# Using probability theory to determine credit spreads

For any bond the lender's required/expected return (= pre-tax cost of debt) will be made up of two elements:

• The risk free rate of return
• A premium (the credit spread) based on the expected probability of default and the expected loss given default

Probability theory can be used to calculate these credit spreads and the risk of default.

## Overview of the method

### Idea

The bank will set an interest rate such that the expected present value of future receipts covers the amount borrowed

### Example

Panda plc is a UK sportswear company which has expanded rapidly over the last 10 years. Its assets are now valued at £55m. In order to fund its most recent project, a new manufacturing facility in a deprived area of the West Midlands, the directors have decided to borrow £10m from the ITCD Bank plc. The money will be repayable in 1 year.

The interest rate quoted on the loan is 5.90% p.a., being base rate plus 65 basis points. The directors are unsure how this rate has been computed. The bank has provided them with the following information by way of explanation:

• Loan from ITCD Bank plc to Panda plc
• Loan value:   £10m
• Asset value:   £55m
• Standard deviation of returns: 40% p.a.
• Base rate:   5.25% p.a.
• Expected level of recovery on default: 70%
• Panda plc Credit Rating: AA
• Credit spread on a 1 year loan: 65 basis points

Required:

Show how the interest rate on the loan of 5.90% has been derived.

Solution

Step 1: probability of default

With assets of £55m and a loan of £10m, the bank will have calculated the likelihood of Panda plc's assets falling below £10m in the next year - i.e. a downward movement of £45m, as follows:

Standard deviation = 40%, which (on an average asset value of £55m) translates to a value of £22m.

Hence, the loan value of £10m lies £45m, or 45/22 = 2.045 Standard Deviations below the mean.

Using normal distribution tables, this means that there is a 0.0204 chance (2.04%) of default, in which case the bank predicts that only 70% of the loan will be recoverable.

The bank will now calculate an interest rate such that the expected receipt in 1 year exactly covers the amount borrowed, as follows:

Assuming a base rate of 5.25%:

Suppose the bank sets its interest rate on the loan as i. It would hope to receive £10m × (1 + i) in one year

Present value of £10m × (1 + i) at time 1 (divide by 1.0525) =   £9,501,188 (1 + i)

Total expected receipt = Expected value if paid + Expected value on default

Expected present value if paid = 0.9796 x £9,501,188 × (1 + i)  =   £9,307,363 × (1 + i)

Expected present value if default = 0.7 x 0.0204 x £9,501,188 × (1 + i) =  £135,677 × (1 + i)

Total expected receipt =     £9,443,040 × (1 + i)

Equating the PV of expected receipts with the PV of the loan made gives:

£10m loan = £9,443,040 x (1 + i)

hence i = 0.05898 or 5.90%, which is base rate plus 65 basis points as quoted.   Created at 9/16/2012 4:29 PM  by System Account  (GMT) Greenwich Mean Time : Dublin, Edinburgh, Lisbon, London Last modified at 11/13/2012 11:46 AM  by System Account  (GMT) Greenwich Mean Time : Dublin, Edinburgh, Lisbon, London  ### Rating : Ratings & Comments  (Click the stars to rate the page) ### Tags: 