R Tutorial : Actuarial equivalence

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How can you use the time value of money concept for calculations that are relevant in everyday life? That's what you will learn in this video. You will meet a superhero and help him with his car loan. Off we go!

Many of the transactions discussed in this course require you to establish an equivalence between two cash flow vectors.
A first example is a loan or a mortgage, for instance, to buy a house. In such an agreement you receive a lump sum of cash from the bank and - in return - you pay back the capital with monthly payments. To calculate these monthly mortgage payments you establish an equivalence relationship between the borrowed capital and the series of mortgage payments.
A second example is an insurance product. Such a product requires an equivalence between the benefits covered by the insurance and the series of premium payments. To calculate these premiums, later on in the course, you will use the exact same concept of equivalence between cash flow vectors.

Please meet Mr. Incredible, an independent superhero living and working in Belgium. He needs a new car worth 20,000 euros.

Mr. Incredible wants to borrow this amount from the bank.

He will pay back the loan with 4 yearly, constant payments equal to, say, K. Mr Incredible will pay his first loan payment at the end of the first year.
The bank will determine an interest rate, covering the interest the bank wishes to receive in return for the 20,000 euro borrowed to our superhero. Can you calculate what the loan payment K should be?

The cash flow from the bank to our superhero is very simple. 20 000 at time 0, allowing Mr. Incredible to buy the car.

The cash flow from the superhero to the bank is 0 at time 0, since his first loan payment is at the end of the first year, followed by four times K, the repayments at time 1, 2, 3 and 4.

So, here is what you will do. The Present Value of the cash flow from the bank to its client is very simple. That's just 20 000 euro. To get the Present Value of the cash flow from the superhero to the bank you discount each of the 4 loan payments, at times 1, 2, 3 and 4, to time 0. You require both present values to be equivalent and solve for the unknown K!

Let's assume the bank charges a constant interest rate of 3%, annually. The discount factors that you need should convert 1 euro at time 0, 1, 2, 3 and 4 to the present moment. Hence, these factors are calculated as negative powers of 1 plus 0-point-03. You then create a vector payments in R that is zero at time 0 followed by four times one. If you multiply this vector with K then you exactly get the cash flow vector from Mr. Incredible to the bank, as the timeline illustrates. Finally, you get the loan payment, K, by dividing 20 000 by the present value of the vector payments. With a rate of 3%, our superhero should annually pay 5380 euro to his bank, over four years, with the first payment in one year from now.

You are almost ready to challenge the many mortgage calculators available online. But, first, let's get your hands dirty on some exercises.

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