present value of annuity

Question

Calculate the present value of an annuity of $25000 paid at the end of each month of 2 years. The annual interest rate is 12%.

Answer 1

We assume that the interest is a compound one calculated monthly and the annuity is a fixed amount.

Let's set the annuity as 'a'. 

Then at the end of the 2nd year, the lump sum value of $25,000 with interest should be equal to the sum of value of annuity paid to you. 

 

***Note that the value of money with the compound interest is,

 

a(1+r/n)^nt

 

where, a is the initial money, r is the annual interest, n is the number of times when interest is calculated per year, and t is the number of years.

 

We need a table to present the methodology used in calculation. 

 

The value of your investment at the end of 12th month of 2nd yr

           0 yr          12th mo of 1st yr                 value of investment

                                                                     at 12th mo of 2nd yr                                                                                                         

            25,000   25,000(1+0.12/12)¹²              25,000(1+0.12/12)²⁴

 

Your annuity            

 

1st yr

 

1st mo   2nd mo   3rd mo   4th mo .........  11th mo   12th mo                                     value of annuity
                                                                                                                         at 12th mo of 2nd yr

 

 

    a ..................................................................                                        a(1+0.12/12)^{((0.12/12)*23)}
                a ............................................................ .                                a(1+0.12/12)^{((0.12/12)*22)}
                             a.................................................                                 a(1+0.12/12)^{((0.12/12)*21)}
                                          a ..................................                                  a(1+0.12/12)^{((0.12/12)*20)}

                                                 ............
                                                                 a ...                                          a(1+0.12/12)^{((0.12/12)*13)}
                                                                                a                               a(1+0.12/12)^{((0.12/12)*12)}                                                

 

2nd yr
1st mo 2nd mo 3rd mo 4th mo ......... 11th mo 12th mo

value of annuity
at 12th mo of 2nd yr

a .................................................................. a(1+0.12/12)((0.12/12)*11)
a ................................................. . a(1+0.12/12)((0.12/12)*10)
a ......................... a(1+0.12/12)((0.12/12)*9)
a .................. a(1+0.12/12)((0.12/12)*8)

............
a ... a(1+0.12/12)((0.12/12)*1)
a a(1+0.12/12)((0.12/12)*0)

So, the sum of the value of annuity at the end of 12th month of the 2nd year should be equal to the value of your investment at the same time.

That is,

25,000(1+0.12/12)24 = a(1+0.12/12)((0.12/12)*23) + a(1+0.12/12)((0.12/12)*22)
+ a(1+0.12/12)((0.12/12)*21) + a(1+0.12/12)((0.12/12)*20) + .....
+a(1+0.12/12)((0.12/12)*1) +a(1+0.12/12)((0.12/12)*0)


****note that

a + ar + ar2 + ... + arn-1 = a(rn-1)/r-1
when r > 1

When we calculate the equation, then

25,000(1+0.12/12)24 = a[(1+0.01)24-1]/[(1+0.01)-1]

so,

a = 1176.84

Your annuity is $1176.84

                                                      

 

Answer 2

A = P[1 + (r/n)]^nt

 

P = Principal amount ($25,000)

r = annual rate of interest (0.12)

t = time to mature (2 years)

n = number of times the interest is compounded each year (in this case, 12)

A = present value

 

A = ($25,000){1 + [(0.12)/(12)]}^[(12)*(2)]

 

A = ($25,000)[(1.01)²⁴] = ($25,000)(1.26973) = $31,743.40