Gordon Freeman wants to have $32,000 in 5 years in order to purchase a new car.
(a) How much should he deposit today in an account earning 6.4%, compounded quarterly, to have the required amount in 5 years?
(b) How much interest will be earned?
(c) If he can only deposit $16,000 now, how short of $32,000 will he be?
(d) Suppose he can deposit $16,000 now in an account that compounds interest continuously. What interest rate would he need to accumulate the $32,000 in 5 years?
(e) Suppose he instead decides to make a deposit at the end of each quarter into an account earning, 6.4%, compounded quarterly. How much should he deposit each quarter to have $32,000 in 5 years?
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(a) To calculate the amount Gordon Freeman should deposit today, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Final amount ($32,000)
P = Principal amount (deposit)
r = Annual interest rate (6.4% or 0.064)
n = Number of times interest is compounded per year (quarterly, so n = 4)
t = Number of years (5)
Plugging in the values, we can solve for P:
32,000 = P(1 + 0.064/4)^(4*5)
Simplifying further,
32,000 = P(1.016)^20
Dividing both sides by (1.016)^20,
P = 32,000 / (1.016)^20
P ≈ $24,689.33
Therefore, Gordon Freeman should deposit approximately $24,689.33 today.
(b) To calculate the amount of interest earned, we subtract the principal amount from the final amount:
Interest = Final amount - Principal amount
Interest = $32,000 - $24,689.33
Interest ≈ $7,310.67
(c) If Gordon Freeman can only deposit $16,000 now, we can calculate how short of $32,000 he will be:
Shortfall = Required amount - Current deposit
Shortfall = $32,000 - $16,000
Shortfall = $16,000
(d) To calculate the interest rate needed to accumulate $32,000 in 5 years with continuous compounding, we can use the formula:
A = P * e^(rt)
Where:
A = Final amount ($32,000)
P = Principal amount ($16,000)
e = Euler's number (approximately 2.71828)
r = Annual interest rate (unknown)
t = Number of years (5)
Plugging in the values, we can solve for r:
32,000 = 16,000 * e^(5r)
Dividing both sides by 16,000:
2 = e^(5r)
Taking the natural logarithm (ln) of both sides:
ln(2) = 5r
Simplifying,
r = ln(2) / 5
r ≈ 0.13863
Therefore, Gordon Freeman would need an interest rate of approximately 13.863% to accumulate $32,000 in 5 years with continuous compounding.
(e) To calculate the amount Gordon Freeman should deposit each quarter to have $32,000 in 5 years, we can use the formula:
A = P(1 + r/n)^(nt)
Where:
A = Final amount ($32,000)
P = Deposit per quarter (unknown)
r = Annual interest rate (6.4% or 0.064)
n = Number of times interest is compounded per year (quarterly, so n = 4)
t = Number of years (5)
Plugging in the values, we can solve for P:
32,000 = P(1 + 0.064/4)^(4*5)
Simplifying further,
32,000 = P(1.016)^20
Dividing both sides by (1.016)^20,
P = 32,000 / (1.016)^20
P ≈ $1,212.07
Therefore, Gordon Freeman should deposit approximately $1,212.07 at the end of each quarter to have $32,000 in 5 years.
To solve these questions, we can use the formula for compound interest:
Future Value (FV) = Present Value (PV) * (1 + r/n)^(n*t)
where:
FV = Future Value
PV = Present Value
r = interest rate (as a decimal)
n = number of compounding periods per year
t = number of years
Now let's solve each part of the question step by step:
(a) How much should he deposit today in an account earning 6.4%, compounded quarterly, to have the required amount in 5 years?
In this case, the Future Value (FV) is $32,000, the interest rate (r) is 6.4% or 0.064 (as a decimal), the compounding periods (n) are 4 (quarterly), and the time (t) is 5 years. We need to find the Present Value (PV).
To find PV, we rearrange the formula:
PV = FV / (1 + r/n)^(n*t)
Substituting the given values:
PV = $32,000 / (1 + 0.064/4)^(4*5)
Calculating this expression will give us the required deposit amount.
(b) How much interest will be earned?
To find the interest earned, we subtract the initial deposit amount (PV) from the Future Value (FV).
Interest = FV - PV
(c) If he can only deposit $16,000 now, how short of $32,000 will he be?
To find the shortfall, we subtract the initial deposit amount (PV) from the desired Future Value (FV).
Shortfall = FV - PV
(d) Suppose he can deposit $16,000 now in an account that compounds interest continuously. What interest rate would he need to accumulate the $32,000 in 5 years?
Let's represent the interest rate as 'r' in this case. The compounding period (n) is not applicable because interest is compounded continuously. The Future Value (FV) is still $32,000, the initial deposit (PV) is $16,000, and the time (t) is 5 years.
We can rearrange the formula and solve for 'r':
PV * e^(r*t) = FV
where 'e' is the mathematical constant approximately equal to 2.71828.
Solving this equation will give us the required interest rate.
(e) Suppose he instead decides to make a deposit at the end of each quarter into an account earning 6.4%, compounded quarterly. How much should he deposit each quarter to have $32,000 in 5 years?
In this case, the Future Value (FV) is still $32,000, the interest rate (r) is 6.4% or 0.064 (as a decimal), the compounding periods (n) are 4 (quarterly), and the time (t) is 5 years. We need to find the equal quarterly deposits (PV) required.
We can rearrange the formula to solve for PV:
PV = FV / [((1 + r/n)^(n*t) - 1) / (r/n)]
Solving this equation will give us the required deposit amount for each quarter.