QuestionPersonal Finance Problem LG2 P5-6 Time value As part of your financial planning, you wish to purchase a nem actly 5 years from today. The car you wish to purchase costs $\$ 14,000$ today, 2 , your research indicates that its price will increase by $2 \%$ to $4 \%$ per year overth next 5 years. a. Estimate the price of the car at the end of 5 years if inflation is (1) $2 \%$ perped and (2) $4 \%$ per year. b. How much more expensive will the car be if the rate of inflation is $4 \%$ rathe than $2 \%$ ? c. Estimate the price of the car if inflation is $2 \%$ for the next 2 years and $4 \%$ s. 3 years after that.

Studdy Solution

STEP 1

1. The current price of the car is $14,000.
2. The price of the car increases due to inflation.
3. Inflation rates can be either 2% or 4% per year.
4. We are estimating the price of the car in 5 years under different inflation scenarios.

STEP 2

1. Calculate the future price of the car with a 2% annual inflation rate over 5 years.
2. Calculate the future price of the car with a 4% annual inflation rate over 5 years.
3. Determine the difference in price between the 4% and 2% inflation scenarios.
4. Calculate the future price of the car with a mixed inflation rate: 2% for the first 2 years and 4% for the next 3 years.

STEP 3

Calculate the future price of the car with a 2% annual inflation rate over 5 years.
The formula for future value with annual compounding is: $\text{Future Price} = \text{Present Price} \times (1 + \text{Inflation Rate})^n$
For a 2% inflation rate: $\text{Future Price} = 14,000 \times (1 + 0.02)^5$
Calculate: $\text{Future Price} = 14,000 \times (1.02)^5$
$\text{Future Price} = 14,000 \times 1.10408$
$\text{Future Price} \approx 15,457.12$

STEP 4

Calculate the future price of the car with a 4% annual inflation rate over 5 years.
For a 4% inflation rate: $\text{Future Price} = 14,000 \times (1 + 0.04)^5$
Calculate: $\text{Future Price} = 14,000 \times (1.04)^5$
$\text{Future Price} = 14,000 \times 1.21665$
$\text{Future Price} \approx 17,033.10$

STEP 5

Determine the difference in price between the 4% and 2% inflation scenarios.
$\text{Difference} = 17,033.10 - 15,457.12$
$\text{Difference} \approx 1,575.98$

STEP 6

Calculate the future price of the car with a mixed inflation rate: 2% for the first 2 years and 4% for the next 3 years.
First, calculate the price after 2 years with 2% inflation: $\text{Price after 2 years} = 14,000 \times (1.02)^2$
$\text{Price after 2 years} = 14,000 \times 1.0404$
$\text{Price after 2 years} \approx 14,565.60$
Next, calculate the price after 3 more years with 4% inflation: $\text{Future Price} = 14,565.60 \times (1.04)^3$
$\text{Future Price} = 14,565.60 \times 1.124864$
$\text{Future Price} \approx 16,384.68$

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