Case 059Distressed and special situationsCore

A hotel loan trades at 72 on a par of 100 with three years and an 8% coupon left. If it either defaults and recovers 40 in year 1 or pays off at par, what is the IRR in each case, and what default probability still gives a 15% expected return?

The situation

Mehfil Hotels owns four city hotels and borrowed Rs 400 crore from a bank three years ago. Occupancy has fallen and the bank wants the loan off its books. It will sell at 72 per 100 of face. The loan pays an 8% annual coupon and matures in three years at par.

A special situations fund sees two outcomes. Scenario A: the company defaults within a year, the hotels are sold, and the loan recovers 40 per 100 at the end of year 1 with no coupon. Scenario B: trading recovers, the coupons are paid, and the loan repays par at maturity. The fund needs a 15% expected return.

Your task

Work out the IRR of each scenario, then find the highest probability of scenario A at which the loan still earns a 15% expected return. Say which way of averaging is right.

Quick check

Scenario B pays 8, 8 and 108 on a price of 72. Roughly what is its IRR?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Scenario A loses 44%; scenario B earns about 22%; the loan still returns 15% in expectation if the chance of A is up to about 24%. A pays 40 on 72 after a year. B pays 8, 8 and 108, a yield to maturity near 22%. Weighting the two IRRs by probability gives the wrong answer, 10%; weight the cash flows instead, year by year, and find the probability at which those expected flows discount to 72 at 15%.

Step 1What does each scenario pay, and what rate does that imply?

Write the cash flows as two strips before touching a rate. Think of paying Rs 72 for a lottery ticket that either pays Rs 40 next year or Rs 8, Rs 8 and Rs 108 over three years; the price makes sense only if the second outcome is likely. Scenario A returns 40 on 72 after one year, an IRR of -44.4%; scenario B returns the coupons and par, an IRR of 21.6%. The second number is the loan's yield to maturityThe single discount rate at which the price equals the present value of all remaining coupons and the principal. at 72, and it is high because the buyer earns both the 8 coupon and the pull from 72 to 100.

Bought at 72, the Mehfil loan returns 40 after one year in scenario A, an IRR of -44.4%, or 8, 8 and 108 over three years in scenario B, an IRR of 21.6%, so the price of 72 is a wager on how likely the second strip is.

Step 2Why is averaging the two IRRs the wrong way to find the expected return?

Because an IRR is not a cash flow, and only cash flows average. Weighting 21.6% and -44.4% by probability says the loan clears 15% only if the chance of default is under 10%; weighting the cash flows year by year and solving for the rate says it clears 15% up to a 24% chance. The gap comes from timing: scenario A's loss arrives in year 1 and scenario B's gain mostly in year 3, and a blended IRR of those timings is not the average of the two IRRs. Build the expected strip, then find its rate.

The relationship

72 = \frac{40p + 8(1-p)}{1.15} + \frac{8(1-p)}{1.15^{2}} + \frac{108(1-p)}{1.15^{3}} \quad \Rightarrow \quad p \approx 24\%

p	the probability of scenario A, default with 40 recovered in year 1
40p + 8(1-p)	the expected year 1 cash flow: 40 if default, the 8 coupon if not
1.15	the fund's required return applied to each year's expected flow

What it says in wordsSet the price equal to the expected cash flow of each year discounted at 15 per cent, then solve for the default probability that makes the two sides equal.

Chance of A	Expected year 1	Expected year 2	Expected year 3	Expected IRR
0%	8.0	8.0	108.0	21.6%
10%	11.2	7.2	97.2	19.1%
20%	14.4	6.4	86.4	16.3%
30%	17.6	5.6	75.6	13.2%
40%	20.8	4.8	64.8	9.7%
50%	24.0	4.0	54.0	5.7%

Per 100 of face on a price of 72. As the chance of default rises from 0% to 50%, the expected cash flows shift towards year 1 and shrink, and the expected IRR falls from 21.6% to 5.7%, crossing 15% near a 24% probability and zero near 62%.

The expected IRR on the Mehfil loan falls as the probability of default rises, from 21.6% with no default to 5.7% at a 50% chance, crossing the fund's 15% requirement at about 24% and turning negative past about 62%, so the buyer at 72 is betting that default is less than one chance in four.

Step 3What does the fund actually underwrite at 72?

The probability, and the recovery behind it. At 72 the fund is saying default is less likely than one in four and that the hotels fetch at least 40 per 100 of debt if it happens; both are claims about the asset, not the loan. So diligence goes to the hotels: what they would sell for in a forced sale, who else ranks ahead, and how long a sale takes, because 40 in year 2 rather than year 1 moves scenario A's IRR further down. The limitation to state: two scenarios is a sketch, and the real analysis spreads recovery across a range and treats the coupons before default as uncertain too.

Where candidates lose it

The usual loss is blending the two IRRs by probability. IRRs are rates on different timings and cannot be averaged; blend the year-by-year cash flows and solve for the rate. The two methods differ here by about 14 points of default probability, which is the whole decision.

The second is quoting the 39% discount to par as a return. It is 28 points of pull spread over three years plus the coupon, about 22% a year, and only if the loan performs.

What the interviewer asks next

Recovery in scenario A arrives at the end of year 2 instead of year 1. What happens to the breakeven probability?
The bank offers the loan at 65. How much default risk can the fund now absorb?
Why might a fund prefer to buy this loan rather than lend new money to Mehfil at 15%?
What rights would you want as a holder of this loan before a default?

Company names and figures are illustrative.