## May you live in Interesting Times

The ancient Chinese curse quoted above is generally denied in China. The story goes that western students studying history have various eras described as of interest to historians — more basically, some eras are interesting to study even for schoolboys (who are more easily bored than girls). As I can imagine a girl putting the point across, these times are all very well to read about, but you wouldn’t actually want to be living then, would you? Hence the perception that such a wish is a mixed blessing.

When you have money to invest, this is an example of Capital. You are supposed to put capital to work. The mathematician is only briefly interested, as he (or she) thinks all the calculations are EASY. Which you might translate as trivial.

Simple interest.

Invest a pile of money, P for principal, and get paid a boring sum of money, M, at the end of every period of time, N of them, paid at interest rate, I. If you wait long enough, you have an amount of money, A. After N periods you have the grand total amount:     A= P(1+I*N)      or, if you only want the extra, A-P=P*I*N.   Think of this as there being interest, but you get it in your pocket. Or bank account.

Example 1: £5000 at 3.5% for 6 years paid annually gives A = 5000*(1+0.035*6) = 5000*1.21 = 6050

Many mathematicians would prefer to use (1+i), but since the bankers don’t, it only serves to confuse. For the few mathematicians reading this I will use j=1+I at some point and call j the interest-plus. That’s a hyphen, not a minus sign.

Example 2: £15000 at 6% p.a. for 6 years paid monthly, so 0.5% per month gives
A = 15000*(1+0.005*(6*12)) = 15000*1.36 = 204000.

Example 3: £15000 at 6% p.a. for 6 years paid monthly, gives
A = 15000*(1+0.06*6) = 15000*1.36 = 204000. This is the same as example 2.

Simple interest is useful if you want to receive it regularly. Imagine you have a lot of money and it is invested at a suitable rate. You could collect the interest every month. This might function as a retirement plan, but it assumes that inflation is not a problem for you.

Example 4: collect interest only from £2 million invested at 3% per annum (p.a.) — collect it every month.
2000000*.03/12 = 5000
Looks good, doesn’t it? All you need is a couple of million first.

Questions:

1   How much will you have after investing a million yuan, ¥1,000,000, at 5% p.a. for nine years?

2   How much interest will you receive each year on \$100 000 at 4% p.a.?

3   How much would you have to invest to receive £1500 per month if the interest rate is 3% per year?

4   As Q3 but at 4%.

5   Back in the mid-70s the interest rate was 2% per month. If you could find someone offering this, how much would you need to receive \$50 per day? Take 365 days per year.

6   How long does it take to recover £1500 from £100,000 at 4.25%?

Moving on, and losing the unnecessary multiply signs like good little mathematicians…

Compound interest

In this case the bank gives you the interest and adds it to your account before it calculates the next chunk of interest. So it matters how often the calculation is done (the period).

A = P J = P(1+i)

Example 5:£5000 at 3.5% for 6 years paid annually gives A= 5000 (1.035) = 6146.28

Example 6: £15000 at 6% p.a. for 6 years paid annually, gives
A= 5000 (1.06) = 5000*1.4185 = 7092.60

Example 7: £15000 at 6% p.a. for 6 years paid monthly, so 0.5% per month gives
A= 5000 (1.005)6*12 = 5000*1.432044 = 7160.22     a small improvement, £70.

Example 8: £2 million invested at 3% per annum but paid daily (use 360 days per year) for just one year gives just how much interest?
2000000 ((1+.03/360)360 -1) = 60906.47  Compare this with £59000 at simple interest, £60770.

Notice the change in numbers between the examples, please. The extra 2.5% made £2800 in Q6; being paid monthly added £200. Q8 £59000x1.03 = £60770, a similar number. But look back at examples 2 & 3: Q1 & Q5 show that the compound interest gave nearly £300 more; Q2 and Q6 differ by nearly £900 and Q3 and Q7 by over £1000. Compound interest is worth having unless you need the money now. Immediately.

Questions: compare each answer with those for simple interest

7   How much will you have after investing ¥1 000 000 at 5% p.a. for nine years calculated annually?

8   How much interest will you receive each year on \$100 000 at 4% p.a. calculated monthly?

9   How much would you have to invest to receive £18000 a year if the interest rate is 3% per year calculated daily? Stick to 360 days per year.

10   As Q9 but at 4%.

11   Back in the mid-70s the interest rate was 2% per month. If you could find someone offering this, how much would you need to receive £50 per day? Take 365 days per year. Is this simple interest? If you didn’t take the £50 out, but there was enough to give you £50 after the first day, how much more than 365x£50 would you have by the end of the year?

12   How long does it take to recover £1500 from £10,000 at 4.25% p.a. if the calculations are done monthly? Take 365 days to a year and remember the answer will be in whole months. In other words, how long does it take £10,000 to become £11500?

A word about precision: Answers are to the nearest penny (hundredth of currency unit). You must use full-length arithmetic on your calculator. You should write numbers down to sufficient length to show what you are doing. Answers without interim figures would lose marks at the level these questions would appear in exams. Ideally, you should write down a precise calculation (using fractions) and then it doesn’t matter what you do on your calculator – you would collect all ‘method’ marks for a correct representation of the problem. If you need to find an answer on the way to answering the question, then you risk the whole answer if you do not use the correct precision at that point. Be very wary of losing precision; I suggest you use 3 or 4 d.p. for interim money answers as a habit and reduce to two only at the end of the question, as if the arithmetic is in real numbers but the money resulting is an approximation.

DJS 20100517

small edits, plus the next section, 20130209

Q7   10.13% difference

Q8    0.07415% difference

Q9    you could say you want £50 per day for 360 days. Simple interest daily says £600,000 is enough for that, so whatever calculation you do should compare nicely. If you did it well, you invested £600k and actually collected £18,000 later, at the end of the 360 days. £591,070 will work. A pound more makes it work better. A teacher’s pension might be this much, so the calculation is relevant as a target for savings, pension fund and the like. Well, it was.

Q10   £450000; £441085

Q11   Written previously with \$. I don’t know that the US had 2% inflation per month in the 70s; I was there in Britain while it happened, 1976. Argentina had about 1% a day at the time. 2% / month is 2/3000 per day, so £75000 will suffice. £75000 x (1+.02/30)^365 -365x50 = 77404.27, or £2404 more than you started with. Of course, the £75000 went down in value by almost as much as you made in total, because interest rates rarely beat inflation by much, if at all. The safer your investment, the less you lose. I had £75000 required to buy a house while waiting for other money to clear various systems; £50/day was what it cost me to borrow the money.

Q12    40 months, not the 39.532 that the calculator says.

The deal the bank gave was that they would lend me £10k and I would pay interest at 4.25% for the time it took me to pay. They recommended three years, which was reasonable, amounting to £11360; they made it clear they expected to make £1500 off me. The small print, when I read it, said £1500 was a minimum charge; this is a disincentive to pay off loans, which cannot be a good thing, socially.

I found other ways of borrowing £10k and paid it all back in 6 months to keep the cost of borrowing down to something reasonable; was I wrong? In the terms of the question, £1500 was the cost of (fee for) borrowing £10k, however fast I paid it back. So I was far better off paying back the money just as fast as I could - and better still is not to borrow at all, but to pay out of savings.  You might argue that ideally I should pay off the loan at the optimum speed so as to benefit most from the rate, so that the interest to be charged equals the fee. I wanted the penalty off my back and regretted agreeing to the deal. My thinking at the time was that I had made a bad decision and that the small print (pay £1500 minimum whatever) was a bad deal. I wrote to the bank and they eventually charged something like 0.5% per month (6% per year) for the time I had use of the money, which was a well-worthwhile letter.

An exception to paying out of savings is where the benefit of the spending (the opportunity cost of having the use of the £10k) is greater than the cost of the borrowing; if your benefit gained from the spend is worth that extra borrowing cost, go for it, but beware of assumptions that all other factors remain the same; if inflation changes, or interest rates change you may well have made a wrong decision.
Opportunity cost is something that econonmists get excited about when they are in business for themselves; my observation is that this very easily becomes an excuse to go spend money they do not have. Remembering that this is the cost of having the opportunity - and keeping track of that cost against gains actually realised, attaches a good value to 'the cost of doing business'.