In: Finance
3. a) Consider an annuity of 6 cash flows of $5,000 payable annually. If the interest rate is 7 per cent per annum, what is the value of this annuity today if the first cash flow is to be paid immediately? [8 marks]
3. b) You are considering the purchase of a home for $700,000. You have available a deposit of $100,000. The bank will lend you money at 7 per cent per annum compounded monthly over a period up to 20 years. If you borrow the required funds over 20 years, what are the monthly repayments? After two years, how much do you still ow the bank? What is the interest component of the 25th repayment? [7 marks]
Sol:
a) Present value of Annuity =
here,
C = cash flow per periods, = $5000
i = interest rate =7% or 0.07
n= number of payments =1 year
so, Present value of Annuity(after 1st year) = $5000 * [1-(1+0.07)-1 ] / 0.07
= $5000 * [1-(1/1.07)] /0.07
= $5000 * [1- 0.9345] / 0.07
= $5000 * 0.0655/0.07
= $4678.57
b) Full amount required to purchase home = $700000
Amount available as deposit = $100000
Required fund from Bank as loan(P) =($700000 - $100000) = $600000
Interest rate(i) = 7% or 0.07
Period(n) = 20 years
Calculation of Monthly Payment:
Monthly payment formula
A = P * [ i (1+i)n / (1+i)n - 1]
Monthly Payments(A)= $600000 * [0.07 (1+0.07)20 / (1+0.07)20 -1]
A = $4651.79
Calculation of interest component of the 25th repayment:
the interest component of the 25th repayment = $3,327.45
Periods | Beginning Balance | Interest | Principal | Ending Balance |
1 | $600,000.00 | $3,500.00 | $1,151.79 | $598,848.21 |
2 | $598,848.21 | $3,493.28 | $1,158.51 | $597,689.69 |
3 | $597,689.69 | $3,486.52 | $1,165.27 | $596,524.42 |
4 | $596,524.42 | $3,479.73 | $1,172.07 | $595,352.36 |
5 | $595,352.36 | $3,472.89 | $1,178.90 | $594,173.45 |
6 | $594,173.45 | $3,466.01 | $1,185.78 | $592,987.67 |
7 | $592,987.67 | $3,459.09 | $1,192.70 | $591,794.97 |
8 | $591,794.97 | $3,452.14 | $1,199.66 | $590,595.31 |
9 | $590,595.31 | $3,445.14 | $1,206.65 | $589,388.66 |
10 | $589,388.66 | $3,438.10 | $1,213.69 | $588,174.97 |
11 | $588,174.97 | $3,431.02 | $1,220.77 | $586,954.19 |
12 | $586,954.19 | $3,423.90 | $1,227.89 | $585,726.30 |
Year #1 End | ||||
13 | $585,726.30 | $3,416.74 | $1,235.06 | $584,491.24 |
14 | $584,491.24 | $3,409.53 | $1,242.26 | $583,248.98 |
15 | $583,248.98 | $3,402.29 | $1,249.51 | $581,999.47 |
16 | $581,999.47 | $3,395.00 | $1,256.80 | $580,742.68 |
17 | $580,742.68 | $3,387.67 | $1,264.13 | $579,478.55 |
18 | $579,478.55 | $3,380.29 | $1,271.50 | $578,207.05 |
19 | $578,207.05 | $3,372.87 | $1,278.92 | $576,928.13 |
20 | $576,928.13 | $3,365.41 | $1,286.38 | $575,641.75 |
21 | $575,641.75 | $3,357.91 | $1,293.88 | $574,347.86 |
22 | $574,347.86 | $3,350.36 | $1,301.43 | $573,046.43 |
23 | $573,046.43 | $3,342.77 | $1,309.02 | $571,737.41 |
24 | $571,737.41 | $3,335.13 | $1,316.66 | $570,420.75 |
Year #2 End | ||||
25 | $570,420.75 | $3,327.45 | $1,324.34 | $569,096.41 |
Calculation of Balance Owe to the bank after 2nd year:
Periods | Beginning Balance | Interest | Principal | Ending Balance |
1 | $600000 | $41547.82 | $14273.66 | $585726.30 |
2 | $585726.30 | $40515.97 | $15305.51 | $570420.75 |