PV of rent payed

rbcc

Junior Member
Joined
Nov 18, 2009
Messages
126
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
 
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?
 
This li'l BASIC program will give you the Future Value f:

100 i = 1.1^(1/12) - 1
110 r = 700 : f = 0
120 FOR y = 1 TO 20
130 FOR m = 1 TO 12
140 x = f * i
150 f = f + r + x
160 PRINT y;m;r;x;b .......OK Mark? :wink:
170 NEXT m
180 r = r * 1.01
190 NEXT y

Ends up with f = 537,169.2787...., and f / (1 + i)^240 = 79,846.7904....
So, as Jeff told you, book is correct.

Btw, rent during 20th year is 845.6762...., or 700(1.01)^19 ; understand?

In case you don't follow the program, here tizz in "English":
100 i = 1.1^(1/12) - 1
Set up monthly rate equivalent to 10% ANNUALLY.

110 r = 700 : f = 0
r = monthly rent; f = future value

120 FOR y = 1 TO 20
Do for 20 y(ears)

130 FOR m = 1 TO 12
Do for 12 m(onths) each year

140 x = f * i
x = current month's interest

150 f = f + r + x
Update the future value

160 PRINT y;m;r;x;b .......OK Mark? :wink:
print each month's results

170 NEXT m
Go do next month

180 r = r * 1.01
calculate monthly rent for next year

190 NEXT y
Go do next year
 
OK (I'm hooked on this one!); here's a formula:

We have the geometric series (TKHunny will be proud of me!):
Code:
k(1.01^0)   k(1.01^1)   k(1.01^2)             k(1.01^19)
--------- + --------- + --------- + ....... + ----------
(1+ i)^0   (1 + i)^12  (1 + i)^24            (1 + i)^228
where k = r[1 - 1/(1 + i)^12] / i
where r = monthly rent ($700)
where i = 1.10^(1/12) - 1

multiplier = m = 1.01 / (1 + i)^12
last term = x = k(1.01^19) / (1 + i)^228

And the formula is:
PV = (xm - k) / (m - 1) = 79846.79...
 


Looks okay, in "general". :wink:

In "particular", here's your program's output using JustBASIC (version 1.01).

1170000
127005.58189830
1370011.20830740
1470016.87958240
...
2010845.6762654162.6930
2011845.6762654202.630440
2012845.6762654242.886350


:wink: ~ Mark

 
rbcc

So, Denis has given you a formulaic way and an algorithmic way to solve this problem. I used a different algorithm. But the important question is whether you now understand why even one of these ways works? The logic of Denis's algorithm is far easier to grasp than mine, but mine works in a spreadsheet, which may be accessible more frequently than BASIC. (Denis loves BASIC, which is a programming language that's easy to love, but the average workplace computer comes equipped with Excel, not BASIC.)
 
Re:

mmm4444bot said:

In "particular", here's your program's output using JustBASIC (version 1.01).

1170000
127005.58189830
1370011.20830740
1470016.87958240

YUK! Plus you missed the b(alance) column:
Code:
01 01  700.00      .00     700.00
01 02  700.00     5.58    1405.58
....
20 12  845.68  4242.88  537169.28
:roll:
 


Denis said:
you missed the b(alance) column

How did you reach the conclusion that I missed something?

It's your program. :?

Denis said:
This li'l BASIC program will give you the Future Value f:

100 i = 1.1^(1/12) - 1
110 r = 700 : f = 0
120 FOR y = 1 TO 20
130 FOR m = 1 TO 12
140 x = f * i
150 f = f + r + x
160 PRINT y;m;r;x;b
170 NEXT m
180 r = r * 1.01
190 NEXT y

 
Thank you for all the replies.

you got me I didn't copy the problem directly, this is only part of a much bigger problem.

Some of the solutions are to complex for me :oops:

Ok so I calculated the PV of the payment for the first year to be 7980.341 then if that is growing by 1% per year then the PV of a growing annuity gives 72587.86 and that is the PV one period behind where we need it so take it forward one year 72587.86*1.10=79846.79. I think that works
 
If it gets the right answer, it obviously works. I must admit that I do not understand what you did to get there, but that is my problem, not yours.
 
rbcc said:
Ok so I calculated the PV of the payment for the first year to be 7980.341 then if that is growing by 1% per year then the PV of a growing annuity gives 72587.86 and that is the PV one period behind where we need it so take it forward one year 72587.86*1.10=79846.79. I think that works
Your 7980.341 is CORRECT! But, like Jeff, I don't follow what you did after...

In the same manner as you calculated your 7980.341, you next need to calculate the PV of $707 (2nd year);
you'll get 8060.144: now you need to PV that PV(!) by dividing by (1 + i)^12, to get 7327.404

Same for 3rd year: PV of 714.07, then divide result by (1 + i)^24

Keep going to 10th year similarly; your net PV will be the sum of the 10 calculations.

NOTE: don't get discouraged; this isn't an easy one :idea:
 
Denis said:
rbcc said:
Ok so I calculated the PV of the payment for the first year to be 7980.341 then if that is growing by 1% per year then the PV of a growing annuity gives 72587.86 and that is the PV one period behind where we need it so take it forward one year 72587.86*1.10=79846.79. I think that works
Your 7980.341 is CORRECT! But, like Jeff, I don't follow what you did after...

In the same manner as you calculated your 7980.341, you next need to calculate the PV of $707 (2nd year);
you'll get 8060.144: now you need to PV that PV(!) by dividing by (1 + i)^12, to get 7327.404

Same for 3rd year: PV of 714.07, then divide result by (1 + i)^24

Keep going to 10th year similarly; your net PV will be the sum of the 10 calculations.

NOTE: don't get discouraged; this isn't an easy one :idea:


Ok, now I get this solution

what I did to get my answer was use the formula

\(\displaystyle \frac{PMT}{k-g} [1-(\frac{1+g}{1+k})^n]\)

so

\(\displaystyle \frac{7980.341}{0.10-0.01} [1-(\frac{1.01}{1.10})^{20}]=72587.8695\)

but that will bring it back to before the payment of 7980.341 so bring it forward one year and that's the answer. I think its the same thing because the formula breaks down in to the sum of future cash flows.
 
rbcc said:
Ok, now I get this solution
what I did to get my answer was use the formula
\(\displaystyle \frac{PMT}{k-g} [1-(\frac{1+g}{1+k})^n]\)
so
\(\displaystyle \frac{7980.341}{0.10-0.01} [1-(\frac{1.01}{1.10})^{20}]=72587.8695\)
but that will bring it back to before the payment of 7980.341 so bring it forward one year and that's the answer. I think its the same thing because the formula breaks down in to the sum of future cash flows.
Well, change your PMT amount to 8778.376 (FV of 12 payments of 700; you have PV)
and you'll get 79,846.79, which is the correct answer.

You do realise (do you?) that this problem can be worded this way:

A loan is negociated at rate of 10% annual, over 20 years.
The monthly payments will be $700 during 1st year.
The payments will increase by 1% each year.
What is the loan amount?
 
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