Answer:
a) To calculate the future value of the GIC at maturity, we can use the formula:
FV = P(1 + r/n)^(n*t)
where:
P = principal amount = $3000
r = annual interest rate = 5.6% = 0.056
n = number of times interest is compounded per year = 1 (annually)
t = number of years = 10
Plugging in the values, we get:
FV = $3000(1 + 0.056/1)^(1*10)
= $3000(1.056)^10
= $5,020.93
Therefore, the future value of the GIC at maturity is $5,020.93.
b) To estimate how long it will take for the GIC to be worth at least $12,000, we can use the same formula as above and solve for t:
FV = $12,000
P = $3000
r = 0.056
n = 1
$12,000 = $3000(1 + 0.056/1)^(1*t)
4 = 1.056^t
t = log(4) / log(1.056)
t ≈ 25.58
Therefore, it will take approximately 25.58 years for the GIC to be worth at least $12,000.
i) If the compounding frequency is monthly, then n = 12 and the formula becomes:
FV = $3000(1 + 0.056/12)^(12*10)
= $5,124.82
The future value of the GIC will be slightly higher because compounding is occurring more frequently.
ii) If the interest rate is 2.8%, compounded semi-annually, then r = 0.028/2 = 0.014 and n = 2. Plugging in the values, we get:
FV = $3000(1 + 0.014/2)^(2*10)
= $4,012.92
The future value of the GIC will be lower because the interest rate is lower and compounding is occurring less frequently.
d) To find the minimum interest rate with daily compounding that would be needed to have a future value that is $100 greater than the future value in part a, we can use the formula:
FV = P(1 + r/n)^(n*t)
where:
FV = $5,120.93
P = $3000
n = 365 (number of times interest is compounded per year with daily compounding)
t = 10
We can solve for r as follows:
$5,120.93 = $3000(1 + r/365)^(365*10)
1.707 = (1 + r/365)^3650
log(1.707) = log(1 + r/365)*3650
log(1.707)/3650 = log(1 + r/365)
0.000108166 = r/365
r = 0.0395
Therefore, the minimum interest rate with daily compounding that would be needed to have a future value that is $100 greater than the future value in part a is 3.95%