Private Wealth Management puzzles, solved step by step

Puzzles
100
Traced to a firm
3
Topics
13
Hard
30
Topic
All topicsCompounding and doubling8Returns arithmetic9Fee and cost drag8Inflation and real return7Tax arithmetic7Probability and risk of loss9Retirement and withdrawal8Fixed income numeracy8Behavioural traps8Estimation and sizing8Options and structured products6Leverage and borrowing6Wealth business economics8
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–1 of 1 · filtered from 100Clear filters
  1. 068An asset returns 6% a year, inflation is 6%, and tax at an illustrative 20% is charged on the nominal gain. What is the real after-tax return, and what effective tax rate is the client paying on his real gain?Inflation and real returnHardIndian wealth management

    Try it first

    What is the real return after tax?

    Show the worked solution

    About -1.1% a year, on a real gain of zero. Tax takes 1.2 points of the 6%, leaving 4.8%, and inflation of 6% then takes more than that: 1.048 over 1.06 less one is -1.13%. Before tax the client only kept pace with prices, so the whole tax bill fell on inflation. His effective tax rate on his real gain is not 20%; it is unbounded, because the real gain was nothing.

    Why does a 20% tax take more than 20% of the real gain?

    Imagine your salary rises exactly as fast as prices, and the tax office treats the whole rise as new income. You are no better off, yet you pay more tax, so you end the year worse off. A tax on nominal gains also taxes the part of the gain that only compensates for inflation, so the tax falls more heavily on the real gain than the headline rate suggests. Here the real gain before tax is zero, so every rupee of tax is a rupee of buying power lost.

    6% earned, 6% inflation, 20% tax on the nominal gain06.0Nominal return-1.2Tax at 20%4.8After tax-6.0Inflation-1.2Real, after taxBefore tax the real return is 6.0 - 6.0 = 0, so the 1.2 of tax is charged entirely on keeping up with prices
    Six per cent earned becomes 4.8% after an illustrative 20% tax, and 6% inflation then leaves a real after-tax return of about minus 1.2 points, exactly -1.13%, although the real return before tax was zero.

    What happens at higher nominal returns?

    The distortion shrinks but does not vanish. The lower the real return, the larger the share of it the tax takes, because the tax is sized on the nominal gain. The table runs the same 6% inflation and 20% tax across nominal returns: at 10% the real gain before tax is 3.77% and after tax 1.89%, an effective tax of about 50% on the real gain.

    Nominal returnReal, before taxReal, after 20% taxEffective tax on real gain
    6%0.00%-1.13%no real gain to tax
    8%1.89%0.38%80%
    10%3.77%1.89%50%
    12%5.66%3.40%40%
    With inflation at 6% and an illustrative 20% tax on nominal gains, the effective tax on the real gain falls from 80% at an 8% nominal return to about 50% at 10% and 40% at 12%, always above the 20% headline.

    Tax systems sometimes correct for this by indexing the cost of an asset to inflation before computing the gain, which taxes only the real part. Whether and where indexation applies has changed over time and differs by asset, so confirm the current rules. The interview answer is the mechanism: without indexation, inflation raises the true tax rate on savers.

    Where candidates lose it

    The trap is answering zero: 6% earned, 6% inflation, nothing lost. It ignores the tax, which is charged on the full nominal 6% whatever inflation does.

    The second miss is subtracting inflation first and taxing the real return, which gives zero again. The tax office sees rupees, not buying power, and the order matters.

    What the interviewer asks next

    • With indexation of the cost to inflation, what would the tax and the real after-tax return be?
    • At what nominal return does the effective tax on the real gain fall to 30%?
    • Why does this matter more for a long-held asset than for a one-year deposit?