Pensions and Proverbs | | DJS

Pensions and Proverbs

Two people, Alf and Bertha, are both 21. They are earning the same pay and they have the same prospects. Alf lives for the moment and is a Now-person but Bertha is more introspective and thoughtful. Alf sees no need to save any money. Bertha disagrees: she starts putting money aside for her old age.

This is called a pension.

Bertha puts an amount b into her pension fund every month. Bertha lives quite comfortably because she has picked amount b to fit with her lifestyle. Alf is spending amount P without thought. It probably disappears in beer and cigarettes. Or cars and holidays.

There’s a bit of a counting problem here. Assume they start work in September; then any savings are paid into a fund after payday, and they probably lose most of two months in various delays. Ten years later, at the start of September, Bertha would have paid 117 or 118 contributions; so think of ‘ten years later’ as being December, 120 payments later, interest due on all of them, contribution 121 due any moment.

Consider what happens to Bertha’s savings. Assume she can gain a decent rate of interest of i% per year. By the time she is ten years older, she has invested 120 b. The first payment has collected interest for the whole ten years, b (1+i/12)^120 and the whole fund, assuming her 121st payment is about to be paid (so payment number 120 has interest due) is, using j = 1+i/12,

Bertha’s Total after ten years  = b* (j120 -1) *j / (j-1)   =   b *[(12+i)/i]* [(1+i/12)120 -1]    

I’m going to call this B(10). You should check my algebra. You should have; I left an error (an extra j) from 201006 to 201302, and an extra closing bracket to 202205.  B(n) =  bj(1-jⁿ)/(j-1) is correct, but every A-level candidate shoudl eb able to show this, more or less as a standard result. The RHS remained wrong until 2022; and nobody wrote! You might like to use (12+i)/i as another intermediate value, say h. When i=5%, h=241. This can make the calculator work a little easier, meaning less prone to error. 

As an example, if b = £100 and i = 5%, then B(10) = £15592.93, when she has invested £12000 – she has, in some people’s minds, added 30% already. Do not confuse money now and money later; there is an issue (known as present value) that must be considered. What matters is the ‘real’ value, and that depends upon the excess of the rate i over inflation.

At this point ten years after Bertha has started saving, Alf settles down, gets the stability message, and starts saving in the same way as Bertha, putting aside £100 every month. In another 30 years, when they are both 61, (let’s say) Alf will have made 360 payments. His total pension fund, Total A(30), will be   £100 *12* [(1-(1.0041666)-360 )/ i] = £83572.64.

At age 61, putting money into his pension from 31 to 61, Alf has £83, 570 in his pension fund.

Bertha adopts Alf’s old habits just three years later than Alf and leaves her pension where it is, perhaps by leaving employment altogether. Being clear, after 13 years of contributions, she ceases to put money into this pension. It still gains compound interest though, and 27 more years later, her B(13), £22002.23, will have grown, at 5% interest, to B(13) x j ^324 = £84634.27.  

At age 61, having put money into her pension from age 21 to 34 and then leaving it to accrue interest, Bertha has £84, 600 in her pension fund.

Which is more than Alf will have. ¹

Obviously, investing early is a good idea. Bertha’s method matches Alf’s for return. She looks to the future early and can then relax — he probably spends the money on raising the children, she might even not be working — but she is going to equal Alf’s savings when they retire.

If they could find i=6%, then Bertha needs only to invest for 11 years and then rest for 30 to exceed Alf investing for 31 years — a single year of overlap. Yes, I picked the numbers to make Bertha have more than Alf  it makes the argument a lot stronger.

If Bertha continues to save (continues to put amount b into her pension every month) then her pension is around 80% more than Alf’s. She will have that much more money available all through retirement. Is Alf a fool for delaying? Will either of them have enough money for their retirement? How much should you put aside? Many people settle for a target of 10% of gross salary. Each and every year.

Here’s a table showing how Bertha’s savings exceed Alf’s:

Alf’s     Bertha’s    Interest      Bertha        |      Alf’s     Bertha’s    Interest      Bertha
years     years       rate p.a.        Alf           |     years     years       rate p.a.        Alf
    30         40            5%          1.833        |        32        42            5%            1.811
    30         40            6%          1.983        |        32        42            6%            1.961

You could easily set up a spreadsheet to explore this more fully, but I hope the point is made:

The earlier you start saving, the better off you will be.

There is a saying in English that applies. There is a very similar proverb in Chinese.

The early bird catches the worm.

DJS 20100625 and ediited 20220513

1  Please check the numbers, as I have also the results  B(13) = £21910.94,  B(13) x j³²⁴ = £84283.10   Trying again in 2020,  Bertha's position after 13 years  would be 156 payments  b *(12/i)* [(1+i/12)¹⁵⁶ -1]    on £100 a month at 5% p.a., is not £21,910.93911 => £84,283.0943. That comes from taking i=0.5 in the factor (1+i)/i  when i is 0.05/12. Silly.

The right number is B913) is 22002.23 and afeter another 27 years £84634.27, thought you may disagree about pennies, and it depends how the instiution does its rounding. £84,600 certainly.

The (12+i)/i term is 241, which is the bit I was struggling with, which confirms the £22,002.2346 figure. The trouble is that the headline rate of 5% is not the value of i. If h is that headline, then (1+i)/i  is (1+h/12) / h/12 = (12+h) / h. When  is itself 1/20 then  (1+i)/i = (12+h) / h = 12x20 +1 = 241.

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