Solved on Feb 07, 2024

The beach house costs $\$ 800000$ now. Inflation is expected to increase the price by $5\%$ annually over 20 years. How much will the house cost after 20 years? If they invest equal end-of-year payments earning $13\%$ annually, how much must they invest each year to buy the house in 20 years?

STEP 1

Assumptions
1. The current cost of the beach house is $800,000.
2. The annual inflation rate is 5%.
3. The time frame for the price increase due to inflation is 20 years.
4. Sami and Lubna will make equal end-of-year payments into an investment account.
5. The investment account has an annual return rate of 13%.
6. The goal is to accumulate enough money to buy the house in 20 years.

STEP 2

First, we need to calculate the future cost of the beach house after 20 years with a 5% annual inflation rate. We can use the formula for compound interest to find the future value:
$Future\, Value = Present\, Value \times (1 + Inflation\, Rate)^{Number\, of\, Years}$

STEP 3

Now, plug in the given values for the present value, inflation rate, and number of years to calculate the future cost of the house.
$Future\, Value = \$800,000 \times (1 + 0.05)^{20}$

STEP 4

Calculate the future value of the house after 20 years.
$Future\, Value = \$800,000 \times (1.05)^{20}$

STEP 5

Compute the future cost of the house.
$Future\, Value \approx \$800,000 \times 2.653297705$

STEP 6

Round the future cost of the house to two decimal places.
$Future\, Value \approx \$2,122,638.16$

STEP 7

Next, we need to calculate how much Sami and Lubna must invest each year to accumulate the future cost of the house. We can use the formula for the future value of an annuity due to calculate the annual payment:
$Future\, Value = Payment \times \left(\frac{(1 + Interest\, Rate)^{Number\, of\, Payments} - 1}{Interest\, Rate}\right)$

STEP 8

Rearrange the formula to solve for the annual payment:
$Payment = \frac{Future\, Value}{\left(\frac{(1 + Interest\, Rate)^{Number\, of\, Payments} - 1}{Interest\, Rate}\right)}$

STEP 9

Now, plug in the values for the future value, interest rate, and number of payments to calculate the annual payment.
$Payment = \frac{\$2,122,638.16}{\left(\frac{(1 + 0.13)^{20} - 1}{0.13}\right)}$

STEP 10

Calculate the denominator of the payment formula.
$Denominator = \left(\frac{(1 + 0.13)^{20} - 1}{0.13}\right)$

STEP 11

Compute the value of the denominator.
$Denominator \approx \left(\frac{(1.13)^{20} - 1}{0.13}\right)$

STEP 12

Round the denominator to two decimal places.
$Denominator \approx \left(\frac{10.834705 - 1}{0.13}\right)$

STEP 13

Continue the calculation of the denominator.
$Denominator \approx \left(\frac{9.834705}{0.13}\right)$

STEP 14

Finish the calculation of the denominator.
$Denominator \approx 75.6515769$

STEP 15

Now, calculate the annual payment.
$Payment = \frac{\$2,122,638.16}{75.6515769}$

STEP 16

Compute the annual payment.
$Payment \approx \$28,069.11$

STEP 17

Since the annual payment must be made at the end of each year, we need to adjust the payment to account for the fact that the first payment will not earn interest in the first year. This means we divide the payment by the interest rate factor for one period.
$Adjusted\, Payment = Payment \times (1 + Interest\, Rate)$

STEP 18

Now, calculate the adjusted payment.
$Adjusted\, Payment = \$28,069.11 \times (1 + 0.13)$

STEP 19

Compute the adjusted annual payment.
$Adjusted\, Payment = \$28,069.11 \times 1.13$

STEP 20

Round the adjusted annual payment to two decimal places.
$Adjusted\, Payment \approx \$31,718.09$
Sami and Lubna must invest approximately $31,718.09 each year to accumulate enough money to buy the house in 20 years.

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The beach house costs $\$ 800000$ now. Inflation is expected to increase the price by $5\%$ annually over 20 years. How much will the house cost after 20 years? If they invest equal end-of-year payments earning $13\%$ annually, how much must they invest each year to buy the house in 20 years?

STEP 1

STEP 2

First, we need to calculate the future cost of the beach house after 20 years with a 5% annual inflation rate. We can use the formula for compound interest to find the future value:
$Future\, Value = Present\, Value \times (1 + Inflation\, Rate)^{Number\, of\, Years}$

STEP 3

Now, plug in the given values for the present value, inflation rate, and number of years to calculate the future cost of the house.
$Future\, Value = \$800,000 \times (1 + 0.05)^{20}$

STEP 4

Calculate the future value of the house after 20 years.
$Future\, Value = \$800,000 \times (1.05)^{20}$

STEP 5

Compute the future cost of the house.
$Future\, Value \approx \$800,000 \times 2.653297705$

STEP 6

Round the future cost of the house to two decimal places.
$Future\, Value \approx \$2,122,638.16$

STEP 7

STEP 8

Rearrange the formula to solve for the annual payment:
$Payment = \frac{Future\, Value}{\left(\frac{(1 + Interest\, Rate)^{Number\, of\, Payments} - 1}{Interest\, Rate}\right)}$

STEP 9

Now, plug in the values for the future value, interest rate, and number of payments to calculate the annual payment.
$Payment = \frac{\$2,122,638.16}{\left(\frac{(1 + 0.13)^{20} - 1}{0.13}\right)}$

STEP 10

Calculate the denominator of the payment formula.
$Denominator = \left(\frac{(1 + 0.13)^{20} - 1}{0.13}\right)$

STEP 11

Compute the value of the denominator.
$Denominator \approx \left(\frac{(1.13)^{20} - 1}{0.13}\right)$

STEP 12

Round the denominator to two decimal places.
$Denominator \approx \left(\frac{10.834705 - 1}{0.13}\right)$

STEP 13

Continue the calculation of the denominator.
$Denominator \approx \left(\frac{9.834705}{0.13}\right)$

STEP 14

Finish the calculation of the denominator.
$Denominator \approx 75.6515769$

STEP 15

Now, calculate the annual payment.
$Payment = \frac{\$2,122,638.16}{75.6515769}$

STEP 16

Compute the annual payment.
$Payment \approx \$28,069.11$

STEP 17

STEP 18

Now, calculate the adjusted payment.
$Adjusted\, Payment = \$28,069.11 \times (1 + 0.13)$

STEP 19

Compute the adjusted annual payment.
$Adjusted\, Payment = \$28,069.11 \times 1.13$

STEP 20

Round the adjusted annual payment to two decimal places.
$Adjusted\, Payment \approx \$31,718.09$
Sami and Lubna must invest approximately $31,718.09 each year to accumulate enough money to buy the house in 20 years.

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STEP 1

STEP 2

STEP 3

Now, plug in the given values for the present value, inflation rate, and number of years to calculate the future cost of the house.Future Value=$800,000×(1+0.05)20Future\, Value = \$800,000 \times (1 + 0.05)^{20}FutureValue=$800,000×(1+0.05)20

STEP 4

Calculate the future value of the house after 20 years.Future Value=$800,000×(1.05)20Future\, Value = \$800,000 \times (1.05)^{20}FutureValue=$800,000×(1.05)20

STEP 5

Compute the future cost of the house.Future Value≈$800,000×2.653297705Future\, Value \approx \$800,000 \times 2.653297705FutureValue≈$800,000×2.653297705

STEP 6

Round the future cost of the house to two decimal places.Future Value≈$2,122,638.16Future\, Value \approx \$2,122,638.16FutureValue≈$2,122,638.16

STEP 7

STEP 8

STEP 9

Now, plug in the values for the future value, interest rate, and number of payments to calculate the annual payment.Payment=$2,122,638.16((1+0.13)20−10.13)Payment = \frac{\$2,122,638.16}{\left(\frac{(1 + 0.13)^{20} - 1}{0.13}\right)}Payment=(0.13(1+0.13)20−1​)$2,122,638.16​

STEP 10

Calculate the denominator of the payment formula.Denominator=((1+0.13)20−10.13)Denominator = \left(\frac{(1 + 0.13)^{20} - 1}{0.13}\right)Denominator=(0.13(1+0.13)20−1​)

STEP 11

Compute the value of the denominator.Denominator≈((1.13)20−10.13)Denominator \approx \left(\frac{(1.13)^{20} - 1}{0.13}\right)Denominator≈(0.13(1.13)20−1​)

STEP 12

Round the denominator to two decimal places.Denominator≈(10.834705−10.13)Denominator \approx \left(\frac{10.834705 - 1}{0.13}\right)Denominator≈(0.1310.834705−1​)

STEP 13

Continue the calculation of the denominator.Denominator≈(9.8347050.13)Denominator \approx \left(\frac{9.834705}{0.13}\right)Denominator≈(0.139.834705​)

STEP 14

Finish the calculation of the denominator.Denominator≈75.6515769Denominator \approx 75.6515769Denominator≈75.6515769

STEP 15

Now, calculate the annual payment.Payment=$2,122,638.1675.6515769Payment = \frac{\$2,122,638.16}{75.6515769}Payment=75.6515769$2,122,638.16​

STEP 16

Compute the annual payment.Payment≈$28,069.11Payment \approx \$28,069.11Payment≈$28,069.11

STEP 17

STEP 18

Now, calculate the adjusted payment.Adjusted Payment=$28,069.11×(1+0.13)Adjusted\, Payment = \$28,069.11 \times (1 + 0.13)AdjustedPayment=$28,069.11×(1+0.13)

STEP 19

Compute the adjusted annual payment.Adjusted Payment=$28,069.11×1.13Adjusted\, Payment = \$28,069.11 \times 1.13AdjustedPayment=$28,069.11×1.13

STEP 20

Round the adjusted annual payment to two decimal places.Adjusted Payment≈$31,718.09Adjusted\, Payment \approx \$31,718.09AdjustedPayment≈$31,718.09Sami and Lubna must invest approximately $31,718.09 each year to accumulate enough money to buy the house in 20 years.

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Now, plug in the given values for the present value, inflation rate, and number of years to calculate the future cost of the house.
$Future\, Value = \$800,000 \times (1 + 0.05)^{20}$

Calculate the future value of the house after 20 years.
$Future\, Value = \$800,000 \times (1.05)^{20}$

Compute the future cost of the house.
$Future\, Value \approx \$800,000 \times 2.653297705$

Round the future cost of the house to two decimal places.
$Future\, Value \approx \$2,122,638.16$

Now, plug in the values for the future value, interest rate, and number of payments to calculate the annual payment.
$Payment = \frac{\$2,122,638.16}{\left(\frac{(1 + 0.13)^{20} - 1}{0.13}\right)}$

Calculate the denominator of the payment formula.
$Denominator = \left(\frac{(1 + 0.13)^{20} - 1}{0.13}\right)$

Compute the value of the denominator.
$Denominator \approx \left(\frac{(1.13)^{20} - 1}{0.13}\right)$

Round the denominator to two decimal places.
$Denominator \approx \left(\frac{10.834705 - 1}{0.13}\right)$

Continue the calculation of the denominator.
$Denominator \approx \left(\frac{9.834705}{0.13}\right)$

Finish the calculation of the denominator.
$Denominator \approx 75.6515769$

Now, calculate the annual payment.
$Payment = \frac{\$2,122,638.16}{75.6515769}$

Compute the annual payment.
$Payment \approx \$28,069.11$

Now, calculate the adjusted payment.
$Adjusted\, Payment = \$28,069.11 \times (1 + 0.13)$

Compute the adjusted annual payment.
$Adjusted\, Payment = \$28,069.11 \times 1.13$

Round the adjusted annual payment to two decimal places.
$Adjusted\, Payment \approx \$31,718.09$
Sami and Lubna must invest approximately $31,718.09 each year to accumulate enough money to buy the house in 20 years.