Present Value of Annuity

Present Value of Annuity (Concept):

   PV of Annuity = C/ (1+r) + C/(1+r)^{^2} + C/(1+r)^{^3}

                             = C [1/(1+r) + 1/(1+r)^{^2}  + 1/(1+r)^{^3}]

                    PV_n = C [1/(1+r)^{^n-2} + 1/(1+r)^{^n-1}  + 1/(1+r)^{^n}]

PV of annuity: Example I

How much money do you need in the bank today so that you can spend $10,000 every year for next 25 years, starting at the end of this year? Suppose r = 5%.

Solution:

              Always draw a timeline first:

             Using excel calculator:

                                               = PV (rate,nper,pmt,[fv],[type])

                                               = PV (0.05, 25, 10000)

                                               = $140,939

PV of Annuity: Example II

You plan to attend a business school and you are forced to take out $100,000 in a loan at 10%. You want to figure out your yearly payments, given that you will have 5 years to pay back the loan.

Solution:

Using excel calculator,

                                    = pmt(rate, nper, pv, [fv], [type])

                                                        = pmt(0.1, 5, 100000)

                                    = $26,379

 Loan Amortization: Example II followed:

Another beauty about finance is suppose if you want to calculate the fourth year balance or if you want to know how much you have to pay at the beginning of 4 year then you might think you have to make the whole amortization table. But you don't have to. If you know,
               rate = 10%
               yearly payment = $26,379
               number of year = 2 ( remember you want to calculate fourth year balance and the year remaining will be 4 and 5 so its only 2 year period left )
Now, use excel calculator,
             = PV(rate, nper, pmt, [fv], [type] )
             = PV(0.1, 4, 26379)
             = $45,781 (little bit down than $45,786 but it is the actual answer)