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I consider myself quite skilled in math, but there is an example in one of my math text books which simply baffles me. I'm sure there is a very simple explanation behind the logic in the example, but I just can't see it.

The example goes as follows:

In order to obtain the rights to manage a river, a company has obliged itself to pay the land owner $75000 annually for as long as the company operates. The company instead offers to pay the land owner a one-time payment. The annual interest rate is 5 %. How much does this one-time payment have to be in order for the two options to be similar in terms of value?

ANSWER: As the 5 % interest rate has to amount to the full annual amount of $75000, we can set up the following equation:

$0,05x=75 000$

Solving for x yields

$x=1500000$

Thus, the one-time amount has to be $1500000.

I really do not understand the logic behind the statement that "the 5 % interest rate has to amount to the full amount of $75000". I see that once we make this assumption, the problem becomes straightforward to solve, but why we can reason like this simply eludes me. If someone can please explain this to me, then I would be very grateful!

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    To pay you a lifetime annuity of $x$, I need to put enough cash in the bank to yield $x$ as an annual interest payment. Ignoring the risk that the interest rate may change over time, we can equate the two transactions. (Note: that risk is real and meaningful, but presumably not meant to be considered here). – 2017-02-23
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    Thanks a lot! Now I see the logic behind this very well :). – 2017-02-23

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