in credit cooperative that offers terms in deferred payment 1year and 5 years to pay find the present different annuity of php 500 every month if the money is worth 10% compunded monthly

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 to1,000atthebeginningofeachyear,for10years.Assuminga106,759.02

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} }PresentValue=A+

r(1+r)

n−1

A((1+r)

n−1

−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} }PresentValue=

r(1+r)

n

A((1+r)

n

−1)

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=0.048809r(semi−annual)=(1+r(annual))

2

1

−1=(1+0.1)

2

1

−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} } =6,759.02PresentValue=1,000+

0.1((1+0.1)

9

1,000((1+0.1)

9

−1)

=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} }=6,601.75PresentValue=500+

0.048809((1+0.048809)

19

500((1+0.048809)

19

−1)

=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} }=6,144.57PresentValue=

0.1(1+0.1)

10

1,000((1+0.1)

10

−1)

=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} }=6,294.52PresentValue=

0.048809(1+0.048809)

20

500((1+0.048809)

20

−1)

=6,294.52

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

Best of luck.