Financial Planning (Basics)

Time value of money in Discrete Time series 

Ever wondered how Banks calculate Annual Payments?  (Without considering Annual Income of an Individual)

This section illustrates the method to calculate annual equal payments for n number of years required to pay $A sum at annual interest rate R. Without using excel files and heavy calculations.

For example: You are borrowing $300,000 to buy a house (at year zero) and you sign a loan contract which fixes the interest rate at 5% per annum and you pay 30 annual equal payments (from year 1 to year 30). What is the amount of money you have to pay each year?

Let's first  illustrate the annual payments, with assumption that the equal annual payment/installment is '$X'.

= $300,000

We know that sum (S) of discrete series (geometric) is given by the formula :

S= (1-B^T+1)/ (1-B)    (Proof is shown at end of this section)

In this case B= 1/(1+r)= 1/(1+0.05) = 1/1.05

Now taking 'X' and '1/(1.05)' out the bracket, the series becomes:

x/(1.05)[1+]=$300,000

(X/1.05)[1+(1/1.05)+ (1/1.05)²)+ (1/1.05)³+ (1/1.05)⁴+ (1/1.05)⁵……………+(1/1.05)²⁹]= $300,000

After converting the sum inside the bracket as per the formula we can write:

[X/ (1.05)]*[ (1-(1/1.05)³⁰)/(1-(1/1.05))]  = $300,000

So value of annual installment ($X) =   ($300,000/[ (1-(1/1.05)³⁰)/(1-(1/1.05))])*1.05 = $19,515.4

This means you will have to pay $19,515.4 as annual installments for $300,000 to be paid in 30 years at 5% annual interest rate. 

Proof of the Summation formula:

Let, 'S' be the sum of discrete series 'B'
S= 1+B+B²+ B³+B⁴+B⁵+B⁶   …………………….B^{T      (i)
Now multiply both the sides by B, so the series becomes}
SB= B+B²+ B³+B⁴+B⁵+B⁶   …………………….B^T+1                 (ii)
^{Now Subtract (ii) from (i)}
S-SB= 1- B^T+1

Now, take ‘S’ common and transfer (1-B) to right hand side. 

Finally, the Sum of discrete series 'B' is given by the formula.
S= (1-B^T+1)/ (1-B)

(It is also possible to calculate in excel, but if you remember this formula you can calculate summation of discrete number upto any length without having to drag the excel cell) 

# Techniques  covered in blog posts labelled AED are taught in courses Mathematics for Applied Economics, and Applied Economics Dynamics offered at Crawford School of Public Policy @ The Australian National University.