## 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- PVIF_{4,5%} 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)xPVIVA _{10%,5} = 800 x(1.10)x 3.79079 x =**

**800 x 4.16987 = $3,335.9**