Summary of Interest Formula. Relationships of Discrete Compounding.
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Transcript of 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,