £70 less per year. If the calculation is done daily but the payment is made monthly, you have a bit more of a problem, solved by finding the monthly equivalent; in this case 1.00166801 = (1 + 2/(100*365))^(365/12).

In this context, in you start off with the sum to invest every period and find how much you will have (the amount A) this is called amortisation (amortization in the US). A good working definition is to call this the process of reducing the value of an asset by a periodic amount. If you work out the total from the periodic payment, this is **a sinking fund.** You ‘sink’ money into it. Sometimes you need to look at the numbers differently because the interest rate you can find changes to a different set of rules to the thing you are saving to buy. It is complicated, but made so by the complexity of the dealings, not by the maths involved. Be particularly aware that the order in which you apply rules can make a difference to the result — this happens a lot with discounts in insurance.

Example: £300 per month for 25 years in a 6% climate (so j=1.005) will provide

p = £300 x 0.005 / (1 - 1.005^{-300}) = £46562.06

Questions:

13 in a 3% climate, £300 per month will provide how much, in 25 years?

14 What is the monthly figure required to have £50000 in 25 years in a 2% climate?

15 What about the same in 15 years?

A mortgage (“maw-gidge”) is a flexible term; the traditional form is where you put money into the mortgage (pay into it) and each monthly payment goes partly towards the interest and partly towards the capital. Technically, your mortgage supplier has given you money and you are paying it back over a fixed period, like 25 years. You can view this in two ways: (i) they (the odd singular ‘they’ that is your bank or building society) have an investment which is making a return and your “Amount” = N monthly payments (probably 25x12=300 of them) (ii) you are putting your payments into a sinking fund to provide the mortgage and all its accrued interest. In a climate where the interest rates go through change, the payments are recalculated quite often. Details vary, but if you are paying a mortgage you should look at what happens if you make an extra payment (it should reduce the capital owed immediately and thus reduce the time to completion); you should enquire what happens if you miss or reduce payments (they won’t be too happy about you extending the period of the mortgage and will usually insist that you pay the interest part as a minimum). If you default (fail to pay in accordance with the rules you agreed to) they will claim the property and sell it to recover their money. So even when you think most of it is yours, failure to pay still costs you your house.

The formula used to calculate the fixed monthly payment (P) required to fully amortize a loan of L kuai (nice general term for currency units) over a term of n months at a monthly interest rate of i. [If the quoted rate is 6% p.a., for example, i = 0.06/12 = .005 and j = 1.005].

P = L[i(1 + i)ⁿ]/[(1 + i)^{n - 1}] or, using j=1+i, P = Li jⁿ / (j^{n - 1})

Example: £50,000 for 25 years at 6%p.a. requires a payment of

£50000 x 0.005 x 1.005^{300} / (1.005^{300} -1) = £ 332.15

You can do this quite neatly in Excel using the PMT (rate, periods, amount) function, so for the above example you would write =PMT(6%/12,25*12,50000); I pasted the above into a cell and it produced -322.15 (you may need to format the result as number or currency; clue is the NUM! error).

Mortgages are, at times, very confusing. In Canada, the compounding period can be different to the payment period, so the rate per payment period and the declared rate are different. In the UK, the interest can be calculated for each separate year (annual compounding) but charged in the monthly payments (as if the calculation is so difficult they don’t want to do it too often, but it may be a case of needing to be able to prove the calculation paperwork).

You can work out equivalent rates and I recommend Wittwer, J.W., "Amortization Calculation," From Vertex42.com, Nov 11, 2008. It explains the function RATE(), which I see little use for.

Questions:

16 £50000 for 30 years at 6%p.a. requires what monthly payment?

17 £50000 for 25 years at 5% p.a. requires what monthly payment?

18 How much mortgage does £300 per month for 25 years buy, at 6%?

You can do this quite neatly in Excel using the PMT function

The next formula is used to calculate the remaining balance (B) of a fixed payment loan after p months:

B = L[(1 + i)ⁿ- (1 + i)p]/[(1 + i)^{n - 1}] or, using j, B = L(j^{n} - j^{p}) / (j^{n - 1})

**Exercise**

19 Mrs Smith is wondering whether she can pay off the remainder of her car loan. She borrowed £15000 at 5% p.a. and she has been paying this for 18 months of the 36 months she budgeted for. What is the balance left?

20 If Mrs Smith had opted for 0% finance but the market is generally offering 5%, what was the value of this £15000 deal to her across the same 3 years? So what sort of figure would you expect as a discount if you were in the same position and able to pay cash for £15000 worth of car? Do you think the money value should be declared differently?

Partial answer: j=1+5/1200; £7780.53; £2422.08 interest paid over 3 years, so £2000 to £2500 off the £15000 price. The present value of the vehicle is not at all the same as the present value of the money, but you are buying the car to have the pleasure of using the money (sorry, the car).

If you were buying school fees, you might save for each year’s fees separately, like this:

I want £20,000 available every year for seven years starting in 12 years time. I will assume 2% above inflation is available, with all payments made on a monthly basis. Looking at the first year we have a calculation very like the last one, £20k x .02/12 / (1 – (1 +2/1200)^{144})) = £123.03. The second year requires , £20k x .02/12 / (1 – (1 +2/1200)^{156})) = £112.37 and the third £103.26, then £95.37, £88.47, £82.40 and £77.00, making a total of £681.90 per month, although you might argue about the last penny. Note that the payments continue until they are needed, so with two complete years to go, £159.40 is still being paid monthly.

You might look at how much is paid out in total here. To raise £140,000 in 12-18 years time (check my counting, please), £120188.16 is put into the fund (please check that, too). If you think you can find the extra £20k more easily then rather than £700 per month now, don’t use a sinking fund. On the other hand, if you can find the money now, you might look upon this as getting a year for free. It isn’t true, but you might think of it that way.

**Questions:**

21 You (think you) want to have a sum of money for things like a house deposit available at the end of University. Imagine £25,000 at age 22. Right now you are 16. Try various models out, starting with five years at 3% p.a. paid monthly and paying monthly. Set up a spreadsheet to model this so you can change the rate easily.

22 Try seeing what happens if you have £100 a month to spare in a climate that is offering 5% per year.

23 Which would you rather have, £100 per month for a year or £1500 in 12 months time?

24 How much money do you need available to you right now so as to have £25,000 in six years time? Try a 2 to 5% climate.

25 What happens if the target is increasing at 2% and you can find 5% as an interest rate? Is that the same as a 3% difference? Look at having €100k in ten years time (but by then it will be around €122k).

DJS 20100517