Annuities and Perpetuities

Perpetuity

 

First, let us start with the latter, so what is a perpetuity?

The boring definition would be: A perpetuity is the present value of a consistent cash flow that continues forever! The formula linked to this lovely definition is straightforward:

 

Present value = cash flow / r (interest rate)

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Nothing is more fun than underlining a dry definition with a cracking example: So let's assume that the owner of the Jolly Brew Ltd. inherits a significant amount of money of no less than 1.200.000GBP; hallelujah! Being a prudent entrepreneur, he doesn't want to blow the entire sum on the world's riches. Instead, he puts some money away to earn a yearly income.

Considering his current lifestyle, he has determined that he wants to take a yearly income of 40,000 GBP, which should provide him with some good support in the coming years. He browses various options and entrusts his money with a fund manager who promises a yearly return of 5% (including fees, etc…).

Now the only question remaining is how much money the owner of the Jolly Brew Ltd needs to entrust his fund manager:

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He uses the formula outlined above:

Present value = (Cash flow / r) or Present value = 40.000 / 0.05 = 800.000

So we need to deposit 800.000GBP with the fund manager, leaving 400.000GBP to spend on fun stuff, hurray!

Annuities

Last but not least, we need to talk about annuities. By definition, an annuity is a financial instrument purchased for an initial sum and then pays out the same amount for a specified number of years. I would say that an annuity is actually an insurance product. In today's world, a real-life practical application of an annuity can be found when you reach your pension age. For example, when you reach age 55 and have some money available after diligently saving up for years, purchasing something like an ordinary annuity might be a good idea. By the way, this is not investment advice, but there is some sense behind such a move. When you plan your pension, you can't take so many risks anymore; hence diversifying into more than just one product makes absolutely sense! There is truth in the saying: Don't put all your eggs in one basket! An annuity can be an excellent addition to your stocks and bond portfolio.

But as you know, I have much more fun explaining things with the help of some examples. So let us assume that after years of hard work, the owner of the Jolly Brew Ltd. has reached retirement age. He is looking around to diversify his retirement options. A pension provider is offering him two types of annuities:

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• Annuity A will pay 20.000GBP each year for 12 years

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• Annuity B will pay out 15.000GBP each year for 20 years

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Both options require the same initial purchase price and interest rate, set at 5%. Now he needs to find out which option works better from a return perspective.

We need to assess how much these yearly payments are worth in today's money to make these options comparable!

Fortunately, there is a formula handy for this scenario which is called the cumulative discount formula!

 

1/r x (1 – (1/ (1+r)^n))

R = the interest rate the bank assumes

N = the number of years the policy runs

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For annuity A, it works as follows:

1 / 0.05 x ( 1 – (1 / (1 + 0.05)^12)

= 20 x (1 – (1/1.796)

 

= 12 x 0.45

 

= 5.4

 

20.000GBP x 5.4 = 108.000GBP

 

Now we have to do the same calculation for annuity B

1 / 0.05 x ( 1 – (1 / (1 + 0.05)^20)

= 20 x (1 – (1 / 2.653)

 

= 20 x 0.623

 

= 12.46

 

15.000GBP x 12.46 = 186.900GBP

 

 

First, I hope this has not blown your head off? Second, looking at the two annuities, it is clear to the owner of the Jolly Brew company that annuity B is the far better option because it has a higher present value! In more straightforward terms, annuity B is worth more in today's money!

I hope that I have not scared you with the examples provided! My only goal is to arm you with the financial savviness to understand the time value of money concept! The basic idea is that cash in hand today is worth more than a promise to get paid in the future. When future payments are involved, we need to find a way to break it down into today's money, where the formula above comes in handy.