COMPOUND INTEREST FORMULAS (Use to learn procedures and for examinations and quizzes)
Frequency of
payments
Direction Values are Carried Through Time
Forward in Time (Compounding)
Single Payment
1 - Final value
(Value as of end point of series)
Vn = V0 (1+i)n
Backward in Time
(Discounting)
2 - Present value
3 - Rate earned when V0 and Vn
(Value as of starting point of
a time period )
V0 = Vn/(1+i)n
=fv(rate, npr, pmt, [pv],[type])
Vn/V0 = (1+i)n
=pv(rate, npr, pmt, fv],[type])
Series ends at a given point of time (Terminal Series)
Annual payments
and annual rate
of interest
4 - Final value of an annuity
(Value as of ending point in time
of a series of annual payments)
Vn = a ((1+i)n -1)/i
are known
Isolate i on right side,
6 - Present value of an
annuity
(Value as of the starting point
in time of a series of annual
payments)
Solve for (1+ i) by taking nth
root of Vn/V0 and subtracting 1
i =( Vn/V0)1/n - 1
Series goes on indefinitely
(Perpetual Series)
10 - Capital value of an Annuity
(Value as of the starting point
in time of a perpetual series of
annual payments)
V0 = a/i
n
5 - Annual payment to a sinking
fund
(Amount of annual payment, a,
needed to have set amount, Vn,
at end of series of payments)
a = Vn [i /((1+i)n -1)]
Periodic
payments
(interest period
different from
payment period
8 - Final value of a series of
periodic payments
(Value as of ending point in time
of a series of periodic payments,
p)
n
V0 = a ((1+i) -1)/(i (1+i) )
7 - Annual installment
payment
(Amount of annual payment,
a, needed to pay off the
amount, V0, over n years)
a = V0 [( i (1+i)n)/((1+i)n -1)]
12 - Annual payment to obtain
a capital value
(Amount of annual payment, a,
obtainable in perpetuity from
the capital value, V0)
a = V0 i
9 – Present value of a series
11 – Capital value of a periodic
of periodic payments, a.
series (Faustman formula)
V0 = p [((1+i)nt -1) / ((1+i)t -1)
((1+i)nt)]
V0 = p /((1+i)t -1)
Vn = p ((1+i)nt -1)/((1+i)t -1)
Interest rate
r ≡ real interest rate
adjustment for
f ≡ rate of inflation
i=r+f+rf
(1+i) = (1+r) (1+f) r = [(1+i)/(1+f)] - 1
inflation
i ≡ nominal interest rate
i ≡ nominal interest rate
n Ξ total number of payments and total number of periods if interest and payment periods are the same
a ≡ amount of annual payment
p ≡ amount of periodic payment
t ≡ number of interest periods between payments when interest and payment periods differ.
nt ≡ total number of interest periods
W.L. Hoover, 2011
COMPOUND INTEREST FORMULAS (Use to learn procedures and for examinations and quizzes)
Compound Interest Formula Assumptions
Time line
Time (years) elapsed, also called point of time
1
0
Year 1
2
3
Year 3
Year 2
n
Year n
Number of the period (year)
Single payment that is compounded - Payment is made at point 0, beginning of the first period, and is compounded for n
years.
Single payment that is discounted – Payment is made at the end of the nth year, and is discounted back n periods to
point 0.
Annuities and Periodic payments –
All payments are made at the end of each period. Therefore, starting at point 0, the first payment occurs at the
end of year 1 and is compounded over n periods. The last payment, made in the nth period, is not compounded
at all.
If there is an initial payment, i.e. at point 0, as well as the series of annuity or periodic payments, the payment in
year 0 must be compounded separately since it’s not included in the formula.
In the annuity and periodic payment formulas that discount payments back to the beginning of period 1 (point
0), the first payment is discounted back 1 period, and the nth payment is discounted back for n periods. If there
is an initial payment it must be added to the value obtained from the formula.
See demonstration of these concepts in Excel file
W.L. Hoover, 2011