Debt Capital Markets puzzles, solved step by step

Puzzles
100
Traced to a firm
16
Topics
13
Hard
30
Topic
All topicsLeverage, coverage and cash flow9Mental maths and numeracy8Estimation and market sizing7Logic and brainteasers8Cost of capital and valuation riddles7Bond pricing and yield7Compounding, PIK and fees6Issuance and refinancing arithmetic8Credit spreads and default probability8Duration and convexity8Capital structure and recovery8Probability and expected value10Yield curve and forward rates6
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–1 of 1 · filtered from 100Clear filters
  1. 089A 10-year government bond yields 7.0% and expected inflation is 4.5%. What is the real yield using the Fisher relation, and how far off is the quick subtraction?Yield curve and forward ratesWarm upFixed income asset management

    Try it first

    Is the exact real yield above or below the 2.5% you get by subtracting?

    Show the worked solution

    The real yield is about 2.39%, against 2.50% from simple subtraction. The Fisher relation divides one plus the nominal yield by one plus inflation: 1.07 / 1.045 - 1 = 2.39%. The shortcut overstates the real yield by about 11 basis points, because inflation also erodes the interest itself, and the gap widens as rates rise.

    Why is subtraction only an approximation?

    Lend Rs 100 for a year at 7% and get Rs 107 back. If prices have risen 4.5%, a basket that cost Rs 100 now costs Rs 104.50. Your Rs 107 buys 107 / 104.5 baskets, 1.0239 of them: a real gain of 2.39%, not 2.5%. Inflation shrinks the purchasing power of everything you get back, the interest as well as the principal, so the real yield is a ratio, not a difference.

    The shortcut overstates the real yield, and more so at higher rates2.50%Shortcut 7.0 - 4.52.39%Fisher 1.07 / 1.045 - 12.0%axis starts at 2.0% to show the gap11 bpGap between shortcut and Fisher, bpnominal and inflation always 2.5 points apart4% and 1.5%4 bp7% and 4.5%11 bp10% and 7.5%17 bp15% and 12.0%32 bp
    At 7.0% nominal and 4.5% inflation the exact real yield is 2.39%, 11 basis points below the 2.50% shortcut, and with the same 2.5 point difference the gap grows from about 4 basis points at low rates to 32 at 15% and 12%.
    The relationship
    1+r=1+n1+π  ⇒  r=1.071.045−1=2.39%1 + r = \frac{1 + n}{1 + \pi} \;\Rightarrow\; r = \frac{1.07}{1.045} - 1 = 2.39\%
    nthe nominal yield, 7.0%
    piexpected inflation, 4.5%
    rthe real yield
    What it says in wordsReal growth is nominal growth divided by price growth.

    When does the shortcut stop being good enough?

    The error is roughly the real yield times inflation, divided by one plus inflation. At low rates it is a few basis points and nobody minds; at high inflation it is tens of basis points and it changes the conclusion. With nominal yields of 15% and inflation of 12%, subtraction says 3.0% but the true real yield is 2.68%. For a desk comparing an inflation-linked bond with a conventional one, a gap of that size is the whole trade. The rates here are illustrations; the relation, not the numbers, is the point.

    Where candidates lose it

    The fast answer is 2.5% with no comment. It is close enough on these numbers, but an interviewer asking for the Fisher relation wants the ratio, and wants you to say which way the shortcut errs.

    The second loss is getting the direction backwards and saying the exact figure is higher. Anchor it with the basket example: Rs 107 buys fewer than 1.025 baskets once they cost Rs 104.50.

    What the interviewer asks next

    • An inflation-indexed bond of the same maturity yields 2.0% real. What inflation is the market pricing in?
    • Why might the gap between nominal and indexed yields not equal expected inflation exactly?
    • If inflation is higher than expected, who gains: the holder of the nominal bond or the issuer?