Summary of Interest Formula. Relationships of Discrete Compounding.

Summary of Interest Formula

),,/(

1),,/(

%%

NiPFNiFP

Relationships of Discrete Compounding

),,/(

1),,/(

%%

NiAPNiPA

),,/(

1),,/(

%%

NiAFNiFA

),,/)(,,/(),,/( %%% NiPFNiAPNiAF

N

k

KiFPNiAP1

),,/(),,/( %%

N

k

KNiPFNiAF1

),,/(),,/( %%

iNiPANiFA ),,/(),,/( %%

Deferred Annuity

• Deferred annuities are uniform series that do not begin until some time in the future.

• If the annuity is deferred J periods then the first payment (cash flow) begins at the end of period J+1.

$6.6074)0373.3(2000)4,12,/( %17 APAP

$46.884)1456.0(6.6074)17,12,/( %170 FPFP

$333,29)8666.5(5000)5,8,/( %5 AFAF

$697,433)7853.14(333,29)35,8,/( %540 PFPF

Multiple Interest Formula

$19.176,5)2998.4(82.1203)8,20,/( %08 PFPF

$73.313)2606.0(82.1203)8,20,/( %0 PAPA

$73.313)0606.0(19.5176)8,20,/( %8 FAFA

HFPAPHAPHP 8839.7)5,10,/)(3,10,/()4,10,/(2 %%%0

QFPQFPQP 3132.0)7,10,/()2,10,/( %%0

HQ 8839.73132.0 HQ 172.25

$63.682)8,12,/()2,12,/(1000 %%1 FPAPP

$75.224)4,12,/(63.682)4,12,/( %%1 PAPAPA

Interest Rate that Vary with Time

Nominal and Effective Interest Rate

• The annual rate is known as a nominal rate.

• A nominal rate of 12%, compounded monthly, means an interest of 1% (12%/12) would accrue each month, and the annual rate would be effectively somewhat greater than 12%.

Consider a principal amount of 1000$ to be invested for a year at a nominal rate 12% compounded semiannually.

Interest rate = 6% per 6 months.

The interest earned during the first 6 months = 1000×0.06 = 60$

Total interest and principal at 6 months = 1000+60 = 1060$

The interest earned during the second 6 months = 1060×0.06 = 63.6$

Total interest earned during the year = 60+63.6 = 123.6$

Effective annual interest rate = 123.6/1000 = 12.36%

11

M

M

ri

M is the number of compounding interest per year

i is effective interest rate per year

r is the nominal interest rate per year

11

M

M

ri

%814.171781.0112

165.01

12

i

Compounding More Often than Once per Year

$4.181)015.1(100)10,5.1,/( 40% PFPF

Example:

A loan of 15,000$ requires monthly payments of 477$ over a 36-month period of time. These payments include both principal and interest.

1.What is the nominal interest rate?

),,/( NiPAPA

)36,,/(15000477 moiPA

monthper75.00318.0)36,,/( iiPA mo

nominal interest rate = 0.75 ×12 = 9%

2. What is the effective interest rate per year

yearper%38.9112

09.0111

12

M

M

ri

3. Determine the amount of unpaid loan principle after 20 month?

59.7166)0243.15(477)16%,75.0,/(47720 APP

Interest Formulas for Continuous Compounding and Discrete Cash Flows

• Interest is typically compounded at the end of discrete periods.

• We can allow compounding to occur continuously throughout the period.

• Continuous compounding assumes that cash flows occurs at discrete intervals, but that compounding is continuous throughout the interval.

1 rei

Chapter 4 Home Work:

1, 3, 5, 7, 8, 10, 12, 14, 18, 20, 22, 25, 31, 34, 38, 47, 49, 54, 55, 60, 62, 64, 66, 68, 72, 95, 99, 100, 103, 107, 112, 113, 115, 116,