MATH SOLVE

3 months ago

Q:
# Peter decides to invest $1,200,000 in a period annuity that earns 4.6% APR compounded monthly for a period of 20 years. How much money will Peter be paid each month?

Accepted Solution

A:

Given is the Present Value of annuity, PV = 1,200,000 dollars.

Given that 4.6% APR compounded monthly i.e. r = 4.6%/12 = 0.003833

Given that 20 years of investment i.e. N = 20x12 = 240

It says to find monthly income from the annuity.

We know the formula for Periodic Payment (when PV is known) is given as follows :-

[tex] Periodic \;payment = \frac{PV}{\frac{1-\frac{1}{(1+r)^{N}}}{r}} \\\\Periodic \;payment = \frac{1,200,000}{\frac{1-\frac{1}{(1+0.003833)^{240}}}{0.003833}} \\\\Periodic \;payment = \frac{1,200,000}{\frac{1-\frac{1}{(1.003833)^{240}}}{0.003833}} \\\\Periodic \;payment = \frac{1,200,000}{\frac{1-\frac{1}{2.504681211}}{0.003833}} \\\\Periodic \;payment = \frac{1,200,000}{\frac{1-0.399252406}{0.003833}} \\\\Periodic \;payment = \frac{1,200,000}{\frac{0.600747594}{0.003833}} [/tex][tex] Periodic \;payment = \frac{1,200,000}{156.7303924} \\\\Periodic \;payment = 7656.460128 \\\\Periodic \;payment = 7656.46 \;dollars \;per \;month [/tex]Hence, Monthly Income would be 7,656.46 dollars.

Given that 4.6% APR compounded monthly i.e. r = 4.6%/12 = 0.003833

Given that 20 years of investment i.e. N = 20x12 = 240

It says to find monthly income from the annuity.

We know the formula for Periodic Payment (when PV is known) is given as follows :-

[tex] Periodic \;payment = \frac{PV}{\frac{1-\frac{1}{(1+r)^{N}}}{r}} \\\\Periodic \;payment = \frac{1,200,000}{\frac{1-\frac{1}{(1+0.003833)^{240}}}{0.003833}} \\\\Periodic \;payment = \frac{1,200,000}{\frac{1-\frac{1}{(1.003833)^{240}}}{0.003833}} \\\\Periodic \;payment = \frac{1,200,000}{\frac{1-\frac{1}{2.504681211}}{0.003833}} \\\\Periodic \;payment = \frac{1,200,000}{\frac{1-0.399252406}{0.003833}} \\\\Periodic \;payment = \frac{1,200,000}{\frac{0.600747594}{0.003833}} [/tex][tex] Periodic \;payment = \frac{1,200,000}{156.7303924} \\\\Periodic \;payment = 7656.460128 \\\\Periodic \;payment = 7656.46 \;dollars \;per \;month [/tex]Hence, Monthly Income would be 7,656.46 dollars.