Question

Ibama received a ten-year annuity. It paid $100 at the end of each quarter for the first four years, and $31 each month for the remaining six years. Express the value at the time of the last payment in terms of annuity symbols introduced in this section if the annual effective interest rate is i for the first four years and j for the following six years.

Evaluate the expression if i = 3% and j = 4.2%.

(Round your answer to the nearest cent.)

Answer 1

Given that,

Ibama received a ten-year annuity.

For the First Four Years.

Each Four Months = $100

No.Of Periods = (4*12)/4

= 48/4

= 12

Rate of Interest Per Year

->The Rate of Interest per period

=i/3 [ 3 Periods in 1 Year]

Let,

p[(1 + r)" - 11/1

Where,

P = Periodic Payment

n = Numeric of Periods

Therefore,

  

100[(1 + i)12 - 1] Fu = 3+100 100[(1 + 300)12 – 1]

Amount = (Fv) 4 Years

3 + 10-[(1+ ) 12 -1]

Rest 6 Years

Payment per Month = &31 = P2

No. of Months = 6*12 = 72 Periods

  $Rate\,\, of \,\,Interest \,\,per\,\, Period \,\,=\,\,\frac{i}{12}$   

(Total Fv) = (Compound Interest for P2) + Future Value of Series

(1 - rn 1. r)" - I = P2 (1+r)" + (Periodic in r) *

1 - - 1) * TE + zlº+ 1) zd =

  $Total\,\,Fv\,\,=\,\,\frac{3*10^{4}[(1+\frac{i}{300})^{12}-1]}{i}\,\,*\,\,[1+\frac{j}{100}]^{72}\,\,+\,\,31\,\,+\,\,\frac{[1+\frac{j}{1200}^{72}-1]}{\frac{j}{1200}}$

Here Given Conditions,

i = 3% and j = 4.2%

  

Fv = 3 * 10'(1 + - 1) [1472 , 31 + 1200 * ( + - 11 4)

  

Divide = سنت انت =

  $10^{4}[(1+\frac{3}{300})^{12}-1]$

3o6 +1)12 Cancel 380 - 200 100+1)12

$Join\,\,\frac{1}{100}+1\,\,=\,\,\frac{101}{100}\,\,=(\frac{101}{100})^{12}$

$10^{4}(\frac{101}{100})^{12}-1$

$10^{4}\,\,=\,\,10000(\frac{101}{100})^{12}-1\\ \\ Decimal\,\,=1268.25030$

$(1+\frac{4.2}{100})^{72}\\ \\ 1+\frac{4.2}{100}\,\,=\,\,1.042\\ \\ (1.042)^{72}\,\,=\,\,19.34085$

  $\frac{31\times \:1200\left(\left(1+\frac{4.2}{1200}\right)^{72}-1\right)}{4.2}$

Multiply 31 * 1200 = 37200

| 37200( +1) 2 -1) 4.2

$\left(1+\frac{4.2}{1200}\right)^{72}\,\,=\,\,(1.0035)^{72}$

$=\,\,\frac{37200\left(1.0035^{72}-1\right)}{4.2}$

1268.25030 * 19.34085 + 2533.40939

$=>\,\,27,061.3608$