A company's 5-year bond trades 450 bps over government with 40% recovery assumed, while a structural model on its equity implies a 3% annual default probability. Back out the market-implied default rate, compare the two, and say which instrument looks mispriced.

The bond prices a default rate of about 7.5% a year, 2.5 times the equity model's 3%, and neither is clearly mispriced. Spread over loss given default is 4.5% over 60%. Of the 450 bps, only 180 bps is expected loss on the equity model's view; the other 270 bps is premium for bearing the risk, illiquidity, or a gap in the equity model. The bond is the better instrument to own only if you trust the 3% and can hold the illiquidity.

Think of a motor insurer. If one car in a hundred is written off each year and a write-off costs 60% of the car's value, the fair premium is 0.6% of the value: probability times severity. A bond spread works the same way in reverse. The spread is roughly the default rate times the loss given default, so the default rate the market is pricing is the spread divided by the loss: 4.5% over 60%, which is 7.5% a year. That is the risk-neutral default intensityThe default rate that makes the bond fairly priced if investors demanded no extra reward for bearing default risk. It is usually higher than the rate at which companies actually default., the rate at which investors would have to expect defaults for the spread to be pure compensation for expected loss. It is not the rate at which the company is likely to default; it is the rate that makes the price fair with no reward for bearing the risk.

Over five years the gap compounds: at 7.5% a year the bond implies about a 31% chance of default before maturity, against about 14% on the equity model. Before calling that a mispricing, list the honest reasons the bond number should be higher. Bond investors are paid for more than expected loss: for the risk that defaults cluster in bad years, for the bond being hard to sell, and for the chance the recovery is worse than 40%. Studies of corporate spreads commonly find the market-implied rate running at a multiple of the rate at which companies actually default, so a ratio of 2.5 is not on its own a signal; confirm the typical multiple for the rating bucket before leaning on it. The structural model, for its part, is only as good as its asset volatility and its leverage inputs, and a 3% figure built from one year of equity volatility is the softer of the two numbers.

Test it, because the interviewer will. The implied rate is spread over loss, so a higher recovery raises it and a lower recovery cuts it. Even at zero recovery the bond implies 4.5% a year, still above the equity model's 3%, so no recovery assumption makes the two agree. Solving for the recovery that would gives 1 minus 4.5% over 3%, which is -50%, an impossible number. The gap is therefore a risk premium, a liquidity premium or a model error, and it cannot be an argument about recovery.

Give a view with its condition attached. If you trust the equity model, the bond is cheap: it pays 270 bps a year above expected loss, and the trade is long the bond, short a small amount of equity as a hedge against the default risk you are now carrying. If you trust the bond market, the equity is too confident, and the trade is the reverse. The honest reading is that the gap is inside the range such premiums usually occupy, so the case does not prove either side wrong; it says the bond is the instrument a patient holder is paid to own, and the equity model needs its asset volatility stress-tested before it drives money. A strong answer also sizes the hedge: the equity's sensitivity to a change in default probability is what sets the ratio, and that comes from the same structural model, so the trade inherits its weakness.

Say the limits. The spread-over-loss formula ignores discounting and the shape of the term structure, which matters little at these levels but grows with the spread. And the two instruments do not share a horizon: the bond's 450 bps is a five-year average, the equity model's 3% is a one-year number, and a company whose risk is rising over time will show exactly this kind of gap without anything being mispriced.