Last Updated on August 28, 2024 by Maged kamel

## Derive the P and F relation as cash flow moment.

### How do we derive the present value and future value of a deposit?

We have a timeline from t=0 to t=n, the Timeline in a year. We have a deposit at time t=0 with the present value P value acting downwards. Using the similarity of the Cash flow and moment diagram, we expect to receive future value inflow at t=n, which is represented by an upward arrow (+F) as income.

The present value does not numerically equal the future value; the presence of an exponential function makes the F value greater than P.

Let us estimate the moment at the pivot point at T=0; we have a moment due to -P, ith arm=(1+i)^0. The pivot point is exactly the same point where the force applies.

The sum of moments at the pivot point at t=0 should equal zero. We have a moment from the present value = (-p)*(1+i) 0, while the moment due to the Future value =+F(1+i)- n since the pivot point is at the left of +F, and the time elapsed is (-n) measured to the left from F.

Considering the sum of moments=0, we have P*(1) because the exponent is set equal to zero, equal to F*(1+i) โn. This can be modified as P=F/(1+i) n, the known relation.

The writing indicates the following points: P is a downward force considered (-P), while F is upward. The pivot point A is at t=0, at a distance =0 from the force (-P). That point is located at the left of F and has a distance =-n.

The pivot point B is at t=n, at a distance =n, from the force (P). The sum of forces in the case of cash flow in the Y- Y-direction does not equal zero, as we will see in the next slide. The P and F relation as cash flow moments is used to get the expression for P.

The pivot point B is at t=n, at a distance =-n from the force (F). We will proceed to the next slide.

### How to derive the expression for the Future value of a deposit?

We want to get the same expression between P and F, which is F=P*(1+i)^n, but this time we will apply another approach by

We are considering the Timeline as expressed in years starting fromt=0 to t=n. We will consider our pivot point at b, where t=n.

Let us check the moment due to (-P), which will be the product of (-P)*(1+i)^n, since point B is at the right of (-P). For the moment, due to F, it is +F(1+i)^0; the point coincides with the force (+F). The sum of moments is zero. we will rearrange the terms , -P(1+i)^n +F(1+i)^0=0.

The P and F relation as cash flow moments are used to get the expression for F. The final value of the future value is F=P(1+i)^n. Please refer to the next slide image.

## The first solved problem uses cash flow moment.

Table 4-2 for the discrete cash flow examples illustrating Equivalence from chapter 4 of William G. Sullivan’s book Engineering Economy. We have an example for a single cash flow diagram F, with the present value P.

The subject can be dealt with in two aspects; the first one is in borrowing ending terminology:

A firm borrows $1000 for eight years. How much must it Repay in a lump sum at the end of the eighth year? But in equivalence terminology, it can be written as What is the future equivalent at the end of eight years of $1000 at the beginning of those eight years?

The borrowed money at t=0 is P of $1000 with an upward arrow. The unknown is the future value, which is F. The given data is the number of years for the loan, n=8 years, and the interest rate i=10%. To get the value of F in terms of P, F=P(F/P, i,n), in that expression P goes with P, so F is equal to F on the right-hand side. Comma i=10% and comma n=8 where n is the number of years.

We will use Table c-13 of interest based on i=10% and get the related F/P, which must be bigger than one. It will be equal to 2.1436. We will multiply F/P by P, and we will get $1000*2.1436=$2143.6.

If we want to get the same result but use the similarity between the cash flow and moment, P=$1000, i=10%,n=8, take the pivot point at t=8 at the right of the upward P, we can write the summation of the moments as 1000*(1+0.10)^8-F(1+0.10)^0=0.

F will move to the right-hand side, with a positive sign, 1000*1.10^8=F*1. This method will give the same value of F by using the calculator, F/P=2.1436. The future value is equal to $2143.588. The slide image shows the four cases for forces and the distance to pivot points.

## The second problem was solved using cash flow and moment relation.

A firm wishes to have $2143.6 eight years from now. What amount should be deposited now to provide for it? In this case, if the future value is inflows, the present value is a deposit and will be with -P, downward arrow.

If we consider this example from the equivalence point of view, what is the present Equivalent of $2143.6 received eight years from now?

The interest rate is 10%. We will consider the pivot point at point t=0. Given F=$2143.6, using the known formula P=F(P/F,i,n), i=10%, and n=8. Where F goes with F and P will equal P, comma 10%, comma 8). Using the table of interest for i=10%, we can get the value of P/F.

P/F must be less than 1, and it is equal to 0.4665, then P=$1000. So the man should deposit $100 to get $2143.6 after eight years by an interest of 10%. For the moment estimation, we will take the summation of all the moments about the pivot point t=0, where P is acting downwards (-P) and is unknown.

-P*(1+0.10)^0+F*(1+0.10)^-8=0, can be written as -P*1.0+$2143.6*(1.10)^-8=0. P will move to the other side, equaling 2143.6/1.1^8=$1000. The result matches the estimation that was made using the formula.

Please click on any image, and you will have a slide show. Each photo will appear more transparent.

This is the video, which has a subtitle and closed caption in English. The video covers this post’s content from 9.42 to the end.

Please refer to this post, Introduction to Economic Equivalence, to review the subject of equivalence.

The next post is 12-Solved problem-Future value for uniform series deposits.