Question

8. You are told today that you’ll inherit a castle from your uncle in 5 years. You’ll need to

pay a property tax of $500,000 on the castle in year 6. Then the property tax will go up

each year by 1%. You and your descendant are going to pay for the property tax each year

forever. Suppose the interest rate is 10% per year. If you want to set aside enough money

today to be able to make all the future property tax payment, how much money do you

need?

Answer 1

= Present Value of Amount of money needed today

= Present Value of Amount of money needed 5 years from now

= Payment in year 6

=

This is a geometric progression with the first term as /1.10 and the multiplier factor as (1.01/1.10) = 0.9182

1.01 is because the tax grows by 1% each year. 1.10 because the of the discounting factor for each successive year.

Sum of this GP = First Term/(1 - multiplier factor) = (500,000/1.10)/(1-0.9182) = 5,555,555.56

Thus, = $5,555,555.56

In order to reach , we must discount this amount by 5 years at 10%.

= $5,555,555.56/(1.10^5) = 3,449,562.91

= 3,449,562.91

This is the amount that needs to be set aside today.

Alternatively, the following formula could have been used to calculate :

= Payment in 6th year

= Rate of interest

g = growth rate of tax payments

= 5,000,000/(0.10 - 0.01) = $5,555,555.56

Rest of the solution remains the same.