0.8 Mathematics of finance  (Page 7/9)

This classic present value problem needs our complete attention because the rationalization we use to solve this problem will be used again in the problems to follow.

Consider for argument purposes that two people Mr. Cash, and Mr. Credit have won the same lottery of $1,000 per month for the next 20 years. Now, Mr. Credit is happy with his $1,000 monthly payment, but Mr. Cash wants to have the entire amount now. Our job is to determine how much Mr. Cash should get. We reason as follows: If Mr. Cash accepts $x$ dollars, then the $x$ dollars deposited at 8% for 20 years should yield the same amount as the $1,000 monthly payments for 20 years. In other words, we are comparing the future values for both Mr. Cash and Mr. Credit, and we would like the future values to equal.

Since Mr. Cash is receiving a lump sum of $x$ dollars, its future value is given by the lump sum formula we studied in [link] , and it is

Since Mr. Credit is receiving a sequence of payments, or an annuity, of $1,000 per month, its future value is given by the annuity formula we learned in [link] . This value is

The only way Mr. Cash will agree to the amount he receives is if these two future values are equal. So we set them equal and solve for the unknown.

The reader should also note that if Mr. Cash takes his lump sum of $119,554.36 and invests it at 8% compounded monthly, he will have $589,020.41 in 20 years.

We have just found the present value of an annuity of $1,000 each month for 20 years at 8%.

Again, consider the following scenario:

Two people, Mr. Cash and Mr. Credit, go to buy the same car that costs $15,000. Mr. Cash pays cash and drives away, but Mr. Credit wants to make monthly payments for five years. Our job is to determine the amount of the monthly payment. We reason as follows: If Mr. Credit pays $x$ dollars per month, then the $x$ dollar payment deposited each month at 9% for 5 years should yield the same amount as the $15,000 lump sum deposited for 5 years. Again, we are comparing the future values for both Mr. Cash and Mr. Credit, and we would like them to be the same.

Since Mr. Cash is paying a lump sum of $15,000, its future value is given by the lump sum formula, and it is

Mr. Credit wishes to make a sequence of payments, or an annuity, of $x$ dollars per month, and its future value is given by the annuity formula, and this value is

We set the two future amounts equal and solve for the unknown.

Therefore, the monthly payment on the loan is $311.38 for five years.