Loan Amount = $ 1000, Planned Annual Payments = $ 100, Interest Rate = 5 %, First Payment comes in at the end of Year 5,
Let the number of years required to pay off the loan be N years (beginning from end of year 5) and the final bigger payment be $ P
Therefore, 1000 = 100 x (1/0.05) x [1-{1/(1.05)^(N-1)}] x [1/(1.05)^(4)] + P / (1.05)^(N) = PV of Level Payments + PV of Final Payment
The above equation can only be solved by means of trial and error as it contains two variable. Further, as each annual payment of $ 100 is 1/10th of the loan amount, the value of N is definitely greater than 10
Hence, when N = 13
PV of Level Payments = 100 x (1/0.05) x [1-{1/(1.05)^(12)}] x [1/(1.05)^(4)] = $ 729.182
Remaining Loan = 1000 - 729.182 = $ 270.818
270.818 = P / (1.05)^(17)
P = $ 620.72
One needs to now progressively increase the value of N such that the value of P approaches the level payment value of $ 100 and the required value of N is that at which the final payment value is greater than the level payment value for the last time.
Now when N = 15
PV of Level Payments = 100 x (1/0.05) x [1-{1/(1.05)^(14)}] x [1/(1.05)^(4)] = $ 814.364
Remaining Loan = 1000 - 814.364 = $ 185.636
185.636 = P / (1.05)^(19)
P = $ 469.094
When N = 19
PV of Level Payments = 100 x (1/0.05) x [1-{1/(1.05)^(18)}] x [1/(1.05)^(4)] = $ 961.705
Remaining Loan = 1000 - 961.705 = $ 38.2948
38.2948 = P / (1.05)^(23)
P = $ 117.623
the answer is 124.72 5. A loan of 1,000 is to be repaid by annual payments...
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