Sam and Alex are planning to retire in Adelaide Hills in 20 years.Currently, the typical house that they're interested in costs $200 000, but they expect inflation to increase the price of the house at a rate of 4% over the next 20 years.In order to buy a house upon retirement, what must they save each year in equal annual end-of-year deposits if they can earn 10% annually?

Question

Quizplus · Accepted Answer

First, we need to figure out the future value of the house in 20 years with inflation. Using the formula FV = PV x (1 + r)^t, where PV is the present value, r is the annual interest rate, and t is the number of years:

FV = $200,000 x (1 + 0.04)^20
FV = $468,235.07

This means that in 20 years, the house they want will cost $468,235.07 due to inflation.

Now, we can use the formula for the annuity payment (A) to figure out how much they must save each year:

A = (FV x r) / ((1 + r)^t - 1)

Plugging in the numbers:

A = ($468,235.07 x 0.10) / ((1 + 0.10)^20 - 1)
A = $23,998.57 / 12.166
A = $1,972.35

So, they need to save $1,972.35 annually in order to afford the house upon retirement. However, the question asks for the equal annual end-of-year deposits, which means they need to save the same amount each year. To find this, we can use the present value of an annuity formula:

PV = A x ((1 - (1 + r)^-t) / r)

Plugging in the numbers:

PV = $1,972.35 x ((1 - (1 + 0.10)^-20) / 0.10)
PV = $30,989.40

Therefore, they need to save $30,989.40 over the next 20 years, which means they need to save $1,549.47 each year (dividing by 20). However, the question asks for the answer in end-of-year deposits, which means we need to subtract the interest earned on the savings. Using the formula for the future value of an annuity (FVA), where PV is the present value, r is the annual interest rate, t is the number of years, and A is the payment:

FVA = PV x (1 + r)^t + A x (((1 + r)^t - 1) / r)

Plugging in the numbers:

FVA = $0 x (1 + 0.10)^20 + $1,549.47 x (((1 + 0.10)^20 - 1) / 0.10)
FVA = $0 + $77,694.81
FVA = $77,694.81

The interest earned on the savings is:

Interest = FVA - PV - (A x t)
Interest = $77,694.81 - $30,989.40 - ($1,549.47 x 20)
Interest = $7,269.91

Therefore, the answer is $1,549.47 - $7,269.91 = $7650.94, or choice B.

Sam and Alex Are Planning to Retire in Adelaide Hills