cimabusiness mathematicsmr. rajesh gunesh future value and compounding –after 1 year, your cd is...

CIMA Business Mathematics Mr. Rajesh Gunesh

Future Value and CompoundingFuture Value and Compounding

– After 1 year, your CD is worth $2,500 (1+0.0675) = $2,668.75

– After 2 years, the CD is worth$2,668.75 (1+0.0675) = $2,500 (1+0.0675)2 = $2,848.89

– After 3 years, the CD is worth$2,848.89 (1+0.0675) = $2,500 (1+0.0675)2 = 3,041.19

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?



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



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

0 1 2 3i = 6.75%

PV = –2500

FV = 2500 (1.0675)3 = 3041.19



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)nm ,

where m is the number of periods per year.– FV = $2,500(1+0.0675/4)34 = $3,055.98.


Present Value and DiscountingPresent 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?

0 1 2 3i = 4.5%

FV = 5000PV = ?

$5,000 = PV (1+0.045)3

PV = $5,000 / (1+0.045)3 = $4,381.48



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



If discounting occurs at a frequency other than annually:

PV = FV / (1+i/m)nm ,

where m = the number of discounting periods per year


AnnuitiesAnnuities

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

0 1 2 3i

PMT PMT PMTPV = ?

FV = ?


Future Value of an AnnuityFuture Value of an AnnuityIf 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?

0 1 2 39%

100 100100

327.81

+ 100(1.09)2 = 118.81+ 100(1.09) = 109.00

= 100.00


Future Value of an AnnuityFuture 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


Future Value of an AnnuityFuture Value of an Annuity

If compounding occurs at a frequency other than annually,

FV = PMT [(1+i/m)nm – 1] / (i/m)


Present Value of an AnnuityPresent Value of an AnnuityHow much is this $100, 3-year annuity worth today,

assuming a 9% discount rate?

0 1 2 39%

100 100100

253.13

91.74 = 100 / (1.09)84.17 = 100 / (1.09)2

77.22 = 100 / (1.09)3


Present Value of an AnnuityPresent 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


Present Value of an AnnuityPresent Value of an Annuity

If compounding occurs at a frequency other than monthly,

PMT [1 – 1 / (1+i/m)nm] / (i/m)


Effective Annual RatesEffective 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 (1+0.0475/12)12 = $104.85


Effective Annual RatesEffective 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

cimabusiness mathematicsmr. rajesh gunesh future value and compounding –after 1 year, your cd is...

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