Time value of Money - Present & Future value of Money

3 Multiple Periods: Uneven and Even (Annuities)

Part II Multiple Periods:  Uneven and Even (Annuities)

♦Periodic Uneven Cash Flows

What is the value of the following set of cash flows today?   The interest rate is 8% for all cash flows.

  Year and  Cash Flow

                    1:    $ 300        2:    $ 500        3:    $ 700      4:    $ 1000

♦Solution:  Find Each Present Value and Add

277.78

428.67

555.68

735.03

=  1997.16

♦Periodic Cash Flow: Even Payments

An annuity is a level series of payments. For example, four annual payments, with the first payment occurring exactly one period in the future is an example of an ordinary annuity. 

A.  Present value of an annuity:  The present value of each of the cash flows is the value of the annuity.  This could be done one at a time, but this might be tedious. Annuity Present Value Interest Factor

PVIFA = [1/(1+i) + 1/(1+i)2 + ... + 1/(1+i)t]

Example:

What is the present value of a 4-year annuity, if the annual interest is 5%, and the annual payment is $1,000?

 

i = 5%; PMT = $1,000; t =4; PV = ?
PV = 1,000 /(1.05) + 1,000/(1.05)2 + 1,000/(1.05)3+ 1,000/(1.05)4 ←LONG WAY

Factor out the single sum interest rate factors:

PV = 1,000 x [1/(1.05) + 1/(1.05)2+1/(1.05)3+ 1/(1.05)4] =

PV = 1,000 x [PVIFA (4,5%)] = ← SHORT WAY

Calculate:  PVIFA(4,5%)  =  1-1/(1+i)t = 1- PVIF4,5%   1- 0.8227 =  3.54595.
                                                i                5%              .05

PV =  1,000 x [3.5460]  = $3,546.

B.  Future value of an annuity:
   

  Annuity Future Value Interest Factor

 FVIFA = [1+ (1+i) + (1+i)2 + ... + (1+i)t-1].

Example: What is the future value of a 4-year annuity, if the annual interest is 5%, and the annual payment is $1,000?

i = 5%; PMT = $1,000; t =4; FV = ?

$1,000x [1+ (1.05) + (1.05)2 + (1.05)3] =

$1,000  x  [FVIFA (4,5%)] =

$1,000 x [4.3101] = $4,310.1

C.Annuity Due

Question:   Compare the payments of the annuity due, above, with those of the ordinary annuity earlier.  What is the difference?  How does this difference affect its value?

Answer:  Each payment in an annuity due occurs one period earlier than it would in ordinary annuity.  Both present value and future value of each payment in an annuity due if (1+i) times greater than it would be for an ordinary annuity.

Question:  What is the present value of the above four-year annuity due?

 $1,000 x [1 + 1/(1+i) + 1/(1+i)2 + 1/(1+i)3]

=          $1,000 x (1+i) x [1/(1+i) + 1/(1+i)2 + 1/(1+i)3+1/(1+i)4]

=          $1,000 x (1+i) x PVIFA i,4

PV interest factor of an annuity due is: (1+i)·PVIFA

FV interest factor of an annuity due is: (1+i)·FVIFA

Problem.What is the present value of an annuity due of five $800 annual              payments discounted at 10%?       

800 x (1.10)xPVIVA10%,5 =                                                                                                                           800 x(1.10)x 3.79079 x  =

800 x 4.16987 =  $3,335.9