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

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

We have a timeline arranged 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 be equal to 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, then we have P*(1) due the exponent is set equal to zero, equal to F*(1+i)^-n, which can be modified to be P=F/(1+i)^n, that is 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 located at t=0, at a distance =0 from the force (-P). For that point, it is a point that is located at the left of F and with 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 are used to get the expression for P.

You can click on any picture to enlarge, then press the small arrow at the right to review all the other images as a slide show.

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 F=P(1+i)^n. Please refer to the next slide image.

## The first solved problem, using cash flow moment.

This is table 4-2 for the discrete cash flow examples illustrating Equivalence from chapter 4 quoted from 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, which is 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 that 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 the table c-13 of interest based on the 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, we get $1000*2.1436=$2143.6.

If we want to get the same result, but by using 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 different four cases for forces and the distance to pivot points.

## The second solved problem, 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, then the present value is a deposit, and will be with -P, downward arrow.

If we deal with this example from the equivalence point of view. What is the present Equivalence 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. by using the known formula P=F(P/F,i,n), i=10%. And n=8. Where F goes with F and P will be equal to 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 8 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 and will be equal to 2143.6/1.1^8=$1000.

The result matches the same estimation done by using the formula.

This is the video, which has a subtitle and closed caption in English. This post content is covered in the video from time 9.42 to the end.

Please refer to this post, Introduction to economic equivalence if you wish to review the subject of equivalence.

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