Question

What are the future value and present value of an annuity of 1300 payable monthly for 30 payments, with the first payment at the end of 19 months from the present time with interest at 12% compounded semiannually?

Answer 1

Compute the effective annual rate (EAR), using the equation as shown below:

$EAR=(1+\frac{Annual\space rate }{compounging\space period}^{compounding\space period}-1$

$=(1+\frac{0.12}{2})^2-1$

$=1.1236-1\\=12.36\%$

Hence, the effective annual rate is 12.36%.

Compute the monthly rate, using the equation as shown below:

$monthly \space rate=\frac{EAR}{12\space months}$

$=\frac{12.36\%}{12\space months}$

$=1.03\%$

Hence, the monthly rate is 1.03%.

Compute the present value annuity factor (PVIFA), using the equation as shown below:

$PVIFA=\frac{1+(1+monthly\space rate^{-Time}}{monthly\space rate}$

$=\frac{1-(1+0.0103)^{-30}}{0.0103}$

$=25.6949451483$

Hence, the present value annuity factor is 25.6949451483.

Compute the value of annuity after 18 months from now, using the equation as shown below:

$Value\space of \space annuity=monthly\space payment\times PVIFA\\=1300\times25.6949451483\\=33403.4286927$

Hence, the value of annuity after 18 months from now is 33403.4286927.

Compute the future value of an annuity, using the equation as shown below:

$Future\space value\\=Value\space of\space annuity\space 18\space months\times (1+monthly\space rate)^{Time}$

$=33403.4286927\times(1+0.0103)^{30}\\=33403.4286927\times1.35991132228\\=45425.7008821$

Hence, the future value of an annuity is 45425.70.

Compute the present value of an annuity, using the equation as shown below:

$present \space value=\frac{value\space of\space annuity \space after\space 18\space months}{1+ monthly\space rates^{Time}}$

$=\frac{33403.4286927}{(1+0.0103)^{18}}$

$=\frac{33403.4286927}{1.20255889154}\\=27776.9587233$

Hence, the present value of the annuity is 27776.96.

The future value of an annuity is 45425.70.

The present value of the annuity is 27776.96.