2.36 Compound interest formula: Total amount = P (1 + (R/100))^{n }where P is principal, R is rate, and n is number of years
$1,200 compounded annually for 5 years at 9% = $1,846.35
$1,000 compounded annually for 4 years at 9% = $1,411.58
$800 compounded annually for 3 years at 9% = $1,036.02
$600 compounded annually for 2 years at 9% =$712.86
Total amount in the fund immediately after the 5^{th} deposit will be $1,846.35 + $1,141.58 + $1,036.02 + $712.86 + $400 (the 5^{th} deposit) = $5,406.81
2.31 Loan payment formula: monthly payment = [rate + (rate/1 + rate)^{months} – 1] x principal
Amortization schedule for $20,000 loan payable annually for 5 years at 10% compound interest
Year |
Interest |
Principal |
Balance |
2012 |
$1,729.28 |
$6,014.85 |
$13,985.15 |
2013 |
$1,099.44 |
$6,644.68 |
$7,340.47 |
2014 |
$403.66 |
$7,340.47 |
$0.00 |
Annual payment = $7,744.13
Interest in second year = $1,099.44
2.48 In the first diagram, the total cash flow in five years is $600. At 10% annual compound interest, the principal amount invested at the beginning of year one is $983. Since the second diagram is said to be equivalent to the first diagram, then the value of X is $120 ($600/5 years). This is because in the second diagram, the annual cash flows are equal for all the five years.
2.49 The total amount of cash flows in the first diagram is $220. The number of investment is 4 years. The value of C that makes the second investment to be economically equivalent at an interest rate of 10% is $36.667 [$220/(c+2c+2c+c)].
2.52 The total amount of inflow is $9,000 (for eight years). An equal amount of outflow is introduced at the beginning of ever year from year 2 to year 8. However, the initial outlay is as twice as the amount invested every year from year 2. This initial amount yield $800 at the end of year 1. Since is interest rate is given as 12% p.a, then the initial amount invested is $6,667 (refer to 2.36 for formula). If 2C = $6,667, then C = $3,333.5.
2.54 Formula for calculating present value of annuity:
Total = Amount [(1 + r)^{n+1} – 1/r] – amount
Present value of annual series of $5,000 for 10 years at 10% compound interest = $30,722.84
The amount $30,722.84 is equivalent to $8,104.5 invested for 5 years at 10 % compound interest.