Calculate NPV of non-constant perpetuity

[SimoneLF]

Hi all,

I need some help with an investment/financial math problem:

"Calculate the net present value of project C, given a discount rate of 7%"

Project C
Time (years)		1	2	3	4	thereafter
Cash flow	-1200	113	113	113	113	120

In this case, "Thereafter" means that the cash flow given is to be received each year from year 5 and onwards.


My first idea was to calculate the NPV of the cash flows from year 0 to year 4 by using the formula NPV = CF₀ + (CF₁/(1+r)¹) + .... + (CF_n/(1+r)ⁿ). In this case, I get NPV = -817,25. Then I wanted to add the PV (using PV = C/(r-g)) after year 4; as the perpetuity from year 5 and onwards is constant g=0.

My thought was that the NPV of the whole project would be equal to the NPV (year 0-4) + PV (after year 4).

However, as I have been working with this problem I realise that the PV formula above assumes that C is the cash payment received at time 1 - which is not the case in my approach.


I am not looking for the answer to the question, but rather some guidance on how I should approach this problem


All help will be greatly appreciated! Thanks!
- Simone

 

[jonah2.0]

WARNING: Beer soaked rambling/opinion/observation ahead. Read at your own risk. Not to be taken seriously. In no event shall Sir jonah in his inebriated state be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the use of his beer (and tequila) powered views.

Hi all,

I need some help with an investment/financial math problem:

"Calculate the net present value of project C, given a discount rate of 7%"

Project C
Time (years)		1	2	3	4	thereafter
Cash flow	-1200	113	113	113	113	120

In this case, "Thereafter" means that the cash flow given is to be received each year from year 5 and onwards.


My first idea was to calculate the NPV of the cash flows from year 0 to year 4 by using the formula NPV = CF₀ + (CF₁/(1+r)¹) + .... + (CF_n/(1+r)ⁿ). In this case, I get NPV = -817,25. Then I wanted to add the PV (using PV = C/(r-g)) after year 4; as the perpetuity from year 5 and onwards is constant g=0.

My thought was that the NPV of the whole project would be equal to the NPV (year 0-4) + PV (after year 4).

However, as I have been working with this problem I realise that the PV formula above assumes that C is the cash payment received at time 1 - which is not the case in my approach.


I am not looking for the answer to the question, but rather some guidance on how I should approach this problem


All help will be greatly appreciated! Thanks!
- Simone

You're on the right track. Your work just need a little bit of tweaking/fine tuning.
Way too drunk again so I'll just give it to you straight (you just give us some feedback that you understood everything o.k.?).
http://m.wolframalpha.com/input/?i=-1200+113*[1-(1.07)^(-4)]/.07+(120/.07)(1.07)^(-4)&x=0&y=0
I'm sure any drunken error on my part will be corrected shortly by Sir Denis, Sir Subhotosh Khan, or Sir Ishuda.

 

[Ishuda]

Almost. You didn't add in the interest the $113 would earn. For problems like these, it is important when you start and stop. Let's take the $113 payment for example. You start at the beginning of year 1 with the initial investment of $113 and continue until the beginning of year 4. How much do you have at the beginning of year 4 (you stop at the beginning of year 4). Well, that's a payment at the beginning of the year for 4 payments, so n = 4 and interest is 7% so we have an amount A1 of
A1 (at beginning of year 4) = 113 * (x⁴ - 1) / (x - 1)
where
x = 1.07
That's just the sum of a geometric series with common ratio x. But we have to bring that back to year zero and that is what your formula did if those CF's had included the interest earned. Well, since A is the amount at the beginning of year 4 we need to bring it back to year 0 which means your n is equal to 4 and we have to bring it back to year zero or
A1 (at beginning of year 0) = 113 * (x⁴ - 1) / (x - 1) / x⁴
or
A1 (at beginning of year 0) = 113 * (1 - 1/ x⁴ ) / (x - 1)
or the usual Present Value formula
PV = PMT ( 1 - 1/(1+i)ⁿ ) / i
with n = 4 and i = 0.07.

For the $120 part of the problem, the n becomes very large, that it n goes to infinity. So, at the beginning of year 5, using the formula for PV, we have
A2 (at the beginning of year 5) = $120 / i
and bringing that back to year zero we have what you gave with a CF of that amount and n = 4 [you only get the interest for 4 years since you start the withdrawals at the beginning of year 5]
A2 (at the beginning of year 0) = $120 / i / (1+i)⁴

That leaves the negative cash flow of $1200 but that is already at year zero. So, add them up.

Edit: Fix that beginning/end time for the $120 part, i.e. you would only get the interest for the 4 years (n=4) since the interest for that fifth year goes to pay for the first withdrawal at the beginning of year 5.

 

[jonah2.0]

WARNING: Beer soaked rambling/opinion/observation ahead. Read at your own risk. Not to be taken seriously. In no event shall Sir jonah in his inebriated state be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the use of his beer (and tequila) powered views.

Ah Lady Simone, why has thou not favored us poor math knight errants with a bit of feedback on our attemps in slaying your math dragon? Thou art ill perchance?
Whether thou art ill or not, I be almost certain that I speak on behalf of some math knight errants here that we pine for your reaction.
Had you been Dulcinea, I might have written a poem for you like the one that follows:

Dulcinea, Dulcinea
Where art thou, mistress of my heart,
Unconscious of thy lover's smart?
Ah me! thou know'st not my distress;
Or thou art false and pitiless.