Private Wealth Management puzzles, solved step by step

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  1. 084A bank fixed deposit quotes 7% a year, compounded quarterly. What is the effective annual yield?Fixed income numeracyWarm upIndian wealth management

    Try it first

    Pick the effective yield.

    Show the worked solution

    About 7.19%. Quarterly compounding pays 7 / 4 = 1.75% each quarter, and each quarter's interest earns interest in the quarters that follow. Rs 100 grows to 101.75, 103.53, 105.34 and 107.19. The effective yield is 1.0175 to the fourth, less 1, which is 7.19%.

    Why is the effective yield above the quoted 7%?

    If a friend repays you in four instalments and you lend each instalment on at once, you end the year with more than if the whole amount came back at year end. The quoted rate is the per-period rate times the number of periods; the effective yield adds the interest earned on interest paid earlier in the year. The more often it compounds, the wider the gap.

    Four quarters of 1.75%, each on a slightly bigger base100.00start+1.75%+1.750101.75end of Q1+1.75%+1.781103.53end of Q2+1.75%+1.812105.34end of Q3+1.75%+1.843107.19end of Q4Interest earned each quarter, Rs: it rises because it is paid on the last quarter's interest tooQuoted rate7.00%Effective yield7.19%The lime sliver is the interest earned on interest: 0.19 of a point
    Rs 100 grows by 1.75% a quarter to Rs 107.19 at the year end, each quarter's interest slightly larger than the last, so the effective yield is 7.19% against the quoted 7%.
    The relationship
    EAR=(1+0.074)4−1=1.01754−1=7.19%\text{EAR} = \left(1 + \frac{0.07}{4}\right)^4 - 1 = 1.0175^4 - 1 = 7.19\%
    0.07the quoted annual rate
    4compounding periods a year
    EARthe effective annual rate, what Rs 100 actually becomes in a year
    What it says in wordsDivide the quoted rate by the periods, compound it that many times, and subtract one.

    When does the 7.19% not apply?

    The effective yield assumes the interest stays in the deposit; on a payout deposit that sends interest to the client each quarter, he earns 7% unless he reinvests it himself. Tax is the second gap: interest on deposits is typically taxed as income each year at the client's slab rate, so the after-tax figure is lower still. Confirm the current treatment before comparing a deposit with a fund.

    Use the effective rate whenever you compare products that compound at different frequencies, such as a quarterly deposit against an annual-coupon bond. Comparing quoted rates across frequencies is comparing different units.

    Where candidates lose it

    The common slip is to say 7% because that is what the bank quoted. The interviewer is checking whether you hear the words "compounded quarterly" and know they change the answer.

    The rarer slip is reading 7% as a quarterly rate and quoting about 31%. Say the per-quarter rate, 1.75%, out loud before compounding.

    What the interviewer asks next

    • What would the effective yield be with monthly compounding?
    • A bond pays 7.1% once a year. Which pays more, the bond or the deposit?
    • How does a 30% tax slab change the comparison?