How to Solve Problems on Installments (Loan Repayment)
Solve installment loan aptitude problems using present-value discounting, with a worked example and practice questions with answers.
Expected Interview Answer
Installment problems are solved by equating the loan’s present value to the sum of each future installment discounted back to today at compound interest, since each installment retires part of a growing, interest-bearing debt.
Each installment amount, when discounted back to the present by the compound interest factor for the number of periods until it is paid, must together sum to the original loan amount — this is the present-value equation. For n equal installments of amount x, the loan equals x/(1+R/100) + x/(1+R/100)^2 + ... + x/(1+R/100)^n, a geometric series that can be solved for x. A quicker approach for two installments is checking that the sum of each installment’s present value equals the loan, rather than tracking a running balance forward. Interviewers probe whether a candidate discounts correctly rather than mistakenly compounding the installments forward.
- One present-value equation handles any number of installments
- Reuses the compound interest formula, so no new formula to memorize
- Clarifies why installments are discounted, not compounded, back to today
AI Mentor Explanation
A cricket board owes a player a signing fee, paid in two yearly installments instead of one lump sum today — to know what that lump sum should have been, you discount each future installment back to today’s value using the same rate the money would have grown at, then add them up. This present-value approach — shrinking future payments back to today rather than growing today’s value forward — is exactly how installment loan problems are solved: the loan equals the sum of each installment’s discounted present value. Getting the discounting direction right, dividing rather than multiplying by the growth factor, is the entire skill being tested.
Worked example (two equal yearly installments)
Present value equation
- 2100 = x/1.1 + x/1.1²
Sum discount factors
- 0.909 + 0.826 = 1.735
Solve for x
- x ≈ 2100/1.735 ≈ 1210
Step-by-Step Explanation
Step 1
Set up the present-value equation
Loan = sum of each installment divided by its own compound discount factor.
Step 2
Discount each installment
Divide installment x by (1+R/100)^k for the k-th payment period.
Step 3
Sum the geometric series
Add all discounted installments and set equal to the loan principal P.
Step 4
Solve for the unknown
Isolate x (or P, or R) algebraically from the summed equation.
What Interviewer Expects
- Correct present-value setup, discounting future installments back to today
- Recognition that each installment is discounted for its own specific period count
- Ability to solve the resulting equation for the unknown installment or principal
- Not confusing discounting (dividing) with compounding (multiplying) forward
Common Mistakes
- Compounding installments forward instead of discounting them back to present value
- Using the same discount period for every installment regardless of when it is paid
- Adding installments without discounting at all, treating them as already-present values
- Algebra errors when solving the geometric-series equation for the unknown
Best Answer (HR Friendly)
“I treat each future installment as a future cash flow that needs to be brought back to today’s value using the compound interest discount factor, then I sum all those present values and set them equal to the original loan. It is the same compound interest formula used in reverse — dividing instead of multiplying — for each installment’s specific delay before payment.”
Follow-up Questions
- How would the equation change for unequal installment amounts?
- How do you compute an EMI for a loan with monthly compounding?
- How would you solve for the interest rate given the loan and installment amounts?
- Why must future installments be discounted rather than compounded to value them today?
MCQ Practice
1. A loan of 2100 at 10% per annum is repaid in 2 equal yearly installments. Each installment is approximately?
2100 = x/1.1 + x/1.1^2 → x ≈ 1210, discounting each installment back to present value.
2. When valuing future loan installments today, you should?
Future cash flows are discounted back to present value by dividing by (1+R/100)^k.
3. For n equal installments of x at rate R, the loan P equals?
The loan equals the sum of the present values of each of the n future installments.
Flash Cards
Core installment present-value equation? — P = sum of x/(1+R/100)^k for each installment period k.
Discount or compound future installments? — Discount them back (divide) — never compound them forward.
What kind of series does the installment equation form? — A geometric series in the discount factor 1/(1+R/100).
How do banks compute EMIs? — By solving the same present-value equality for the equal installment amount.