Three deposit options and a sales claim that needs auditing
A branch is pitching three fixed-deposit options to a customer who has exactly Rs 1,00,000 to place for exactly two years, with no withdrawals in between:
- Scheme A: 8% per annum, compounded annually.
- Scheme B: 9% per annum, simple interest.
- Scheme C: 8% per annum, compounded half-yearly. Interest under A and C is reinvested at the same rate, and no TDS applies at this level of interest. The branch's sales note reads: 'Scheme B is best because 9% beats 8%.' The customer, a retired teacher, wants the maturity value and the effective annual yield of each option, and asks what simple rate Scheme B would have needed to merely match the best compounded option. A second customer with the same Rs 1,00,000 wants total interest of Rs 20,000 at the end of two years and will accept a lower headline rate if the money stays locked for the full term.
1. Compute the maturity value and interest for each scheme, and assess whether the sales note's conclusion and its reasoning are correct.
Read the worked answer
Computations (Rs 1,00,000, 2 years):
| Scheme | Working | Interest | Maturity |
|---|---|---|---|
| A: 8% p.a. annual | 1,00,000 x 1.08^2 = 1,00,000 x 1.1664 | 16,640 | 1,16,640 |
| B: 9% p.a. simple | 1,00,000 x 9% x 2 | 18,000 | 1,18,000 |
| C: 8% p.a. half-yearly | 1,00,000 x 1.04^4 = 1,00,000 x 1.16985856 | 16,985.86 | 1,16,985.86 |
Verdict on the note. The conclusion is right - B pays the most, by Rs 1,014.14 over C - but the reasoning is wrong and would mislead on another day. Compounding frequency does matter: at the same nominal 8%, C beats A by Rs 345.86 because interest is capitalised twice a year. B wins here only because its 100 basis point rate premium outweighs C's frequency advantage; the note treats the rate and the frequency as if only one of them existed. A correct note would say 'B wins because a 100 bps premium is worth more than semi-annual capitalisation over two years'.
Also relevant: B's return is contractual and locked, whereas C depends on reinvestment at 8% throughout; and B's advantage shrinks the moment the tenure shortens, since frequency effects accumulate with time.
2. The customer asks for the effective annual yield of each option and for the simple rate Scheme B would have needed to just match Scheme C. Explain both.
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Effective annual yield - the single annually-compounded rate that reproduces the same two-year outcome, i.e. (1 + total interest / principal)^(1/2) - 1:
- A: sqrt(1.1664) - 1 = 8.00% (annual compounding is already the benchmark).
- C: sqrt(1.16985856) - 1 = 8.16%, which is simply (1.04)^2 - 1 - the semi-annual rate doubled into annual terms.
- B: sqrt(1.18) - 1 = 8.63%, not 9%. The advertised 9% simple is a nominal figure; because no interest is earned on the first year's Rs 9,000, the true annualised return is 8.63%.
Rate B would need to match C. Matching C means earning Rs 16,985.86 over two years on Rs 1,00,000, so the simple rate r solves 1,00,000 x r x 2 = 16,985.86, giving r = 8.493% p.a. (to match A, the figure is 8.32%). So B's 9% simple is roughly 50 bps above break-even against C - a real but thin margin, which is the number the customer should be shown.
Judgement point: comparing 9% simple with 8% compounded is a category error unless both are converted to the same effective-annual basis. On that basis C (8.16%) still beats A (8.00%), and B (8.63%) still leads the field.
3. For the second customer seeking Rs 20,000 interest in two years on Rs 1,00,000, find the required nominal rate under annual compounding and under half-yearly compounding, and interpret the gap.
Read the worked answer
Target: maturity must be Rs 1,20,000, i.e. a growth factor of 1.20 over two years.
Annual compounding: (1 + r)^2 = 1.20, so 1 + r = sqrt(1.20) = 1.095445, giving r = 9.5445% p.a. (about 9.54%).
Half-yearly compounding: (1 + r/2)^4 = 1.20, so 1 + r/2 = 1.20^(1/4) = 1.046635, giving r/2 = 4.6635% and r = 9.327% p.a.
Interpretation. The half-yearly product reaches the same Rs 1,20,000 with a headline rate about 22 bps lower, because each half-year's interest starts earning in the following half-year. The practical message for the branch is that a 9.33% half-yearly scheme and a 9.54% annual scheme are the same offer dressed differently - so quoting the higher headline number as 'better' would be a mis-sale. Two sanity checks matter before quoting either figure: confirm the rate is guaranteed for the full 24 months rather than reset after 12, and confirm the customer will not break the deposit early, since premature-withdrawal penalties are applied to the contracted rate and would knock the effective yield below both figures.