Present Worth Using Annuity Gradient

J

Jill

New member
Joined
Mar 4, 2009
Messages
11
I am stumped on this problem: 5. Determine the present worth of the following cash flow at 10% interest. Include at least one gradient. Show calculations.

Years Cash Flow
Y4 : $50
Y5: $100
Y6: $150
Y7: $200
Y8: $100
Y9: $100
Y10: $50
Y11: $50

This is what I have assumed so far: I have to account for the money for all of the yeas and I think this is an annuity gradient problem. It might be a differred annuity because it starts at year 4. I thought that the gradient was 50 but looking at year 7 and 8 it increases by 100.

I started my caluclations like this but I have a feeling I am going in the wrong direction. I used an online calculator which showed that the PW is 409 and this is the answer I have, if that is the correct answer. What does include at least one gradient mean?

I also looked on line and the closest formula that I see is (F/P, 10%, 8) so I am thinking that if that is the gradient for NPW then I should be using this formula.

Any how I went back to using the NPV formula: NPV = CF4/(1+rate)^4 +CF5/(1+rate)^5......CF11/(1+rate)^11

Y4: 50/1.10^4 = 34.150
Y5: 100/1.10^5 = 62.09
Y6: 150/1.10^6 = 84.67
Y7: 200/1.10^7 = 102.63
Y8: 100/1.10^8 = 46.65
Y9: 100/1.10^9 = 42.41
Y10: 50/1.10^10 19.28
Y11: 50/1.10^11 =17.52

NPW= 34.15 + 62.09+ 84.67 +102.63+46.65+42.41+ 19.28 +17.52 = 409.4
NPW = 409.

If this is correct how do I include at least one gradient?
Thanks again for all your help.
 
Your NPV of 409.40 is CORRECT :wink:

Don't know what a "gradient" means here; dictionary:
gra·di·ent
noun
..........
Mathematics: A vector having coordinate components that are the partial derivatives of a function
with respect to its variables.

However, I think all it means here is show (with your answer) one of the calculations,
like your "Y7: 200/1.10^7 = 102.63

Perhaps someone else will confirm...or will tell me I'm wrong :cry:
 
These are the definitions that I am going by:
Gradient annuity: Is similar to an annuity but its periodic payouts increase at a constant rate of growth.

* This is the one I think applies to the problem:

Arithmetic gradient: conversion factor which is used to find annuity amount with gradient, where gradient is the period changes of payment each year,

To find the present worth of an arithmetic gradient series, we use the realtion P= G(P/G, i,n).

A gradient that starts at any time that is not the end of second year is called a shifted gradient.
 
I am still struggling on this one, the teacher gave a hint this is what he said:

Here is my hint for your questin:

Assuming annuities, the first annuity grows at a fixed rate of $50 for its 4 periods

(4 - 7), an arithmetic gradient.

What is the percentage that your annuity grows in years 4-7?

I still don't know how to comput it.

Does anyone have an idea????
 
I'm a little unclear on what it is with which you are struggling.

You can find a Present Value any way you like.

50(v^4 + 2v^5 + 3v^6 + 4v^7 + 2v^8 + 2v^9 + v^10 + v^11)

or

50(v^4 + v^5 + v^6 + v^7 + v^8 + v^9 + v^10 + v^11) + 50(v^5 + 2v^6 + 3v^7 + v^8 + v^9)

or

50(v^4 + 2v^5 + 3v^6 + 4v^7 + 5v^8 + 6v^9 + 7v^10 + 8v^11) - 50(3v^8 + 4v^9 + 6v^10 + 7v^11)

or

50(v^4)(1 + 2v^1 + 3v^2 + 4v^3 + 2v^4 + 2v^5 + v^6 + v^7)

What PERCENT did it increase between 4 and 5? 2/1 = 2 ==> 100% increase
What PERCENT did it increase between 5 and 6? 3/2 = 1.5 ==> 50% increase
What PERCENT did it increase between 6 and 7? 4/3 = 1.333 ==> 33.33% increase
What PERCENT did it increase between 7 and 8? 5/4 = 1.25 ==> 25% increase
What PERCENT did it increase between 4 and 8? 5/1 = 5 ==> 400% increase

After you manage the setup, the rest is algebra. Is your algebra good enough? If not, it IS time to upgrade!
 
I REALLY don't like your methodology. WAY too much rounding error.

I don't see how you used a "Gradient". You still haven't told us what that is.

I get 409.408
 
You must log in or register to reply here.
Top