rbcc said:
Hi, I'm having trouble with this question
You currently rent an apartment for $700 a month and you expect the rent to increase by 1% a year. You expect to earn 10% on your investments and to live in the apartment for 20 years. Calculate the PV of the rent payed.
So this would be a case of calculating the PV of a growing annuity right?
700*1.01= 707 per month which would be 707*12=8484 per year.
then
8484/(0.10-0.01)* [ 1- (1.01/1.10)^20]
=94266.67 *0.8186
=77169.19
but the solutions say that the PV is 79 846.79, what am I doing wrong?
Thank You
rbcc
If you have copied the problem statment exactly, it is VERY badly worded and has an implicit and quite unrealistic assumption. Given that assumption, I get the same answer as your book.
Now I used excel to get the answer. If we have to use formulas, I may not remember all the ones most convenient for this problem.
Present value is always about interest rates. The opportunity rate of interest = i = 0.1 = 10% per year.
In practice, rent is usually paid monthly in advance, but as nearly as I can tell, the problem is assuming that rent is paid at the end of every month.
So, we need the monthly opportunity rate = m in addition to the annual opportunity rate.
What is the monthly rate that corresponds to an annual rate of 10%? In other words, m = ? Be careful with this question.
Now that we have m, we can calculate the present value of the first year's rent at the start of the first year.
The formula for that is 700 * {1 - [(1 + m)[sup:1sh7n0f2]-12[/sup:1sh7n0f2]]}/m, right?
Let's call that F[sub:1sh7n0f2]1[/sub:1sh7n0f2].
But the rent goes up every year by 1%.
So the present value of the n[sup:1sh7n0f2]th[/sup:1sh7n0f2] year's rent at the start of the the n[sup:1sh7n0f2]th[/sup:1sh7n0f2] year goes up 1% from the year before, right? Nothing has changed except the monthly rent, and it has changed by 1% from the year before.
So, if the present value of the n[sup:1sh7n0f2]th[/sup:1sh7n0f2] year's rent at the start of the n[sup:1sh7n0f2]th[/sup:1sh7n0f2] year= F[sub:1sh7n0f2]n[/sub:1sh7n0f2], then F[sub:1sh7n0f2]n[/sub:1sh7n0f2] = ?
Do you see how to proceed from here, or do these future present values give you a headache?