A fund was established in 2001, whose history is recorded below: Deposit of $1000 on 1/1/2001, Deposit of $2000 on 1/1/2003, Withdrawal of $500 on 7/1/2003, Deposit of $1400 on 7/1/2004, Withdrawal of $1000 on 1/1/2005. The fund earned interest at the rate of 4% compounded semiannually until the end of 2003. At that date, the interest rate was augmented to 5% compounded semiannually. What was the principal in the fund at the end of 2007?
The principal in the fund at the end of 2007 would be $3,845.
fund was established in 2001, whose history is recorded below: Deposit of $1000 on 1/1/2001, Deposit of $2000 on 1/1/2003, Withdrawal of $500 on 7/1/2003, Deposit of $1400 on 7/1/2004, Withdrawal of $1000 on 1/1/2005. The fund earned interest at the rate of 4% compounded semiannually until the end of 2003. At that date, the interest rate was augmented to 5% compounded semiannually. What was the principal in the fund at the end of 2007?
Well, it seems like this fund has quite a history! Let's break down the timeline and calculate the principal at the end of 2007 step by step, shall we?
1. Deposit of $1000 on 1/1/2001.
This means the principal at the beginning of 2001 is $1000.
2. Deposit of $2000 on 1/1/2003.
The principal at the beginning of 2003 is the previous principal + the deposit = $1000 + $2000 = $3000.
3. Withdrawal of $500 on 7/1/2003.
Ouch! That's quite an early withdrawal. So the principal is now $3000 - $500 = $2500.
4. The interest rate was 4% compounded semiannually until the end of 2003.
Let's calculate the interest for this period. Since the compounding period is semiannually, we divide the interest rate by 2 and calculate for each year.
For the first year, the interest is $2500 * (4% / 2) = $2500 * 0.02 = $50.
For the second year, the interest is $2550 * (4% / 2) = $2550 * 0.02 = $51.
So, at the end of 2003, the principal + interest is $2550 + $51 = $2601.
5. The interest rate was augmented to 5% compounded semiannually.
Now, let's calculate the interest for the remaining years, 2004 to 2007, with the new interest rate.
For each year, the interest is simply the principal at the beginning of the year multiplied by (5% / 2).
Year 2004: $2601 * (5% / 2) = $2601 * 0.025 = $65.025
Year 2005: ($2601 + $1400) * (5% / 2) = $4001 * 0.025 = $100.025
Year 2006: ($4001 - $1000) * (5% / 2) = $3001 * 0.025 = $75.025
Year 2007: $3001 * (5% / 2) = $3001 * 0.025 = $75.025
6. Finally, let's add up the principal and interest for each year.
2003: $2601
2004: $2601 + $65.025 = $2666.025
2005: $2666.025 + $100.025 = $2766.05
2006: $2766.05 + $75.025 = $2841.075
2007: $2841.075 + $75.025 = $2916.1
And there you have it! The principal in the fund at the end of 2007 is approximately $2916.1.
To calculate the principal in the fund at the end of 2007, we need to calculate the future value of the initial deposits and subsequent transactions using the given interest rates.
Here is a step-by-step breakdown of the calculations:
1. Calculate the future value of the initial deposit of $1000 on 1/1/2001 at an interest rate of 4% compounded semiannually:
- Time period: 6 years (from 1/1/2001 to the end of 2006)
- Interest rate per period: 4%/2 = 2% (semiannual)
- Number of periods: 6*2 = 12 (semiannual)
- Future value = $1000 * (1 + 0.02)^12 = $1000 * (1.02)^12 = $1261.62
2. Calculate the future value of the deposit of $2000 on 1/1/2003 at an interest rate of 4% compounded semiannually:
- Time period: 4 years (from 1/1/2003 to the end of 2006)
- Interest rate per period: 4%/2 = 2% (semiannual)
- Number of periods: 4*2 = 8 (semiannual)
- Future value = $2000 * (1 + 0.02)^8 = $2000 * (1.02)^8 = $2258.31
3. Calculate the future value of the withdrawal of $500 on 7/1/2003 at an interest rate of 4% compounded semiannually:
- Time period: 3.5 years (from 7/1/2003 to the end of 2006)
- Interest rate per period: 4%/2 = 2% (semiannual)
- Number of periods: 3.5*2 = 7 (semiannual)
- Future value = $500 * (1 + 0.02)^7 = $500 * (1.02)^7 = $541.22
4. Calculate the future value of the deposit of $1400 on 7/1/2004 at an interest rate of 5% compounded semiannually:
- Time period: 2.5 years (from 7/1/2004 to the end of 2006)
- Interest rate per period: 5%/2 = 2.5% (semiannual)
- Number of periods: 2.5*2 = 5 (semiannual)
- Future value = $1400 * (1 + 0.025)^5 = $1400 * (1.025)^5 = $1521.39
5. Calculate the future value of the withdrawal of $1000 on 1/1/2005 at an interest rate of 5% compounded semiannually:
- Time period: 2 years (from 1/1/2005 to the end of 2006)
- Interest rate per period: 5%/2 = 2.5% (semiannual)
- Number of periods: 2*2 = 4 (semiannual)
- Future value = $1000 * (1 + 0.025)^4 = $1000 * (1.025)^4 = $1103.81
6. Calculate the total future value at the end of 2006 by summing up all the future values from the previous calculations:
- Total future value = $1261.62 + $2258.31 - $541.22 + $1521.39 - $1103.81 = $3396.29
7. Calculate the principal at the end of 2007 by compounding the total future value from the end of 2006 for one year at an interest rate of 5% compounded semiannually:
- Time period: 1 year (from the end of 2006 to the end of 2007)
- Interest rate per period: 5%/2 = 2.5% (semiannual)
- Number of periods: 1*2 = 2 (semiannual)
- Principal = $3396.29 * (1 + 0.025)^2 = $3396.29 * (1.025)^2 = $3518.20
Therefore, the principal in the fund at the end of 2007 is approximately $3518.20.
To determine the principal in the fund at the end of 2007, we need to track the deposits, withdrawals, and the interest earned on the fund.
First, let's calculate the interest earned during the different periods:
1. From 1/1/2001 to 1/1/2003 (2 years):
The initial deposit of $1000 earns interest compounded semiannually at 4%. Using the compound interest formula, the value at the end of this period is:
Principal * (1 + (interest rate/number of compounding periods))^(number of compounding periods * number of years)
= $1000 * (1 + (0.04/2))^(2*2)
= $1000 * (1 + 0.02)^4
= $1000 * 1.02^4
≈ $1082.43
2. From 1/1/2003 to 7/1/2003 (0.5 years):
No deposits or withdrawals during this period, so no interest is earned.
3. From 7/1/2003 to 1/1/2005 (1.5 years):
The deposit of $2000 and the withdrawal of $500 both occur at the beginning of this period, so they do not affect the interest earned during this period.
The principal at the beginning of this period is $1082.43.
The interest earned during this period is:
Principal * (1 + (interest rate/number of compounding periods))^(number of compounding periods * number of years)
= $1082.43 * (1 + (0.04/2))^(2*1.5)
= $1082.43 * (1 + 0.02)^3
= $1082.43 * 1.02^3
≈ $1149.86
4. From 1/1/2005 to 12/31/2007 (3 years):
The deposit of $1400 and the withdrawal of $1000 both occur at the beginning of this period, so they do not affect the interest earned during this period.
The principal at the beginning of this period is $1149.86.
The interest earned during this period is:
Principal * (1 + (interest rate/number of compounding periods))^(number of compounding periods * number of years)
= $1149.86 * (1 + (0.05/2))^(2*3)
= $1149.86 * (1 + 0.025)^6
= $1149.86 * 1.025^6
≈ $1324.51
Now, let's calculate the principal at the end of 2007:
Principal = (Previous Principal + Deposits) - Withdrawals + Interest Earned
Principal at the beginning of 2007:
= $1324.51
Deposits in 2007 (none):
= $0
Withdrawals in 2007 (none):
= $0
Interest earned in 2007:
= Principal at the beginning of 2007 * (1 + (interest rate/number of compounding periods))^(number of compounding periods * number of years)
= $1324.51 * (1 + (0.05/2))^(2*1)
= $1324.51 * (1 + 0.025)^2
= $1324.51 * 1.025^2
≈ $1361.85
Principal at the end of 2007:
= ($1324.51 + $0) - $0 + $1361.85
≈ $2686.36
Therefore, the principal in the fund at the end of 2007 is approximately $2686.36.