Kate is thinking about investing $45000 for 5 years. She deposits her money into an account which earns interest paid quarterly at a rate of 3.99% p.a. After 1½ years, Kate withdraws her investment (including interest) and deposits the full amount into a different account that pays interest at 4.29% p.a. resting semi-annually. She then leaves her investment untouched for the remainder of the 5 years.
(a) How much interest was accrued within the second year of the investment?
(b) What will be the value of Kate’s investment at the end of the five years?
b) amount after 1½ years
= 45000(1 + .0399/4)^6
= 47761.31
so for the next 3½ the above is invested at .29% pa compounded semi-annually
final amount
= 45000(1 + .0399/4)^6 (1 + .0429/2)^7
= 55411.01
a) If I read this correctly we need
amount she has after 2 years - amount she has after 1 year
= 45000(1 + .0399/4)^6 (1+ .0429/2)^1 - 45000(1 + .0399/4)^4
= 48785.79 - 46822.54
= 1963.25
For questions like this, I usually draw a "time-line" and mark on it the different critical times and interest rates.
To calculate the interest accrued within the second year of the investment, we need to calculate the interest earned during the first half of the second year and the interest earned during the second half of the second year separately.
For the first half of the second year (from 1½ years to 2 years), we can use the formula for compound interest, which is:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (including interest)
P = the principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case:
P = $45000
r = 3.99% p.a. = 0.0399
n = 4 (quarterly compounding: 4 times per year)
t = 0.5 years (1½ years to 2 years)
Using these values in the formula, we can calculate the amount at the end of the first half of the second year (A1):
A1 = 45000(1 + 0.0399/4)^(4*0.5)
A1 = 45000(1 + 0.009975)^(2)
A1 ≈ $46756.11
To calculate the interest earned during the first half of the second year, we subtract the initial investment amount:
Interest1 = A1 - P
Interest1 = $46756.11 - $45000
Interest1 ≈ $1756.11
Now, for the second half of the second year (from 2 years to 2½ years), we use the same formula with the new interest rate of 4.29% p.a.:
r = 4.29% p.a. = 0.0429
n = 2 (semi-annually compounding: 2 times per year)
t = 0.5 years (2 years to 2½ years)
Using these values in the formula, we can calculate the amount at the end of the second half of the second year (A2):
A2 = A1(1 + 0.0429/2)^(2*0.5)
A2 = $46756.11(1 + 0.02145)^(1)
A2 ≈ $48410.86
The interest earned during the second half of the second year is:
Interest2 = A2 - A1
Interest2 = $48410.86 - $46756.11
Interest2 ≈ $1654.75
Therefore, the total interest accrued within the second year of the investment is:
Total Interest = Interest1 + Interest2
Total Interest ≈ $1756.11 + $1654.75
Total Interest ≈ $3410.86
Now, to calculate the value of Kate's investment at the end of the five years, we need to calculate the amount in the second account after adding the interest earned in the second year.
For the remaining 3 years, we use the formula for compound interest with the interest rate of 4.29% p.a., compounded semi-annually, and the initial amount of $48410.86 (amount at the end of the second year):
P = $48410.86
r = 4.29% p.a. = 0.0429
n = 2 (semi-annually compounding: 2 times per year)
t = 3 years
Using these values in the formula, we can calculate the amount at the end of the five years:
Total Amount = P(1 + r/n)^(nt)
Total Amount = $48410.86(1 + 0.0429/2)^(2*3)
Total Amount ≈ $56191.76
Therefore, the value of Kate's investment at the end of the five years will be approximately $56,191.76.