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Andy looks at his account and notices that if the current monthly interest rate stays constant he is expected to have $54,000 in 6 years (i.e. once 6 years have elapsed) and $67,000 in 8 years. a. How much money does he have now (at time 0)? b. If his predictions are correct, except after 7 years, the nominal rate halves and then stays at that value, how much money will he have in 8 years? Assume the interest rate is compounded monthly.

Respuesta :

Answer:

initial principal amount is $ 28272.36

after 7 year interest rate  = 0.4514 %

amount after 8 year is = $63489.85

Explanation:

given data

amount = $54000 for 6 year i.e 72 months

amount = $67000 for 8 year i.e 96 months

to find out

How much money does he have now and after 7 years, the nominal rate halves and then stays at that value, how much money will he have in 8 years

solution

we consider here principal amount P = x

then amount equation will be

amount = P ×[tex]( 1+r)^{t}[/tex]   ......................1

here P is initial principal amount and r is rate and t is time

54000 = x ×[tex]( 1+r)^{72}[/tex]

x = [tex]\frac{54000}{( 1+r)^{72}}[/tex]      .......................3

67000 = x ×[tex]( 1+r)^{96}[/tex]  

x = [tex]\frac{67000}{( 1+r)^{96}}[/tex]       .........................4

so from equation 3 and 4 we get

[tex]\frac{54000}{( 1+r)^{72}}[/tex] = [tex]\frac{67000}{( 1+r)^{96}}[/tex]

solve it we get r

r = 0.9028 %

and x will be then from equation 3

x = [tex]\frac{54000}{( 1+0.009028)^{72}}[/tex]  

x = 28272.36

so initial principal amount is $ 28272.36

and

interest rate after 7 year is

interest rate = [tex]\frac{0.009028}{2}[/tex] = 0.4514 %

and

amount after 8 year will be

amount = 28272.36 × [tex](1+0.009028)^{84}[/tex] × [tex](1+0.004514)^{12}[/tex]

amount after 8 year is = $63489.85

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