Expected Return of a Defaultable Zero Coupon Bond

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1. Introduction

Yield and expected return are not the same thing — especially for credit-risky bonds. In this post, we walk through a rigorous numerical example involving a 1-year zero-coupon bond that is subject to default risk.

2. Watch the Explainer

3. Bond Setup

Let’s consider the following:

  • Face value: 100
  • Maturity: 1 year
  • Risk-free rate: 3%
  • Credit spread: 4%
  • Probability of default ($PD$): 6%
  • Recovery rate ($RR$): 45%

4. Market Price of the Bond

The bond has a promised yield to maturity (YTM) of: $$ r + s = 3\% + 4\% = 7\% $$ Being a zero-coupon bond, its price is: $$ P = \frac{100}{1.07} = 93.46 $$ So the market asks you to pay \$93.46 today.

The market promises a yield of 7% — but only if the bond survives to maturity. That’s the Yield to Maturity (YTM).

5. Expected Payoff

The bond has a 94% chance of paying the full face value of \$100, and a 6% chance of defaulting, in which case you recover only 45%: $$ \text{Expected Payoff} = 0.94 \times 100 + 0.06 \times 45 = 94 + 2.7 = 96.7 $$

6. Expected Return

You pay \$93.46 today and expect to receive \$96.7. The expected return is: $$ \text{Expected Return} = \frac{96.7}{93.46} – 1 = 3.47\% $$ So your expected return is not 7%, but 3.47%. There’s a quick formula that gets you very close: $$ \begin{align} \text{Expected Return} &= r + s – (\text{PD} \times (1 – \text{RR})) \\ &= 3\% + 4\% – (6\% \times 55\%) \\ &= 7\% – 3.3\% = 3.7\% \end{align} $$ Slightly higher than 3.47%, because this is an approximation — but the intuition is intact.

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