Lesson SUMMARY OF TIME VALUE OF MONEY FINANCIAL FORMULAS

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******************************* IMPORTANT ************************************

please read these notes below before using the formulas.
otherwise you could very easily get the wrong answer.

NOTES SECTION

if you have already finished reading the notes, then you can go directly to the start of the formula section by clicking on the following link.

<a href= "#formulas">go to start of formula section</a>

CONTACT

if you have any questions about the material in this lesson, please send an email to dtheophilis@gmail.com and i will answer your question as soon as i can.

TIME PERIODS

time periods are normally specified as annually or semi-annually or quarterly or monthly or daily or continuous.

the interest rate per time period is the annual interest rate divided by the number of compounding periods per year.

the number of time periods is the number of years multiplied by the number of compounding periods per year.

annual time periods means you are compounding 1 time per year.
semi-annual time periods means you are compounding 2 times per year.
quarterly time periods means you are compounding 4 times per year.
monthly compounding means you are compounding 12 times per year.
daily compounding means you are compounding either 365 or 360 days per year, depending on what is specified as part of your problem - if in doubt, then ask your instructor how many days per year you are supposed to assume.
continuous compounding means you are compounding an infinite number of time periods per year - this is a different formula than discrete compounding formulas that have just been described above, as you will see at the end of this summary.

in order to use these formulas correctly, you need to convert interest rate as a percent to interest rate as a decimal.

the conversion is very simple.
interest rate as a decimal is equal to interest rate as a percent divided by 100.
the verse is also very simple.
interest rate as a percent is equal to interest rate as a decimal multiplied by 100.

if you are given the interest rate as a percent, you need to divide it by 100 to obtain the interest rate as a decimal.
for example, an interest rate of 15% is divided by 100 to get you an interest rate of .15.

the formulas require interest rate as a decimal.
do not forget to convert percent to rate or the formulas will not give you the correct answer.

a typical problem might tell you that you have 15% interest rate for 9 years compounded monthly.

the interest rate per time period in this problem would be 15% / 100 / 12 = .0125 per month.

the number of time periods in this problem would be 9 * 12 = 108 months.
you would make r = .0125 and you would make n = 108.

you divide the interest rate by the number of compounding periods per year.
you multiply the number of years by the number of compounding periods per year.

ANNUITIES

an annuity is the same as a recurring series of equal payments.
it can involve periodic payments on a personal loan, or payments on a mortgage for a house, or payments on a car loan, or payments that you receive from your account, or any other kind of situation that assumes equal periodic payments where the number of compounding periods per year is the same as the number of payments per year.

when you are dealing with annuities, you need to make sure you are dealing with payments at the beginning of each time period or payments at the end of each time period.

the default situation is payments at the end of a time period, but sometimes the problem will require payments at the beginning of a time period.

most personal loans or car loans or mortgages involve payments at the end of each time period.

you should assume end of time period payments unless the problem specifies otherwise.

payments made at the beginning of a time period use the same formulas as payments made at the end of a time period except that an adjustment is made at the end to account for the difference.

you will see this when you look at the annuity formulas where the payments are made at the beginning of the time period rather than at the end of the time period.  it's the same formula with an adjustment made at the end to account for the fact that the payments are made at the beginning of the period rather than at the end.

PARENTHESES

in this tutorial, the parentheses are inserted so that you can easily enter the problems in your calculator exactly as shown and be reasonably confident you will get the right answer.

when you enter the formula in your calculator, (1+r)^n, where r = .015 and n = 30, can be entered as either (1.015)^30 or (1+.015)^30.

the examples would enter it as (1.015)^30.
when entered this way, you could eliminate the surrounding parentheses, but they are kept there so as to make the examples look more like the formula without adding any confusion as to when you should use them or when you shouldn't.  you should just follow the formula as is.  if you use (1+.015) in your calculator instead of (1.015), the calculator will be able to handle it the exact same way because (1+.015) is mathematically equivalent to (1.015).

ONLINE DISCRETE COMPOUNDING FINANCIAL CALCULATOR

there is an online calculator that you can use to check your results when you use discrete compounding.

that calculator can be found at <a href = "http://arachnoid.com/finance/index.html" target = "_blank">http://arachnoid.com/finance/index.html</a>

this calculator, as well as many other calculators, assumes interest rate as a percent rather than interest rate as a decimal.

the formulas in this lesson assume interest rate as a decimal.
the calculators in this lesson assume interest rate as a percent.

for example:

15% compounded monthly in the calculator would be entered as 15/12 = 1.25.
15% compounded monthly using the formulas in this lesson would be entered as 15/100/12 = .0125

you should be able to duplicate your answer using the financial calculator that you got manually by using the formulas in your scientific calculator.
if you don't, then you did something wrong, either in using the financial calculator, or in using the formulas via the scientific calculator.

in that case, send an email to dtheophilis@gmail.com telling me what the problem is and i'll let you know what i think the answer should be with an explanation of how i derived it, and what you may have done wrong.

THINGS TO KNOW ABOUT USING THE ONLINE FINANCIAL CALCULATOR PROVIDED WITH THIS LESSON

future value of a present amount requires payment amount to be equal to 0.
present value of a future amount requires payment amount to be equal to 0.

annuity for a present amount requires future value to be equal to 0.
annuity for a future amount requires present value to be equal to 0.

annuities and payments mean the same thing.

when dealing with annuities, you need to specify whether payments are at the beginning of the time period of at the end of the time period.

ONLINE CONTINUOUS COMPOUNDING FINANCIAL CALCULATOR

there is an online calculator that you can use to check your results when you use continuous compounding.

that calculator can be found at <a href = "http://www.ultimatecalculators.com/continuous_compounding_calculator.html" target = "_blank">http://www.ultimatecalculators.com/continuous_compounding_calculator.html</a>

this calculator assumes interest rate as a percent rather than interest rate as a decimal, as do most online financial calculators.

A WORD ABOUT THE EXAMPLES IN THIS TUTORIAL

an example of the use of each of these formulas is shown with each formula.
these examples have been tested through the use of the online financial calculator provided, or the use of the online continuous compounding calculator provided, to ensure you get the same answer whether you use the formulas or the calculator.

you should try to duplicate these examples using the formulas and your scientific calculator to make sure you get the same answer you see in the tutorial

you should also try to duplicate these examples using the online calculators provided, to make sure you can get the same answers you see in the tutorial.

if you cannot duplicate the answers in these examples using the online calculators provided, then send an email to dtheophilis@gmail.com and i will assist you as soon as i can.

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<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

FORMULA SECTION

to go to the specific formula, click on the link provided.

FUTURE VALUE OF A PRESENT AMOUNT <a href = "#f1">go to formula</a>
PRESENT VALUE OF A FUTURE AMOUNT <a href = "#f2">go to formula</a>
FUTURE VALUE OF AN ANNUITY WITH END OF TIME PERIOD PAYMENTS <a href = "#f3">go to formula</a>
FUTURE VALUE OF AN ANNUITY WITH BEGINNING OF TIME PERIOD PAYMENTS <a href = "#f4">go to formula</a>
PRESENT VALUE OF AN ANNUITY WITH END OF TIME PERIOD PAYMENTS <a href = "#f5">go to formula</a>
PRESENT VALUE OF AN ANNUITY WITH BEGINNING OF TIME PERIOD PAYMENTS <a href = "#f6">go to formula</a>
ANNUITY FOR A FUTURE AMOUNT WITH END OF TIME PERIOD PAYMENTS <a href = "#f7">go to formula</a>
ANNUITY FOR A FUTURE AMOUNT WITH BEGINNING OF TIME PERIOD PAYMENTS <a href = "#f8">go to formula</a>
ANNUITY FOR A PRESENT AMOUNT WITH END OF TIME PERIOD PAYMENTS <a href = "#f9">go to formula</a>
ANNUITY FOR PRESENT AMOUNT WITH BEGINNING OF TIME PERIOD PAYMENTS <a href = "#f10">go to formula</a>
FUTURE VALUE OF A PRESENT AMOUNT USING CONTINUOUS COMPOUNDING <a href = "#f11">go to formula</a>
PRESENT VALUE OF A FUTURE AMOUNT USING CONTINUOUS COMPOUNDING <a href = "#f12">go to formula</a>

FUTURE VALUE OF A PRESENT AMOUNT

f = p*(1+r)^n

f is the future value.
p is the present amount.
r is the interest rate per time period.
n is the number of time period.

example:

you are investing 5000 at 9% per year compounded monthly for a period of 12 years.
how much do you have at the end of the investment period?

p = 5000
r = 9% per year / 100 / 12 = .0075 per month.
n = 12 years * 12 = 144 months.

formula of:
f = p*(1+r)^n becomes:
f = 5000*(1.0075)^144 which is equal to 14664.18387.

if you use the online discrete compounding financial calculator provided, you would do the following:
present value = 5000
future value = 0
number of periods = 20 * 12 = 144
payment amount = 0
interest rate per period = 9% / 12 = .75% per month entered as .75
payment at is left as is.
calculator will tell you that fv = -14,664.18.
disregard the sign of the result.

<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

PRESENT VALUE OF A FUTURE AMOUNT

p = f/(1+r)^n

p is the present value.
f is the future amount.
r is the interest rate per time period.
n is the number of time periods.

example:

you need to have 5000 in 12 years.
you want to know how much you have to invest today at 9% interest rate per year for 12 years compounded monthly.

f = 5000
r = 9% per year / 100 / 12 = .0075 per month
n = 12 years * 12 = 144 months.

formula of:
p = f/(1+r)^n becomes:
p = 5000/(1.0075)^144 which is equal to 1704.834052

if you use the online discrete compounding financial calculator provided, you would do the following:
present value = 0
future value = 5000
number of periods = 12 * 12 = 144
payment amount = 0
interest rate per period = 9% / 12 = .75% per month entered as .75
payment at is left as is.
click on pv.
calculator will tell you that pv = -1,704.83
disregard the sign of the result.

<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

FUTURE VALUE OF AN ANNUITY WITH END OF TIME PERIOD PAYMENTS

f = (a*((1+r)^n-1))/r

f is the future value of the annuity.
a is the annuity.
r is the interest rate per time period.
n is the number of time periods

example:

you will be investing 500 at the end of each month into an account for 20 years earning 6% per year compounded monthly.
how much will you have in 20 years?

a = 500
r = 6% per year / 100 / 12 = .005 per month.
n = 20 years * 12 = 240 months.

formula of:
f = (a*((1+r)^n-1))/r becomes:
f = (500*((1.005)^240-1))/.005 which is equal to 231020.4476.

if you use the online discrete compounding financial calculator provided, you would do the following:
present value = 0
future value = 0
number of periods = 20 * 12 = 240
payment amount = 500
interest rate per period = 6% / 12 = .5% per month entered as .5
payment at is set to end.
click on fv and the calculator will tell you that fv = -231,020.45
disregard the sign of the result.

<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

FUTURE VALUE OF AN ANNUITY WITH BEGINNING OF TIME PERIOD PAYMENTS

f = ((a*((1+r)^n-1)/r))*(1+r)

f is the future value of the annuity.
a is the annuity.
r is the interest rate per time period.
n is the number of time periods

example:

you will be investing 500 at the beginning of each month into an account for 20 years earning 6% per year compounded monthly.
how much will you have in 20 years?

a = 500
r = 6% per year / 100 / 12 = .005 per month.
n = 20 years * 12 = 240 months.

formula of:
f = ((a*((1+r)^n-1)/r))*(1+r) becomes:
f = ((500*((1.005)^240-1))/.005)*(1.005) which is equal to 232175.5498

if you use the online discrete compounding financial calculator provided, you would do the following:
present value = 0
future value = 0
number of periods = 20 * 12 = 240
payment amount = 500
interest rate per period = 6% / 12 = .5% per month entered as .5
payment at is set to beginning.
click on fv and the calculator will tell you that fv = -232,175.55
disregard the sign of the result.

<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

PRESENT VALUE OF AN ANNUITY WITH END OF TIME PERIOD PAYMENTS

p = (a*(1-1/(1+r)^n))/r

p is the present value of the annuity.
a is the annuity.
r is the interest rate per time period.
n is the number of time periods.

example:

your loan payments are 500 due at the end of each month for a period of 6 years at 18% interest rate per year compounded monthly.
how much did you borrow?

a = 500
r = 18% per year / 100 / 12 = .015 per month.
n = 6 years * 12 = 72 months.

formula of:
p = (a*(1-1/(1+r)^n))/r becomes:
p = (500*(1-1/(1.015)^72))/.015 which is equal to 21922.33339

if you use the online discrete compounding financial calculator provided, you would do the following:

present value = 0
future value = 0
number of periods = 6 * 12 = 72
payment amount = 500
interest rate per period = 18% / 12 = 1.5% per month entered as 1.5
payment at is set to end.
click on pv.
calculator will tell you that pv = -21,922.33
disregard the sign of the result.

<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

PRESENT VALUE OF AN ANNUITY WITH BEGINNING OF TIME PERIOD PAYMENTS

p = ((a*(1-1/(1+r)^n))/r)*(1+r)

p is the present value of the annuity.
a is the annuity.
r is the interest rate per time period.
n is the number of time periods.

example:

your loan payments are 500 due at the beginning of each month for a period of 6 years at 18% interest rate per year compounded monthly.
how much did you borrow?

a = 500
r = 18% per year / 100 / 12 = .015 per month.
n = 6 years * 12 = 72 months.

formula of:
p = ((a*(1-1/(1+r)^n))/r)*(1+r) becomes:
p = ((500*(1-1/(1.015)^72))/.015)*(1.015) which is equal to 22251.16039

if you use the online discrete compounding financial calculator provided, you would do the following:

present value = 0
future value = 0
number of periods = 6 * 12 = 72
payment amount = 500
interest rate per period = 18% / 12 = 1.5% per month entered as 1.5
payment at is set to beginning.
click on pv.
calculator will tell you that pv = -22,251.17
disregard the sign of the result.

<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

ANNUITY FOR A FUTURE AMOUNT WITH END OF TIME PERIOD PAYMENTS

a = (f*r)/((1+r)^n-1)

a is the annuity.
f is the future amount.
r is the interest rate per time period.
n is the number of time periods.

example:

you will retire in 50 years and you will want to have 1 million dollars in the bank at that time that you can draw from.

how much do you need to invest at the end of each month for 40 years at 6% interest per year compounded monthly so that you will have 1 million dollars at the end of that period?

f = 1000000
r = 6% per year / 100 / 12 = .005 per month.
n = 40 years * 12 = 480 months

formula is:
a = (f*r)/((1+r)^n-1)
formula becomes:
a = (1000000*.005)/((1.005)^480-1) which is equal to 502.136406

if you use the online discrete compounding financial calculator provided, you would do the following:
present value = 0
future value = 1000000
number of periods = 40 * 12 = 480
payment amount = 0 
interest rate per period = 6% / 12 = .5% entered as .5
payment at is set to end.
click on pmt.
calculator will tell you that pmt = -502.14
disregard the sign of the result.

<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

ANNUITY FOR A FUTURE AMOUNT WITH BEGINNING OF TIME PERIOD PAYMENTS

a = ((f*r)/((1+r)^n-1))*(1/(1+r))

a is the annuity.
f is the future amount.
r is the interest rate per time period.
n is the number of time periods.

example:

you will retire in 40 years and you want to have 1 million dollars in the bank at that time that you can draw from.

how much do you need to invest at the beginning of each month for 40 years at 6% interest per year compounded monthly so that you will have 1 million dollars at the end of that period?

f = 1000000
r = 6% per year / 100 / 12 = .005 per month.
n = 40 years * 12 = 480 months

formula is:
a = ((f*r)/((1+r)^n-1))*(1/(1+r))
formula becomes:
a = ((1000000*.005)/((1.005)^480-1))/(1.005) which is equal to 499.638215

if you use the online discrete compounding financial calculator provided, you would do the following:
present value = 0
future value = 1000000
number of periods = 40 * 12 = 480
payment amount = 0 
interest rate per period = 6% / 12 = .5% entered as .5
payment at is set to beginning.
click on pmt.
calculator will tell you that pmt = -499.64
disregard the sign of the result.

<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

ANNUITY FOR A PRESENT AMOUNT WITH END OF TIME PERIOD PAYMENTS

a = (p*r)/(1-(1/(1+r)^n))

a is the annuity.
p is the present amount.
r is the interest rate per time period.
n is the number of time periods.

example:

you have 1 million dollars invested in an account that you will be drawing from at the end of each year for the next 30 years at 12% compounded annually.
how much will you be drawing each year?

p = 1000000
r = 12% per year / 100 / 1 = .12 each year.
n = 30 years * 1 = 30 years.

formula is:
a = (p*r)/(1-(1/(1+r)^n))
formula becomes:
a = (1000000*.12)/(1-(1/(1.12)^30)) which is equal to 124143.6576

if you use the online discrete compounding financial calculator provided, you would do the following:
present value = 1000000
future value = 0
number of periods = 30 * 1 = 30
payment amount = 0 
interest rate per period = 12% / 1 = 12% entered as 12
payment at is set to end.
click on pmt.
calculator will tell you that pmt = -124,143.66
disregard the sign of the result.

<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

ANNUITY FOR PRESENT AMOUNT WITH BEGINNING OF TIME PERIOD PAYMENTS

a = ((p*r)/(1-(1/(1+r)^n)))*(1/(1+r))

a is the annuity.
p is the present amount.
r is the interest rate per time period.
n is the number of time periods.

example:

you have 1 million dollars invested in an account that you will be drawing from at the beginning of each year for the next 30 years at 12% compounded annually.
how much will you be drawing from each year?

p = 1000000
r = 12% per year / 100 / 1 = .12 each year.
n = 30 years * 1 = 30 years.

formula is:
a = ((p*r)/(1-(1/(1+r)^n)))*(1/(1+r))
formula becomes:
a = ((1000000*.12)/(1-(1/(1.12)^30)))*(1/(1.12)) which is equal to 110842.5514

if you use the online discrete compounding financial calculator provided, you would do the following:
present value = 1000000
future value = 0
number of periods = 30 years * 1 = 30 years
payment amount = 0 
interest rate per period = 12% per year / 1 = .12 per year
payment at is set to beginning.
click on pmt.
calculator will tell you that pmt = -110,842.55
disregard the sign of the result.

<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

FUTURE VALUE OF A PRESENT AMOUNT USING CONTINUOUS COMPOUNDING

f = p*e^(r*n)

f is the future value.
p is the present amount.
e is the scientific constant equal to 2.718281828.
r is the interest rate per time period.
n is the number of time periods.

example:

you are investing 5000 at 9% per year compounded continuously for a period of 12 years.
how much do you have at the end of the investment period?

p = 5000
r = 9% per year / 100 / 1 = .09 per year.
n = 12 years * 1 = 12 years.

formula is:
f = p*e^(r*n)
formula becomes:
f = 5000*e^(.09*12) which is equal to 14723.39776

if you use the online continuous compounding financial calculator provided, you would do the following:
go to the future value portion of the calculator.
present value = 5000
r (annual interest rate) = 9% per year entered as 9
t (number of years invested) = 12
the calculator will tell you that the future value is $14,723.40

<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

PRESENT VALUE OF A FUTURE AMOUNT USING CONTINUOUS COMPOUNDING

p = f/(e^(r*n))

p is the present value.
f is the future amount.
e is the scientific constant equal to 2.718181828.
r is the interest rate per time period.
n is the number of time periods.

example:

you need to have 5000 in 12 years.
you want to know how much you have to invest today at 9% interest rate per year for 12 years compounded continuously.

f = 5000
r = 9% per year / 100 / 1 = .09 per year.
n = 12 * 1 = 12 years

formula is:
p = f/(e^(r*n))
formula becomes:
p = 5000/(e^(.09*12) which is equal to 1697.977628

if you use the online continuous compounding financial calculator provided, you would do the following:
go to the present value portion of the calculator.
future value = 5000
r (annual interest rate) = 9% per year entered as 9
t (number of years invested) = 12
the calculator will tell you that the present value is equal to $1,697.98

<a href= "#formulas">go to start of formula section</a>

<a href = "#notes">go to start of notes section</a>

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