Mr. Rajesh Gunesh

1
Future Value and Compounding
Suppose you have 2,500 that you can put in a
three-year bank CD yielding 6.75 annually. How
much money will you have when this CD matures?

• After 1 year, your CD is worth
• 2,500 (10.0675) 2,668.75
• After 2 years, the CD is worth
• 2,668.75 (10.0675) 2,500 (10.0675)2
  2,848.89
• After 3 years, the CD is worth
• 2,848.89 (10.0675) 2,500 (10.0675)2
  3,041.19

2
Future Value and Compounding

• More generally,
• FV PV (1i)n ,
• where
• FV the future value of a lump sum
• PV the initial principal, or present value of
  the lump sum
• i the annual interest rate
• n the number of years interest compounds

3
Future Value and Compounding

• A good way of understanding this process is
  through the use of a time line

i 6.75
PV 2500
4
Future Value and Compounding

• How much would this CD be worth at maturity if
  interest compounds quarterly?

• The trick is to convert the interest rate into a
  periodic rate and compound each period, rather
  than annually.

FV PV (1i/m)n?m , where m is the number of
periods per year.

• FV 2,500(10.0675/4)3?4 3,055.98.

5
Present Value and Discounting

• Suppose you will receive 5,000 three years from
  now. If you can earn 4.5 on your savings, how
  much is this worth to you today?

i 4.5
FV 5000
PV ?
5,000 PV (10.045)3
? PV 5,000 / (10.045)3 4,381.48
6
Present Value and Discounting

• More generally,
• PV FV / (1i)n ,
• where
• FV the future value of a lump sum
• PV the initial principal, or present value of
  the lump sum
• i the annual discount rate
• n the number of years

7
Present Value and Discounting

• If discounting occurs at a frequency other than
  annually
• PV FV / (1i/m)n?m ,
• where
• m the number of discounting periods per year

8
Annuities

• An annuity is a series of payments or receipts
  made at regular intervals for a determined period
  of time

9
Future Value of an Annuity

• If you will receive 100 at the end of each of
  the next 3 years and can invest it at 9, how
  much will it be worth at the end of the 3 years?

9
100.00
10
Future Value of an Annuity

• More generally,
• FV PMT PMT(1i) PMT(1i)2
• PMT(1i)3 PMT(1i)n1

n1 PMT ? (1i)nt1
t0
PMT (1i)n 1 / i
11
Future Value of an Annuity

• If compounding occurs at a frequency other than
  annually,

FV PMT (1i/m)n?m 1 / (i/m)
12
Present Value of an Annuity

• How much is this 100, 3-year annuity worth
  today, assuming a 9 discount rate?

13
Present Value of an Annuity

• More generally,
• PV PMT / (1i) PMT / (1i)2
• PMT / (1i)3 PMT / (1i)n

n PMT ? 1 / (1i)t
t1
PMT 1 1 / (1i)n / i
14
Present Value of an Annuity

• If compounding occurs at a frequency other than
  monthly,

PMT 1 1 / (1i/m)n?m / (i/m)
15
Effective Annual Rates

• Which provides the highest total return, a
  savings account that pay 5.00 compounded
  annually or one that pays 4.75 compounded
  monthly?
• One way to answer this is to calculate the future
  value of 100 invested in each
• PV1 100 (1.05) 105.00
• PV2 100 (10.0475/12)12 104.85

16
Effective Annual Rates

• Alternatively, you can calculate the effective
  annual rate associated with each account
• EAR (1 i/m)m 1
• EAR1 (1 0.05/1)1 1 0.0500 5.00
• EAR2 (1 0.0475/12)12 1 0.0486 4.86
• Effective annual rates are directly comparable in
  terms of total yield