What is the Compound Annual Interest Rate for a 5-Year Annuity?

C

To solve this problem, we need to use the formula for the present value of an annuity, which is:

PV = C x [1 - (1+r)^(-n)] / r

where:

• PV = present value of the annuity (i.e., the amount you pay upfront to purchase the annuity)
• C = annual payment amount
• r = annual interest rate
• n = number of years

We know that the PV of the annuity is $1,000, the annual payment amount is $263.80, and the number of years is 5. So we can plug those values into the formula and solve for r:

$1,000 = $263.80 x [1 - (1+r)^(-5)] / r

Multiplying both sides by r and rearranging, we get:

$1,000r = $263.80 x [r - r(1+r)^(-5)]

Expanding the right-hand side, we get:

$1,000r = $263.80r - $263.80r(1+r)^(-5)

Adding $263.80r(1+r)^(-5) to both sides, we get:

$1,000r + $263.80r(1+r)^(-5) = $263.80r

Dividing both sides by $263.80r, we get:

(1+r)^(-5) + 3.79 = 1

Subtracting 3.79 from both sides, we get:

(1+r)^(-5) = -2.79

Taking the reciprocal of both sides, we get:

1+r = (-2.79)^(-1/5)

Simplifying, we get:

1+r = 0.896

Subtracting 1 from both sides, we get:

r = -0.104

This is a negative interest rate, which doesn't make sense in this context. So we made a mistake somewhere in our calculations. Checking our work, we notice that we made an error in the expansion of the right-hand side of the equation:

$1,000r = $263.80r - $263.80r(1+r)^(-5)

The correct expansion should be:

$1,000r = $263.80 x [(1 - (1+r)^(-5)) / r]

So let's start over and use the correct equation:

$1,000 = $263.80 x [(1 - (1+r)^(-5)) / r]

Multiplying both sides by r and rearranging, we get:

$1,000r = $263.80 x [1 - (1+r)^(-5)]

Expanding the right-hand side, we get:

$1,000r = $263.80 - $263.80(1+r)^(-5)

Adding $263.80(1+r)^(-5) to both sides, we get:

$1,000r + $263.80(1+r)^(-5) = $263.80

Dividing both sides by $263.80, we get:

3.789r + (1+r)^(-5) = 1

Subtracting 1 from both sides, we get:

3.789r + (1+r)^(-5) - 1 = 0

We can use numerical methods to solve this equation for r. One approach is to use the Newton-Raphson method, which involves iteratively improving an initial guess for r. We can start with an initial guess of 0.1 (i.e., 10%) and apply the following formula to obtain a new guess:

r_new = r_old - f(r_old) /