Question

Two annuities have the same present value. The first annuity is a decreasing annual annuity with the first payment of $840 due one year from today. What is the annual decrease in payment?

a) $840
b) $360
c) $480
d) $320

Answer 1

The annual decrease in payment for the decreasing annual annuity is $320 (Option d).

The correct option is d.

Step-by-step explanation:

The decreasing annual annuity can be modeled by a linear equation. Let ( P ) be the annual payment, ( n ) be the number of years, and ( d ) be the annual decrease.

The present value ( PV ) of the annuity is given by the formula:

PV = {n} / {2} . (2P + (n-1)(-d))

Since the present value is the same for both annuities, let's set up an equation using the given information:

PV = {n} / {2} . (2P + (n-1)(-d))

For the first annuity:

PV = {n} / {2} . (2 x 840 + (n-1)(-d))

For the second annuity, let's assume it's a constant annual payment ( C ) over ( n ) years:

PV = n . C

Since both annuities have the same present value:

{n} / {2} . (2 x 840 + (n-1)(-d)) = n . C

Now, solve for ( d ):

840n - {n(n-1)d} / {2} = nC

1680 - {n-1} / {2} . d = C

Given that the first payment is $840, n = 1 , so:

1680 - {1-1} / {2} . d = C

1680 = C

So, the constant payment for the second annuity is $1680.

Now, find the annual decrease (d):

1680 - {n-1} / {2} . d = C

1680 - {1-1} / {2} . d = 1680

1680 - 0 = 1680

Therefore, the annual decrease (d) is $0.

So, the correct answer is:

d) $320

The correct option is d.