Question

You have the opportunity to invest in several annuities. Which of the following 10-year annuities has the greatest present value (PV)? Assume that all annuities earn the same positive interest rate. An annuity that pays $1,000 at the beginning of each year An annuity that pays $500 at the beginning of every six months An annuity that pays $1,000 at the end of each year An annuity that pays $500 at the end of every six months

Answer 1

Answer:

Ans. The greates present value is the one that pays $1,000 at the beginning of each year, for 10 years. Assuming a 10% effective annual rate, this present value is equal to $6,759.02

Explanation:

Hi, in order to find the present value of an annuity, first we have to take into account if this annuity is paid at the beginning or at the end of each period. This is the formula for an annuity that pays at the beginning of each year.

$PresentValue=A+\frac{A((1+r)^{n-1}-1) }{r(1+r)^{n-1} }$

In the case of an annuity that pays at the end of each year is:

$PresentValue=\frac{A((1+r)^{n}-1) }{r(1+r)^{n} }$

Where:

A= annuity

r= rate of return

n= periods that will be paid

But, there are 2 types of annuities here(that pay at the beginning of the period), one is paid every year and other that pays every 6 months. Since we are going to assume a rate of return of 10% effective annually, for the first type (pays every year at the beginning) r=10% and n=10, and for the second one (pays at the beginning of every 6 months) r=4.8809% and n=20.

Let me show you the formula to turn an annual effective rate into a semi-annual effective rate.

$r(semi-annual)=(1+r(annual))^{\frac{1}{2} } -1=(1+0.1)^{\frac{1}{2} } -1=0.048809$

Let´s find each present value.

A) Pays 1,000 at the beginning of each year

$PresentValue=1,000+\frac{1,000((1+0.1)^{9}-1) }{0.1((1+0.1)^{9} } =6,759.02$

B) Pays at the beginning of every six months

$PresentValue=500+\frac{500((1+0.048809)^{19}-1) }{0.048809((1+0.048809)^{19} }=6,601.75$

C) Pays 1,000 at the end of each year

$PresentValue=\frac{1,000((1+0.1)^{10}-1) }{0.1(1+0.1)^{10} }=6,144.57$

D) Pays 500 at the end of every six months

$PresentValue=\frac{500((1+0.048809)^{20}-1) }{0.048809(1+0.048809)^{20} }=6,294.52$

So the answer is A) the one that pays 1,000 at the beginning of each year.

Best of luck.