Recovery and one-period risky present value
What you will be able to do
Section titled “What you will be able to do”After this lesson, you should be able to:
- interpret recovery as a fraction of a named base paid at a named time;
- calculate loss given default as the complement of recovery;
- calculate present value in one explicitly labeled recovery-of-par-paid-at-maturity model.
Name the recovery convention before calculating
Section titled “Name the recovery convention before calculating”This lesson uses one recovery convention. A claim has a positive local recovery-par-amount and one local recovery-payment-time , measured from valuation-time .
- If the reference entity survives through , the holder receives at .
- If the reference entity defaults at or before , the holder receives at , where the recovery-rate is deterministic.
The survival payment and the default payment are both paid at . This timing is a model assumption, not a description of an actual instrument. This lesson does not teach recovery paid at default, recovery of market value, settlement delay, or a CDS settlement mechanism.
Tuckman and Serrat define a bond recovery rate relative to the face amount. They also identify the complementary loss fraction in their credit-loss discussion [1]. Payment of the recovery at maturity is a convention of this lesson. The lesson does not claim that it is the only recovery convention.
Recovery and loss given default are complements here
Section titled “Recovery and loss given default are complements here”The loss-given-default fraction is defined as:
For example, gives in this toy model. The value 0.40 is chosen only for the arithmetic. The lesson makes no claim that 40% is a market convention, a forecast, or an appropriate assumption for any real entity.
Calculate the expected maturity amount, then discount it
Section titled “Calculate the expected maturity amount, then discount it”The input survival-probability is under the risk-neutral-probability-measure . In the one-default model, the probability of default by is .
For the single interval from through , is the interval-default-probability. The expected amount paid at under is the survival payment and the default payment, each multiplied by its probability:
This is the two-event case of the expectation from disjoint events in the probability lesson: survival pays , and default pays . Shreve gives the expectation of a finite-valued random variable[2]. Tuckman and Serrat weight credit cash flows by survival and default probabilities and then discount them[3].
The survival payment and the default payment are both paid at . So one input discount-factor applies to both. The recovery-of-par-present-value is the discounted expected amount:
This present-value is calculated from the perspective of the holder of the claim: the modeled future receipts and their value are positive. A purchase price paid by the holder would have the opposite cash-flow sign.
Partial recovery
Let , , , and . The expected fraction of par paid at maturity is:
Multiplying this fraction by the par amount and the discount factor gives the present value:
The expected amount at maturity, before discounting, is USD 952.00. The discount factor converts this amount to its value at valuation time.
Zero-recovery boundary
Let , , , and . The default payment is zero, so the present value is:
With zero recovery, the present value is the discounted expected survival payment. This example makes no statement about real recovery outcomes.
Full-recovery boundary
Keep , , and , but set . The survival payment and the default payment are now both the full par amount, paid at :
With full recovery, the present value does not depend on the survival probability. The reason is that both modeled events pay the same amount on the same date. This result holds only under the timing convention of this lesson.
Check your understanding
Section titled “Check your understanding”The assessment tests the meaning of recovery, loss-given-default arithmetic, and direct and transfer calculations under the same one-period timing convention.
Knowledge check 6.2.1 Recovery and risky present value
Link to Knowledge check 6.2.1: Recovery and risky present valueIn the lesson's one-period recovery-of-par model, what does a recovery rate of 40% mean if default occurs by maturity?
Check your answer to reveal the explanation.
A one-period claim has USD 2,000 par, a current price of USD 1,850, and a 25% recovery-of-par rate. Under the lesson model, which positive recovery cash flow is used after default?
Check your answer to reveal the explanation.
Enter loss given default as a decimal when the recovery rate is 35% (0.35).
Check your answer to reveal the explanation.
A recovery-of-par model pays USD 900 after default on a claim with USD 1,500 par. Enter loss given default as a decimal.
Check your answer to reveal the explanation.
A one-period defaultable zero-coupon claim has USD 1,000 par, survival probability 0.90 to maturity, recovery-of-par rate 0.40 after default, and maturity discount factor 0.95. Both the survival payoff and any recovery are paid at maturity. What is its time-zero present value in USD?
Check your answer to reveal the explanation.
A one-period defaultable zero-coupon claim has USD 2,500 par, cumulative default probability 0.08, loss given default 0.55, and maturity discount factor 0.92. The recovery-of-par amount after default is paid at maturity. What is its time-zero present value in USD?
Check your answer to reveal the explanation.
Model boundary and review note
Section titled “Model boundary and review note”The model has one maturity date, one deterministic recovery rate, one input survival probability, and one input discount factor. Recovery is a fraction of par paid at maturity, also when default occurs earlier. The lesson omits:
- settlement at the default time;
- coupons and multiple intervals;
- stochastic recovery and recovery of market value;
- dependence between interest rates and default;
- calibration;
- liquidity, funding, taxes, and counterparty risk.
All materially changed content and source locators remain `draft` pending independent human review.
References
Section titled “References”- Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). Ch. 14 §14.2, printed p. 353. draft ↩
- Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (2004). Ch. 1 §1.3, printed pp. 13–18. draft ↩
- Tuckman & Serrat, Fixed Income Securities: Tools for Today's Markets (4th ed., 2022). Ch. 14 §14.7, printed pp. 371–372, eq. 14.8 and Table 14.11. draft ↩
Credit default swap, the credit derivative the CDS lessons define and value.
The name of a family of standard credit default swap indices, each a standard portfolio of single-name contracts.
Duration times spread, a spread-risk measure for bonds.
Financial Industry Regulatory Authority.
International Money Market. In the CDS lessons, IMM dates are the standard maturity dates on the twentieth of March, June, September, and December.
International Swaps and Derivatives Association.
International Organization for Standardization.
Jump to default, the loss on an immediate default of the reference entity.
Coordinated Universal Time, the time standard the date arithmetic counts calendar days in.