Mrs. Hart, at age 65, can expect to live for 25 years. If she can invest at 5% per annum compounded monthly, how much does she need now to guarantee herself $900 every month for the next 25 years?

PV = Integral from 0 to T of R(t)e^(-rt)dt

where R(t) is dollars per year continuously for T, T is years, r is interest rate.

PV = Integral 0 to 25 of (900*12)e^(-.05t)dt
** Take anti-derivative **
= (10800/-.05)e^(-.05t) evaluated from 0 to 25

= (10800/-.05)e^(-.05(25)) - (10800/-.05)e^(-.05(0))

PV = $154114.96

PV = Integral from 0 to T of R(t)e^(-rt)dt

where R(t) is dollars per year continuously for T, T is years, r is interest rate.

PV = Integral 0 to 25 of (900*12)e^(-.05t)dt
** Take anti-derivative **
= (10800/-.05)e^(-.05t) evaluated from 0 to 25

= (10800/-.05)e^(-.05(25)) - (10800/-.05)e^(-.05(0))

PV = $154114.96

Reiny, 153,954.05 is coming up wrong. any suggestions?

To calculate the amount Mrs. Hart needs to invest now to guarantee herself $900 every month for the next 25 years, we need to use the present value of an ordinary annuity formula.

The formula for the present value of an ordinary annuity is:

PV = PMT * ((1 - (1 + r)^(-n)) / r)

Where:
PV = Present Value (the amount Mrs. Hart needs to invest now)
PMT = Monthly payment ($900)
r = Interest rate per period (5% per annum compounded monthly, or 5% / 12)
n = Total number of periods (25 years * 12 months per year)

Let's plug in the values and calculate the present value:

r = 5% / 12 = 0.05 / 12
n = 25 years * 12 months = 300

PV = $900 * ((1 - (1 + (0.05 / 12))^(-300)) / (0.05 / 12))

Now, let's calculate it step by step:

1. Calculate the value inside the parentheses:

(1 + (0.05 / 12)) = 1.004167

2. Calculate the exponent:

(-300)

3. Calculate the value inside the brackets:

(1.004167)^(-300) = 0.246305

4. Calculate the numerator:

(1 - 0.246305) = 0.753695

5. Calculate the denominator:

(0.05 / 12) = 0.004167

6. Calculate PV:

PV = $900 * (0.753695 / 0.004167) = $162,453.26

Therefore, Mrs. Hart needs to invest approximately $162,453.26 now to guarantee herself $900 every month for the next 25 years, assuming a 5% annual interest rate compounded monthly.

Present Value

= 900(1 - 1.0041666..)^-300)/.0041666...
= $153,954.05