CFA Level II Exam · Credit Analysis Models

Valuing Risky Bonds and Credit Valuation Adjustment (CVA)

Updated 7 October 2026 · Fact-checked

Credit valuation adjustment (CVA) is the present value of expected credit loss on a claim. For each date, multiply expected exposure by loss given default (1 − recovery rate), the probability of default and the discount factor, then sum. A risky bond's value is its risk-free value minus CVA.

Understand Valuing Risky Bonds and Credit Valuation Adjustment

A bond that can default is worth less than the same bond with no default risk. The gap is the price of credit risk. The Level II approach splits the problem in two: first value the cash flows as if they were risk-free, then subtract the present value of the expected loss from default.

That expected loss has three drivers. Expected exposure is how much you stand to lose if default happens at a given date. For a bond, it is the value at that date of all remaining promised cash flows. Loss given default (LGD) is the share of exposure you do not recover: LGD = 1 − recovery rate. Probability of default (POD) is how likely default is at that date. Each loss is then discounted with risk-free rates.

The default probabilities used for pricing are risk-neutral, not historical. They are backed out of market prices, so they already include compensation for bearing default risk. Risk-neutral PODs are usually higher than real-world ones. Use them for valuing, and do not read them as forecasts.

The credit spread links all of this. A risky bond yields more than a risk-free bond with the same cash flows. For a short horizon, spread ≈ POD × LGD, so a higher POD, a lower recovery rate or both widen the spread. Rearranged, POD ≈ spread ÷ LGD. This is only an approximation. When the question gives prices, solve the exact equation.

In a vignette, the data are usually a table of risk-free rates or discount factors, conditional or unconditional default probabilities, a recovery rate and the bond's cash flows. Your job is to line them up date by date.

Key formulas to remember

Loss given default: LGD = 1 − Recovery rate; Recovery rate is a percentage of exposure. Use LGD, not recovery, in the CVA formula.
Credit valuation adjustment: CVA = Σ [ Expected exposure(t) × LGD(t) × POD(t) × DF(t) ]; Sum over every date t on which default can occur. POD(t) is the unconditional probability of default at t. DF(t) is the risk-free discount factor.
Risky value: Value of risky bond = Value of risk-free bond − CVA; Risk-free value uses the same promised cash flows discounted at risk-free rates.
Expected exposure for a bond: Exposure at t = Cash flow at t + PV at t of all later cash flows; This assumes default occurs just before the payment at t, so the payment at t is also at risk.
Unconditional default probability: POD(t) = Survival probability to t−1 × Conditional POD(t); Survival probability to t = Π (1 − conditional POD) over earlier periods. Check which type of POD the question gives.
One-period risk-neutral POD from prices: Risky price × (1 + r) = Face × (1 − p) + Face × Recovery × p, so p = [Face − Risky price × (1 + r)] ÷ [Face × LGD]; r is the risk-free rate for the period. Here risky price = Face ÷ (1 + risky yield). Exact for a one-period zero-coupon bond.
Spread approximation: Credit spread ≈ POD × LGD, so POD ≈ Spread ÷ LGD; A short-horizon approximation. It is not exact.

How to solve Valuing Risky Bonds and Credit Valuation Adjustment questions

Use this order for any CVA or risky-bond valuation question. Build a small table with one row per date.

1List each date on which default can occur, with the promised cash flow at each date.
2Get the risk-free discount factor for each date. If you are given spot rates, compute DF = 1 ÷ (1 + spot)^t.
3Compute expected exposure at each date: that date's cash flow plus the PV at that date of the later cash flows, using forward or risk-free rates as given.
4Compute LGD = 1 − recovery rate. Check whether recovery is the same at every date.
5Confirm the POD is unconditional for each date. If conditional, convert it using survival probabilities.
6For each date, multiply exposure × LGD × POD × DF. Sum the rows to get CVA.
7Value the bond at risk-free rates, then subtract CVA to get the risky value.
8If the question asks for a default probability from prices or spreads, set up the expected-payoff equation. Use the spread approximation only if the question allows it.

Quickest way: Row-by-row table with LGD factored out

When to use it: Use it when LGD is the same at every date and the bond has two or three dates, which is typical in an item set.

Compute exposure × POD × DF for each date and sum them.
Multiply the total by LGD once at the end.
If a later question changes the recovery rate, rescale: new CVA = old CVA × (new LGD ÷ old LGD). No recalculation is needed.
For a quick check on a one-period bond, test whether POD ≈ spread ÷ LGD is close to your exact answer. A big gap means an error.

Common mistakes in Valuing Risky Bonds and Credit Valuation Adjustment

Using the recovery rate instead of LGD in the CVA formula.
The vignette gives recovery (for example 40%), and it is the number in front of you.
Fix: Write LGD = 1 − recovery as the first line of your working. With 40% recovery, LGD is 60%.
Using conditional default probabilities as if they were unconditional.
Both are called 'probability of default' and appear in similar tables.
Fix: If the table says 'conditional' or 'given survival', multiply by the survival probability to the prior date. The unconditional year-2 POD is (1 − PD1) × conditional PD2.
Leaving out the payment due at the default date from exposure.
Students count only future cash flows after that date.
Fix: Exposure at t includes the cash flow due at t plus the PV of later flows, unless the question defines exposure differently. Follow the vignette's definition.
Discounting the loss at the risky yield instead of the risk-free rate.
The risky yield is the rate usually associated with the bond.
Fix: CVA discounts expected losses at risk-free rates. The credit risk is already in the PODs and LGD, so using the risky yield would count it twice.
Treating the spread approximation as exact and treating risk-neutral PODs as real-world forecasts.
POD ≈ spread ÷ LGD is quick and gives a plausible number.
Fix: When prices are given, solve the exact payoff equation. Remember that risk-neutral PODs include a risk premium and are normally higher than historical default rates.
Adding CVA to the risk-free value.
CVA is reported as a positive number, so the sign is easy to forget.
Fix: CVA is a deduction for the holder of the bond. Risky value is always below risk-free value when POD and LGD are positive.

Worked examples

Example 1

Vignette: An analyst values a 2-year, 6% annual-coupon bond with face value 100. The risk-free rate is a flat 4% for all maturities. Unconditional risk-neutral default probabilities are 1.0% at the end of year 1 and 1.5% at the end of year 2. Recovery is 40% of exposure. Default can occur only at the payment dates, just before each payment. Questions: (1) What is the expected exposure at year 1? (2) What is the CVA? (3) What is the value of the risky bond?

Show the solution

LGD = 1 − 0.40 = 0.60. Discount factors: DF1 = 1 ÷ 1.04 = 0.961538. DF2 = 1 ÷ 1.04² = 0.924556.
Exposure at year 2 = 106 (final coupon plus principal).
Exposure at year 1 = coupon 6 + PV at year 1 of 106 = 6 + 106 ÷ 1.04 = 6 + 101.9231 = 107.9231.
Year 1 row: 107.9231 × 0.60 × 0.010 × 0.961538 = 0.6226.
Year 2 row: 106 × 0.60 × 0.015 × 0.924556 = 0.8820.
CVA = 0.6226 + 0.8820 = 1.5047, or about 1.50.
Risk-free value = 6 ÷ 1.04 + 106 ÷ 1.0816 = 5.7692 + 98.0030 = 103.7722.
Risky value = 103.7722 − 1.5047 = 102.2675.

Answer: (1) Expected exposure at year 1 is about 107.92. (2) CVA is about 1.50. (3) The risky bond is worth about 102.27, compared with 103.77 if risk-free.

Example 2

Vignette: A 1-year zero-coupon bond with face value 100 trades at a yield of 5.00%. The 1-year risk-free rate is 3.00%. The analyst assumes a recovery rate of 40% of face value. Questions: (1) What is the bond's price? (2) What is the exact risk-neutral probability of default? (3) What would the risk-neutral probability be if recovery were 60%, with the price unchanged?

Show the solution

Price = 100 ÷ 1.05 = 95.2381.
Under risk-neutral pricing, price × (1 + risk-free rate) = expected payoff. So expected payoff = 95.2381 × 1.03 = 98.0952.
Expected payoff = 100 × (1 − p) + 40 × p = 100 − 60p. Set 100 − 60p = 98.0952, so 60p = 1.9048 and p = 0.031746, or 3.17%.
Check with the approximation: spread 2.00% ÷ LGD 0.60 = 3.33%. This is close to the exact 3.17% but not equal.
With 60% recovery, expected payoff = 100 − 40p. Set 100 − 40p = 98.0952, so p = 1.9048 ÷ 40 = 0.047619, or 4.76%.

Answer: (1) The price is 95.24. (2) The exact risk-neutral default probability is about 3.17%. (3) With 60% recovery, it rises to about 4.76%, because a smaller loss per default needs a higher default probability to explain the same spread.

Exam tips

Read the vignette for three things first: whether the PODs are conditional or unconditional, how recovery is defined, and what exposure means. Most wrong answers come from misreading these.
Build the date-by-date table even for two dates. It keeps the discount factors, exposures and PODs from getting mixed up across rows.
Use the direction of change to eliminate options: higher recovery lowers CVA, higher POD raises CVA, and a higher CVA lowers the risky bond's value.
For questions on interpreting results, remember that spread widens when POD or LGD rises, and that risk-neutral PODs include a risk premium and are not forecasts.
There is no penalty for wrong answers, so answer every question. If time is short, rescale a prior CVA (for example for a changed recovery rate) rather than recomputing.

Valuing Risky Bonds and Credit Valuation Adjustment in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Valuing Risky Bonds and Credit Valuation Adjustment: frequently asked questions

How do I calculate CVA for the CFA Level II exam?

For each date, multiply expected exposure by LGD, the unconditional probability of default and the risk-free discount factor. Add up the results across dates. LGD is 1 minus the recovery rate.

What is the difference between risk-neutral and real-world default probability?

Risk-neutral probabilities are implied by market prices and include compensation for bearing default risk, so they are used for valuation. Real-world probabilities come from historical default experience. Risk-neutral figures are usually higher.

How is the value of a risky bond related to CVA?

The value of a risky bond equals the value of an identical risk-free bond minus CVA. CVA is the present value of the expected loss from default, so it always reduces the bond's value.

Can I use credit spread to find probability of default?

As an approximation, POD ≈ spread ÷ LGD for a short horizon. If the question gives bond prices and a risk-free rate, solve the expected-payoff equation instead, which is exact for a one-period bond.