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 ( ) = $2,668.75
After 2 years, the CD is worth
$2, ( ) = $2,500 ( )2 = $2,848.89
After 3 years, the CD is worth
$2, ( ) = $2,500 ( )2 = 3,041.19
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2. Future Value and Compounding
More generally,
FV = PV (1+i)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
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3. Future Value and Compounding
A good way of understanding this process is through the use of a time line: 1 2 3
i = 6.75%
PV = –2500
FV = (1.0675)3 =
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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 (1+i/m)nm ,
where m is the number of periods per year.
FV = $2,500( /4)34 = $3,
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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? 1 2 3
i = 4.5%
PV = ?
FV = 5000
$5,000 = PV ( )3
 PV = $5,000 / ( )3 = $4,381.48
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6. Present Value and Discounting
More generally,
PV = FV / (1+i)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
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7. Present Value and Discounting
If discounting occurs at a frequency other than annually:
PV = FV / (1+i/m)nm ,
where
m = the number of discounting periods per year
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8. Annuities
An annuity is a series of payments or receipts made at regular intervals for a determined period of time 1 2 3 i
PMT
PV = ?
FV = ?
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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? 1 2 3 9%
100 =
+ 100(1.09) =
+ 100(1.09)2 =
327.81
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10. Future Value of an Annuity
More generally,
FV = PMT + PMT(1+i) + PMT(1+i)2
+ PMT(1+i)3 + … + PMT(1+i)n–1
n–1
= PMT  (1+i)n–t–1
t=0
= PMT [(1+i)n – 1] / i
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11. Future Value of an Annuity
If compounding occurs at a frequency other than annually,
FV = PMT [(1+i/m)nm – 1] / (i/m)
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12. Present Value of an Annuity
How much is this $100, 3-year annuity worth today, assuming a 9% discount rate? 1 2 3 9%
100
91.74 = 100 / (1.09)
77.22 = 100 / (1.09)3
84.17 = 100 / (1.09)2
253.13
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13. Present Value of an Annuity
More generally,
PV = PMT / (1+i) + PMT / (1+i)2
+ PMT / (1+i)3 + … + PMT / (1+i)n n
= PMT  1 / (1+i)t
t=1
= PMT [1 – 1 / (1+i)n] / i
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14. Present Value of an Annuity
If compounding occurs at a frequency other than monthly,
PMT [1 – 1 / (1+i/m)nm] / (i/m)
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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 ( /12)12 = $104.85
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16. Effective Annual Rates
Alternatively, you can calculate the effective annual rate associated with each account
EAR = (1 + i/m)m – 1
EAR1 = ( /1)1 – 1 = = 5.00%
EAR2 = ( /12)12 – 1 = = 4.86%
Effective annual rates are directly comparable in terms of total yield
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