Lesson THE RELATIONSHIP BETWEEN PRESENT VALUE, FUTURE VALUE, AND PAYMENTS

This lesson defines Present Value, Future Value, and Payments and ties them together to hopefully give you a better understanding of the nature of each and how they relate to each other.

INVESTMENT PERIOD

The investment period is the period of time between when an investment starts and when it ends.

The investment period is divided into time periods.

In between those time periods are time points.

Each time period has 2 time points associated with it.

1 time point is at the beginning of the time period.
1 time point is at the end of the time period.

Time periods are defined as years, months, quarters, or whatever other measure is being used. Years, months, and quarters, however, are the time periods most used.

If the time periods of your study are defined one way, but the requirements of your study dictate that they be described in another way, then you need to convert your time periods to the requirements of your study.

The most common of this is the study is defined in years, but you need to convert your time periods to months because your payments are going to be monthly and / or your interest will be compounded monthly.

The most common conversion factors are as follows:

number of monthly time periods = number of yearly time periods * 12
number of quarterly time periods = number of yearly time periods * 4

interest rate per month = interest rate per year / 12
interest rate per quarter = interest rate per year / 4

If the study is already defined in the units of measure to be used, then no adjustment is necessary.

Sometimes you are given the number of time periods and the interest rate per time period. They may not even be defined. If so, then no adjustment is necessary and they can be taken as is.

An example of a study with 3 time periods in it follows:
-----
time point 0 = beginning of time period 1 = start of study
time point 1 = end of time period 1 = beginning of time period 2
time point 2 = end of time period 2 = beginning of time period 3
time point 3 = end of time period 3 = end of study

Present value calculations are made to the start of the study period which is the beginning of time period 1 which is also time point 0.

Future value calculations are made to the end of the study period which is the end of time period 3 which is also time point 3

The first time point in the study is always time point 0 which is always the beginning of time period 1 which is always the start of the study (at least for what we are discussing).

The last time point in the study is always the end of the last time period in the study is always numbered the same as the number of time periods in the study (at least for what we are discussing).

Example:
Study length = 6 time periods.
Time point 0 = beginning of time period 1 = start of study.
Time point 6 = end of time period 6 = end of study.

STUDY PERIOD VERSUS INVESTMENT VERSUS LOAN PERIOD

For practical purposes, the terms are used interchangeably.
They pretty much mean the same thing for purposes of discussing these financial formulas and how to use them.
They all mean the period of time from when the investment period starts and when the investment period ends.

PRESENT VALUE

Present Value is the value of an investment at the beginning of the investment period.
This could be:

The amount of a loan.
The amount of money in an account of some sort.
The present value of a series of payments that could be payments on a loan or periodic contributions to an account or periodic withdrawals from an account.
The present value of the future value of a loan or of an account.

If you are talking about a loan, then the Present Value is the amount of the loan. This is what the borrower has to pay back to the lender with interest.

If you are talking about making contributions to an account, or withdrawing from the account, this is the value of the account at the beginning of the investment period.

Sometimes you are given the present value and sometimes you are asked to calculate the present value from either the future value or from the payments.

FUTURE VALUE

Future Value is the value of an at investment at the end of the investment period.
This could be:

The remaining balance of a loan.
The amount of money made by a lender.
The amount of money that has been earned on an investment.
The amount of money remaining in an account after withdrawing from that account.

If you are talking about a loan, then:

From the perspective of the borrower, the future value of the loan would be the remaining balance on the loan account at the end of the investment period which should be 0 if the loan is fully paid off. In order to calculate this, the payments are made as negative amounts of money to indicate money is being withdrawn from the account.
Assuming the loan was $100,000, then the balance of the loan is $100,000 at the start of the investment period and the balance of the loan is $0.00 at the end of the investment period.
The payments include principal (the amount of the loan) plus interest charged to the borrower of the loan.

From the perspective of the lender, the future value of the loan would be how much the lender made on the loan. In order to calculate this, the payments are made as positive amounts of money to indicate money is being added to the account.
The balance of the loan is $0.00 at the start of the investment period.
At the end of the investment period the account will contain the principal amount of the loan plus the interest charged to the borrower of the loan plus additional interest earned by re-investing the received payments on the loan at the interest rate of the loan.

If you are talking about an account, then the same person is usually adding to the account or withdrawing from the account. The process is very similar to a loan except the same person is acting as the lender and the borrower.

Sometimes you are given the future value and sometimes you are asked to calculate the future value from the present value or the payments.

PAYMENTS

Payments are amounts of money that are either being added to or withdrawn from an account at periodic intervals of time.

If the payments are made on a loan, then the payments are made by the borrower and given to the lender.
If the payments are made to or withdrawn from an account, then the payments are added into or taken from the account usually by the owner of the account.

In the basic financial formulas, each payment is always the same amount of money.

Payments are made at the beginning of each period of time, or at the end of each period of time.

The basic financial formulas assume payments made at the end of each period of time. If payments are to be made at the beginning of each period of time, then special adjustments to the formulas are made to accommodate that requirement. These are covered in the lessons on each formula.

Sometimes you are given the payments and sometimes you are asked to calculate the payments from either the future value or the present value.

PAYMENTS VERSUS ANNUITIES

Annuities and Payments are similar.
What I call payments, somebody else might call an annuity.
The same formulas are used.
Present Worth of a Payment could also be called Present Worth of an Annuity, etc.

PRESENT AMOUNT VERSUS FUTURE AMOUNT

I use the term present amount in the formula Future Value of a Present Amount.
Other might use Future Value of Capital, or Future Value of an Amount.
They mean the same thing and the same formula is used, regardless of the terminology.
Same thing with Present Value of a Future Amount.

FORMULAS USED IN THIS LESSON

The formulas used to provide the results in this lesson are the 6 basic financial formulas that have their own lessons so only the results of the calculations will be shown. The actual calculations will not. If you wish to follow the details of the calculations, you may go to the formulas and try to duplicate them. I cheated by using a financial calculator. I did, however, make sure that the formulas and the calculator provide the same answer, which they do, as they should.

All you really need to be aware of is that there is nothing being done here that you cannot duplicate using those formulas.

The titles of these formulas and their associated lessons are:
-----
FUTURE VALUE OF A PRESENT AMOUNT
PRESENT VALUE OF A FUTURE AMOUNT
FUTURE VALUE OF A PAYMENT
PRESENT VALUE OF A PAYMENT
PAYMENT FOR A FUTURE VALUE
PAYMENT FOR A PRESENT VALUE

There is also a general discussion of financial formulas in the lesson titled:

BASIC ASSUMPTIONS AND FORMULAS USED IN FINANCIAL FORMULAS

PAYMENT VERSUS SERIES OF PAYMENTS VERSUS PAYMENTS

The following terms are used interchangeably and are usually referring to the same thing.
Payment
Payments
Series of Payments

RELATIONSHIP BETWEEN PV AND FV AND PAYMENT

These are related in the following way:

The Future Value of a series of payments is also the Future Value of the Present Value of that same series of Payments

The Present Value of a series of payments is also the Present Value of the Future Value of that same series of payments.

The series of payments for a Future Value is also the series of payments for the Present Value of that Future Value.

The series of payments for a Present Value is also the series of payments for the Future Value of that Present Value.

The same interest rate per time period and the same number of time periods is assumed.

EXAMPLE OF THE RELATIONSHIP BETWEEN PV AND FV AND PAYMENT

Assume a Payment of $1,000 being made monthly for a period of 5 years at an interest rate of 20% a year compounded monthly.

Investment Period is 5 years.
You will need to convert the time periods from years to months because the interest is compounded monthly and the payments are being made monthly.

You will need to convert the interest rate per year to interest rate per month.
You will need to convert percent interest to interest rate.

End of period payments are assumed.

Number of time periods = number of years * 12 months per year = 5 * 12 = 60.
interest rate per year = 20% / 100% = .2
interest rate per month = .2 / 12 = .0166666666

The Present Value of these Payments = PVPMT (1000) for 60 time periods at .0166666666 interest rate per time period = 37744.56062 = $37,744.56

The Future Value of these Payments = FVPMT (1000) for 60 time periods at .0166666666 interest rate per time period = 101758.2084 = $101,758.21

The Present Value of the Future Value = PVFV (101758.2084) for 60 time periods at .0166666666 interest rate per time period = 37744.56062 = $37,744.56. This is the same as the Present Value of the Payments.

The Future Value of the Present Value = FVPV (37744.56062) for 60 time periods at .0166666666 interest rate per time period = 101758.2084 = $101,758.21. This is the same as the Future Value of the Payments.

The Payments for a Future Value = PMTFV (101758.2084) for 60 time periods at .0166666666 interest rate per time period = $1,000. This is the same as the payments for the Present Value.

The Payments for a Present Value = PMTPV(37744.56062) for 60 time periods at .0166666666 interest rate per time period = $1,000. This is the same as the payments for the Future Value.

As long as you are dealing with the same time periods and the same interest rate per time period, the PV and FV and PMT are all related and you will get the same answer for the same variable regardless of which formula you use.

FVPMT = FVPV
PVPMT = PVFV
PMTFV = PMTPV

EXAMPLE FROM THE BORROWER'S PERSPECTIVE OF TAKING OUT A LOAN

You need $100,000
You go to the bank and they will let you have it but you have to pay them back in 3 years and the interest rate will be 30% a year.
Payments are made at the end of each time period.
Each time period is assumed to be one year.

Present Value of the loan is $100,000
Number of Time Periods = 3
Interest Rate per Time Period = 30% / 100% = .3

PMTPV (100000) = 55062.65664 = $55,062.66 payment per time period.
FVPMT (-55062.65664) = $0.00 = future value of the loan account at the end of the investment period.

You have made a total of 3 payments of $55,062.66 each which equals $165,187.97.
$100,000 was the amount of the loan.
$65,187.97 was interest on the loan.

Note that the payments were negative indicating a withdrawal from this account. Every time a payment was made to the lender, this account was reduced by the amount of the payment.

At the end of the investment period, the balance in your loan account was $0.00

Details for the account transactions follow:

time point 0 = $100,000.00 = Amount of the loan at the start of the investment period.
time point 1 = $100,000.00 * 1.3 = $130,000 - $55,062.66 = $74,937.34
time point 2 = $74,937.35 * 1.3 = $97,418.55 - $55,062.66 = $42,355.89
time point 3 = $42,355.89 * 1.3 = $55,062.66 - $55,062.66 = $0.00

Your (the borrower's) loan account is zero at the end of the loan period (same as investment period).
PV = $100,000
PMT = $55,062.66
FV = 0

EXAMPLE FROM THE LENDER'S PERSPECTIVE OF GRANTING A LOAN

You just changed hats.
You are now a representative of the bank.
The borrower comes to you and requests a loan of $100,000. You let the borrower have it under the condition that the borrower must pay you back in 3 years at the interest rate of 30% per year.

Payments are made at the end of each time period.
Each time period is assumed to be one year.

Present Value of the loan is $100,000
Number of Time Periods = 3
Interest Rate per Time Period = 30% / 100% = .3

PMTPV (100000) = 55062.65664 = $55,062.66 payment per time period.
FVPMT (+55062.65664) = 219700.0000 = $219,700.00 = future value of the loan account at the end of the investment period.

The borrower has made a total of 3 payments of $55,062.66 each which equals $165,187.97.
$100,000 was the amount of the loan which the borrower paid back to you.
$65,187.97 was interest on the loan which the borrower paid to you.
The future value of the loan was $219,700.00.
$219,700.00 minus the $100,000 pay back of the money lent minus $$65,187.97 interest paid by the borrower on the loan = $54,512.03 which is the additional money you made by re-investing the payments made by the borrower on the loan at the same interest rate of the loan.

Note that the payments were positive indicating an addition to this account. Every time a payment was made to you, this account was increased by the amount of the payment. This is not the borrower’s account now. This is the lender’s account for the same loan.

At the end of the investment period, the balance in your loan account was $219,700.00

Details for the account transactions follow:

time point 0 = $0.00 = Amount in your account (lender account) at the start of the investment period.
time point 1 = $0.00 * 1.3 = $0.00 + $55,062.66 = $55,062.66
time point 2 = $55,062.66 * 1.3 = $71,581.;45 + $55,062.66 = $126,644.11
time point 3 = $126,644.11 * 1.3 = $164,637.34 + $55,062.66 = $$219,700.00

The lender's loan account is equal to the future value of the loan payments at the end of the loan period (same as investment period).

INVESTING INTO A RETIREMENT ACCOUNT

To save myself a lot of work, I will make an unrealistic assumption.

This assumption is that you will be making the exact same payments as the loan into a retirement account for the exact same period of time as the loan and that the interest on the account will be the exact same interest rate as the loan.

Since the number crunching has already been done, the only thing left to do is relate the events to a retirement account rather than a loan.

The interpretation here is from the perspective of the lender, i.e. the retirement account is treated the same as if it was the loan account of the lender.

You make payments of $55,062.66 into the account for a period of 3 years earning an interest rate of 30% year.
At the end of the 3 year period you have $219,700.00 in the account assuming re-investment of the payments at the same interest rate of the account.

WITHDRAWING FROM A RETIREMENT ACCOUNT

I won't be able to save myself a lot of work this time because the beginning balance of the loan account was $100,000 and the beginning balance of the retirement account is $219,700.00

I will, however, use the same number of time periods (3), and the same interest rate per year (30%).

The figures come out to be:
PV = $219,700.00
FV = $0.00
PMT = 210,972.66

The interpretation here is from the perspective of the borrower, i.e. the retirement account is treated the same as if it was the loan account of the borrower now that you will be withdrawing from it.
You withdraw payments of $55,062.66 from the account for a period of 3 years earning an interest rate of 30% year.
At the end of the 3 year period you have nothing left in the account.
Your retirement account, which was equivalent to the lender's account when you were putting money into it, has now become equivalent to the borrower's account when you are withdrawing money from it.

SUMMARY OF RETIREMENT ACCOUNT TRANSACTIONS

The transactions of making payments into a retirement account is equivalent to the transactions of making payments into a lender's account. They look the same as shown below:

Lender’s account / investing into retirement account.

time point 0 = $0.00 = Amount in your account at the start of the investment period.
time point 1 = $0.00 * 1.3 = $0.00 + $55,062.66 = $55,062.66
time point 2 = $55,062.66 * 1.3 = $71,581.;45 + $55,062.66 = $126,644.11
time point 3 = $126,644.11 * 1.3 = $164,637.34 + $55,062.66 = $$219,700.00

The transactions of withdrawing payments from a retirement account is equivalent to the transactions of withdrawing payments from a borrower's account. They look the same as shown below:

borrower's account:

time point 0 = $100,000.00 = Amount of the loan at the start of the investment period.
time point 1 = $100,000.00 * 1.3 = $130,000 - $55,062.66 = $74,937.34
time point 2 = $74,937.35 * 1.3 = $97,418.55 - $55,062.66 = $42,355.89
time point 3 = $42,355.89 * 1.3 = $55,062.66 - $55,062.66 = $0.00

retirement account:

time point 0 = $219,700.00 = Amount in the retirement account at the start of the investment period.
time point 1 = $219,700.00 * 1.3 = $285,610.00 - $120,972.66 = $164,637.34
time point 2 = $164,637.34 * 1.3 = $214,028.55 - $120,972.66 = $93,055.89
time point 3 = $93,055.89 * 1.3 = $120,972.66 - $120,972.66 = $0.00

The numbers in the retirement account are different, but the process is the same.

A BONUS LESSON

Suppose when you withdrew from the retirement account you want to leave a certain amount of money in there, presumably for your beneficiaries.

The formula for calculating this is as follows:
PMTPV (money in the account) - PMTFV (money you want to leave in the account) = PMT(how much you need to withdraw each time period in order to do that).

An example:

You have $219,700 in the retirement account.
You want the account to be left with $100,000 at the end of the 3 year period.
The interest rate on the account is 30% per year.

First you calculate the PMT from the Present Value of the money in the account.

That would be $219,700.

You use the basic formulas as if $0.00 would be in the account at the end of the investment period.

PMTPV($219,700) for 3 years at 30% interest rate per year = $120,972.66

Second you calculate the PMT from the Future Value of the money you want to be left in the account at the end of the investment period.

PMTFV($100,000) for 3 years at 30% interest rate per year = $25,062.66

Third you subtract PMTFV($100,000) from PMTPV($219,700) to get:
PMT = $120,972.66 - $25,062.66 = $95,910.00

This payment value will be withdrawn from the retirement account each year for 3 years and you will be left with $100,000 at the end of the investment period as shown below:

time point 0 = $219,700.00 = Starting Amount in the Retirement Account.
time point 1 = $219,700.00 * 1.3 = $285,610.00 - $95,910.00 = $189,700.00
time point 2 = $189,700.00 * 1.3 = $246,610.00 - $95,910.00 = $150,700.00
time point 3 = $150,700.00 * 1.3 = $195,910.00 - $95,910.00 = $100,000.00

Questions or comments regarding this lesson can be directed to dtheophilis@yahoo.com