This web page will allow you to calculate the present value of
multiple periodic cash flows each defined by a variety of parameters.
It is also able to compute a parameter of one or more of these cash
flows if the desired present value is given. The resulting cash flows
are summarized individually as well as in a spreadsheet form.
I've developed a functional notation to describe a series of related
cash flows compactly. This may seem confusing at first, but I've
given numerous examples and it allows for a great deal of flexibility
in describing cash flows.
The cash flows are all defined in terms of a period. The actual
length of that period is up to you and isn't explicitly known by the
calculator. The interest rates specified must all be effective
rates for the period you are using, so you must adjust discount and
growth rates accordingly. In other words, these calculations assume
that the interest rates given are for one period with compounding
based on that period. If you need to convert an interest rate from
one period to another (for example, converting a yearly discount rate
to a monthly one), I have a converter
that does this.
You may specify the final present value and solve for a specific
parameter in the cash flow functions. This is done by substituting 'X'
for the parameter for which you want to solve. If you use an 'X' in more
than one cash flow function, it must stand for the same parameter in
each one. This is illustrated by one of the examples.
I have attempted to insure that the calculations performed here are
correct, however this calculator is provided without guarantee.
All of the parameters after amt in these
functions have default values, and you may leave off those on the
right side whose default values are what you want. For example, if
you want: PVI(120,5,0,0,1,0,0):Title you can type: PVI(120,5). The
label at the end of a cash flow function is optional. Here are the
cash flow functions and a description of the parameters:
Cash Flow Function Definitions: (See examples below.)
You have two choices as to what you want this to compute. This
program can either compute the present value of a set of cash flows or
the value of a parameter given the total present value. Check the one
you want and if you choose to compute the value of a parameter, input
the present value to use. Also, if you choose to calculate a
parameter, put an X in place of that parameter in each appropriate
cash flow function. If you choose to solve for either
flows or fy
it will find the nearest whole number.
Cash flows to Analyze:
Enter the title to be used on the resulting pages:
Enter the positive cash flows here, one per line:
Enter the negative cash flows here, one per line:
Maximum number of spreadsheet rows displayed:
Cash Flow Function Summary:
You win a million dollar lottery paid over 20 years starting now.
Each yearly check will be for $50,000. Unfortunately, you'll have
to pay 20% ($10,000) of it each year in taxes. Additionally, you hire
an accountant to keep track of this windfall. You pay him $500 the
first year, and his fee will increase at a rate of 5% a year. What's
the present value (discounted at 6%) of your lottery winnings minus
the taxes and accounting fees?
The answer is: $477,172.13
Now let's say that each year there's a 1% chance that the organization
paying the lottery goes bankrupt, and you won't get any more payments.
There's also a 2% chance that they will default on any year's payment.
These factors will affect the lottery payments and the taxes, but not
the accountant because he has to take care of your previous earnings too.
These factors are taken into account as follows:
The answer is: $433,063.30
What is the present value (from the Bank's point of view) of a 15 year
mortgage at 7.2% interest with a monthly payment of $1200?
First, figure out the number of periods, which will be 15 (years)
times 12 (months in a year) = 180. Then figure out the monthly
interest rate, which will be 7.2 divided by 12 or 0.6. Note that this
mortgage rate is compounded monthly, so the effective yearly rate is a
little larger than 7.2%, but usually it is written as a 7.2% yearly
rate. See the example below on converting interest rate periods.
Generally mortgages are paid at the start of the month beginning the
month after they are set up, so in this case
sy will be 1.
The answer is: $131,861.36
Your friend Prickly Pete sets up his own lottery. He's selling 1000
tickets at $1.00 a piece. The grand prize winner will receive $100 at
the drawing and each year for the next 5 years (a total of six
payments). There will be two second prize winners who will receive
$50 at the drawing and each year for the next 5 years. The drawing is
later today and he offers to sell you the last ticket. What's the
present value of buying the ticket using a discount rate of 5%,
ignoring taxes and the questionable reputation of your friend?
Here we'll use the pl parameter to
indicate the probability of winning. More specifically, the pl parameter is the probability that the
cash flow won't happen, so we have to subtract the probabilites of
winning from 1 to get the appropriate pl parameter to use. There is a one in one
thousand chance of winning the Grand Prize (0.001), so the probability
of not winning it is 0.999. There is a two in one thousand chance of
winning the Second Prize (0.002), so the probability of not winning
that is 0.998.
The answer is: $0.07. You actually stand to gain 7 cents a ticket
on average by participating in this lottery. Perhaps Pete isn't
so prickly after all.
A More Contrived Example: The Adoption
Venus Moneybags has just given birth to a baby girl named Athena. The
pregnancy was unplanned, and she doesn't want to keep the baby--she
wants you to adopt her. However, even though Venus doesn't want to
raise Athena, she does want her daughter to have an expensive
upbringing. Venus offers you the following enticements and
stipulations to adopt Athena:
She will pay you $20,000 at the beginning of each of the next 21
years, and, as a bonus for a job well done, at the start of the 22nd
year, she will pay you an additional lump sum of $40,000 if you manage
to flip a coin three times in a row and not get three tails. You may
decide how you spend the annual $20,000 allowance, but at the
beginning of each of those first 21 years, you must transfer some of
this annual sum into a trust fund for the girl. The first year you
will deposit $2,000 into the account, and you will increase that
amount by 10% each subsequent year. Additionally, the title to a
piece of timberland will be transferred to you at the time of the
adoption. You can never sell it, but because the land will be managed
through sustainable forestry practices, it will generate an income of
$5,000 at the start of each year in perpetuity.
Of course, you will be required to pay for Athena's tuition at the
elite Marbury School for Girls from first grade (age 6) through
twelfth grade (age 18). You don't know exactly how much that will
cost, but you do know that the current tuition is $8,000 per year and
historically the price has grown at a rate of 6% per year. Venus
thinks that after twelve years at such a prestigious academy, the girl
ought to pay for college herself.
This all sounds pretty good to you, but Venus informs you that there
is one more payment you have to make. Because Athena's father was a
champion race walker, each year of the Summer Olympics until Athena is
21, you must make a contribution to them in his honor in the amount of
$10,000. (Suppose the next Summer Games are the year after next.)
And there's one last catch. Venus herself is an avid hang-glider,
which is a risky sport. If Venus dies while hang-gliding, you will no
longer receive the $20,000 yearly payment. There is a 2% chance per
year that she will die in such an accident.
What is the present value of this whole transaction for you (ignoring
taxes) if you use a discount rate of 5%?
The answer is: $139,567.43
Computing the Value of a Parameter
Your neighbor asks to borrow $10,000 from you with the following
repayment plan. He'll pay $1,000 at the end of each year for ten
years, and additionally will pay you $4,000 at the end of the tenth
year. You can get 6% interest in your bank's money market account.
Are you better off lending to your neighbor or keeping your money in
the bank? How would you change the terms of the deal so it would be
equivalent to leaving your money in the bank? Assume that you're
certain that you will be paid back and that you won't need the money
before the ten years is up.
There are several ways to decide if this is a better deal than putting
your money in the bank. First let's try figuring out the present
value of those payments given a discount rate of 6%. If the present
value is greater than $10,000 then it's a better deal than leaving
your money in the bank and if it's less, then it's not.
Note that this calculator assumes that a cash flow occurs at the
beginning of a period. In this example, the cash flows are at the end
of the ten years starting in the first year (year 0). For calculation
purposes, a cash flow at the end of one period is the same as one at
the beginning of the next. Because of this, convert the cash flows in
this example to start at the beginning of year 1 (the second period)
and go for the next ten years.
This reveals that the present value is only $9,593.67, so you're
better off leaving the money in the bank. Now let's try setting the
present value to $10,000 and solving for the discount rate. If the
computed rate is less than 6%, then we know that by lending the money
to the neighbor, we're getting a lower rate than we would by leaving
it in the bank. Here's what to enter to figure out the discount rate:
The discount rate calculated is 5.295%, so this result agrees with the
one above, and you're better off leaving your money in the bank. Now,
how could your neighbor change his terms to make them equivalent to
the bank's as far as earnings?
First, let's try seeing what his yearly payment would have to be to bring
up the present value to $10,000 at a 6% discount rate.
He'd have to pay you $1,055.21 yearly instead of $1,000. If he didn't
want to increase his annual payments, by how much would he have to
increase the lump sum?
So, the lump sum must be increased to $4,727.68 to be equivalent.
Converting Interest Rate Periods
Often you'll want to convert an interest rate from one period to another.
Since this calculator requires that the discount and growth rates be
effective for the period used, in these cases you'll need to convert.
Your friend and neighbor Willet Hale wants to borrow $5,000 to
replace the roof on his house. He says that he'd like to pay it off
in equal monthly installments over the next 24 months and that you can
choose the amount. You decide that since he's your friend and you see
his tattered roof out your front window, you'll make the loan at the
same rate the bank gives you, which is 5%.
You want to solve for 'X' to get the monthly payment, but you can't
use 5% for the discount rate, because you need a discount rate for a
month not a year. Dividing it by 12 isn't correct because that
doesn't take into account compounding. You can use my
rate converter calculator to convert the
5% per year rate to 0.407412% per month. Now just plug it in and solve
The answer is: $218.22
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