using the formula for present value of ordinary annuity or the amortization formula to solve this problem.
PV=13000
I= .015
PMT=550
n?
Is I the annual interest or monthly interest?
It will be assumed monthly interest.
Present Value, P = 13000
Payment (monthly) A = 550
interest (monthly) i = 0.015
The amortization formula would equate future value with the sum of all the payments, all increased at rate of interest i.
Future value = sum of all payments
Let R=1+i
PRn = A + AR + AR² + AR³ + ... + ARn-1
=A(Rn+1)/(R-1) (by factoring)
Hence
(Rn-1)/((R-1)*Rn) = P/A
To solve for the period n, there is no explicit formula to calculate.
The easiest way is to calculate the payment for a given period n.
If the payment matches 550, then the estimated n is correct.
For example,
The equation can be converted into a formula for the monthly payment, A
A=P(R-1)R^n/(R^n-1)
For
P=13000
R=1.015
we make a first estimate from
13000/550=23.6
We know n>23.6, so try 30
A=13000(.015)1.015^30/(1.015^30-1)
=541.3 < 550
So we try 29 payments
A=556.1
We then know that the period n lies between 29 and 30, and for all practical purposes, we would put it at 30.
The right-hand side of the amortization formula should read:
A(Rn-1)/(R-1) (by factoring)
To solve for 'n' in the present value of an ordinary annuity formula, you'll need to rearrange the formula to isolate 'n'. The formula is:
PV = PMT * [(1 - (1 / (1 + I)^n)) / I]
In this case, you have the following values:
PV = 13000
I = 0.015
PMT = 550
Now, let's solve for 'n':
1. Start by rearranging the formula to isolate 'n':
PV = PMT * [(1 - (1 / (1 + I)^n)) / I]
PV * I = PMT * (1 - (1 / (1 + I)^n))
Divide both sides of the equation by (PMT * I):
(PV * I) / (PMT * I) = 1 - (1 / (1 + I)^n)
2. Simplify the equation:
PV / PMT = 1 - (1 / (1 + I)^n)
3. Substitute the given values:
13000 / 550 = 1 - (1 / (1 + 0.015)^n)
23.64 = 1 - (1 / (1.015)^n)
4. Subtract 1 from both sides:
23.64 - 1 = - (1 / (1.015)^n)
5. Simplify the equation:
22.64 = - (1 / (1.015)^n)
6. Multiply both sides by -1 to remove the negative sign:
-22.64 = 1 / (1.015)^n
7. Take the reciprocal of both sides:
-1 / 22.64 = (1.015)^n
8. Solve for 'n' by taking the logarithm of both sides to base 1.015:
log base 1.015 (-1 / 22.64) = log base 1.015 ((1.015)^n)
n = [log base 1.015 (-1 / 22.64)] / [log base 1.015 (1.015)]
Now, using a scientific calculator or software, evaluate the right-hand side of the equation to find the value of 'n'.