Formulas for Loans, Mortgages
and Savings accounts
Bernard Liengme
January 2012
The interest rates banks charge on credit card
Interest
Rate
balances is criminal! Somewhere around 26%
• When you borrow money you pay interest; when
you save money you earn interest.
• The interest is computed as a percentage rate
and in normally quoted as a yearly (per annum)
value. It is then called the APR – annual
percentage rate.
• But you are generally charged (or earn) on a
monthly basis
• If rate is the per annum percentage then the
monthly percentage is rate/12
Number of Periods (nper)
• A period is the interval between which you
make a payment on the load (or the bank
deposits earned interest into your account)
• All of our examples will use monthly periods
• If nper is the number of years over which you
will pay of the load then in our formulas we
replace nper by nper * 12.
Principal
• The starting amount of a load to distinguish it
from the interest
• Get the spelling right it is NOT principle
• This word does not appear in Excel formulas
Present Value PV
• A $100 bill is worth (surprise!) $100. That is its
present value.
• You have won a prize that pays you $100 a month
for 10 years. What is it worth? Or what is its
present value?
• Its PV is the amount of money someone would
need to invest and get $100 a month for 10 years
leaving nothing in the bank at the end of that
time
• We have to assume the interest rate remains the
same as today’s value.
Future Value FV
• A $100 bill is worth $100 today and will be
worth $100 in 10 years time (It may not buy as
many pints of beer but it will still have a value
of $100). That is it PV and its FV.
• I deposit $100 in the bank, wait 10 years and
then withdraw it. How much will I get? Or
what will be its future value?
• The catch phrase is the time value of money.
Payment pmt
• You borrow $1,000 from the bank for 1 year.
• Every month you give the bank $86.99
(normally this will be deducted from you
regular bank account automatically)
• After 12 months the loan is paid off
• When you began the loan: PV = 1000, FV = 0,
rate = 8%/12 and nper = 1 * 12. We will see
soon how to compute the payment pmt.
How are these thing related?
• By the money equation shown here
nper

1

rate
)
 1 

nper
  fv  0
pv *(1  rate)  pmt *(1  rate * type) * 


rate


• But we will not have to worry about it
Money has a value and a direction
• There is a difference between giving and
receiving money.
• Money received is a credit and we give it a
positive value
• Money paid into the bank is a debit and we
give it a negative sign
Excel formulas
• All the formulas for personal finance use the
terms PV, FV, rate, nper and pmt
• For everything but rate, we can rearrange the
money equation to get just one term on the
left.
nper

1

rate
)
 1 

nper
  fv  0
pv * (1  rate)  pmt * (1  rate * type) * 


rate


nper
pmt * (1  rate * type)  1  rate)  1 
fv

pv  
*
nper

 (1  rate)nper
(1  rate)
rate


PV
• Present value =PV(rate,nper, pmt, [fv],[type])
• Arguments in brackets [] can be omitted when
zero
• My prize gives me $100 each month for 10
years, what is its PV? If the rate is 8% pa then
we compute the present value with:
=PV(8%/12,10*12,100)
What if I will also get $500 at the end
of the 10 years?
=PV(8%/12,10*12,100, 500)
And the answer is
• Open the workbook Finance and on the PV
worksheet note the way we do the
calculations. Why are the two PV values
shown in red?
• Because it someone went to the bank to set
up a price like this they would have to give
money to the bank. Money would flow away
from them.
FV
• Future value =FV(rate, nper, pmt, [pv],[type])
• I deposit $100 each month into a saving account. The
interest rate is 8% pa.
• PV = 0 since the saving account had nothing in it before the
first payment. What formula should I use to find how much
I can expect after 3 years?
=FV(8%/12,3*12,-100)
Why is pv given the value -100 with a negative sign?
See worksheet FV
Payment
• Payment: =PMT(rate, nper, pv, [fv], [type])
• I borrow $1,000 at 8% pa and pay off the loan
in 12 monthly installments. How much must I
pay?
• See worksheet PMT
Number of Periods
• I plan to save $250 each month until I have
$100,000. Assuming the APR is 8%, how many
monthly payments must I make?
• Number of periods
=NPER(rate, pmt, pv, fv, type)
• Rate=APR/12; PMT = -250 (note the negative)
• PV = 0, FV = 100,000, type = 0 (end of month)
• Worksheet NPER
Rate
• Rate is found with
RATE(nper,pmt,pv,fv,type,guess)
• When the syntax is shown like this the bold
arguments are required and the others are
optional – in the other form we used [] for
optional arguments
• The money equation cannot be solved for rate, so
Excel uses a ‘trial-and-error’ method like
GoalSeek. Hence the guess argument: it
sometimes helps to give Excel a start!
Rate (cont)
• I plan to save $400 each month for 25 years to
give myself a pension. If my target is $500,000
what must the APR be during this time?
• The answer is 9.65% which is unlikely at the
present time. See worksheet Rate
IPMT and PPMT
• Where as PMT computes the payment on a
loan, IPMT computes the interest part of this
payment and PPMT computes how much
went to paying down the principal.
• See worksheet IPMT