Bond prices and probability of default

We learn in Finance 101 that the price of a bond is the present value of future cash flows. There is no mention of default risk. Still, bond prices move each day, without a change in the payment schedule. Isn’t there a conflict? Must it not be that any change in the price of a bond is the result of a change in the perceived risk of the bond — which in turn implies that the price of a bond must reflect probability of default and not just be the present value of future cash flows? Furthermore, I see some threads talk about implied probability of default from bonds, which leads me to the question: what happens if such probability differs from those implied by CDS spreads? I would imagine there is an arbitrage opportunity there but don’t know what it would look like in practice (not least because we don’t know which is “right”).

• Credit risk does get reflected in the price of bonds. The interest rate used to discount the cash flow is inversely related to price. And the interest rate includes a credit risk premium which compensates investors for taking on the default risk.
– nyk
Oct 21, 2020 at 3:25
• Thank you for your response. My question then is: how much of that premium is credit risk? Isn’t there something to be said for supply / demand? For example: suppose a company in dire straits needs \$1bn to finance a safe project that will boost its credit rating. Wouldn’t creditors ask for a premium, knowing how important the project is to the firm, even though the firm becomes less risky by taking on the project? Oct 21, 2020 at 3:39
• Credit risk premium is determined by market forces (i.e. supply & demand).
– nyk
Oct 21, 2020 at 5:08

Let's start with the "safest" bonds in the world, and work our way down the credit quality curve. In Europe, the safest and virtually "credit-risk free" bonds are the German Bunds. If you look at the 10y yield of the German bunds, these are negative 60 bps as of this morning. The ECB deposit rate is negative 50 bps: from the fact that the German 10y yield is even 10 bps lower than the ECB rate, you could either conclude that the bond market is pricing in further ECB rate cut, or you could conclude that the market is worried about stock-market decline greater than 0.6%, and some hedgers would rather lose 0.6% annually by holding the German bunds than hold exposure in the stock market.

Why does the price of the German Bund fluctuate every day, even though there is (almost) no credit risk? Because market participants reassess their need to hedge various risks (if more risks perceived, Bund price goes up -> yield goes down, if less risk perceived, Bund price goes down -> yields go up). In addition, as already mentioned, near ECB monetary policy meetings, the Bund might also reflect the market's view on ECB deposit rate cuts or hikes.

When you look at government bonds such as France or Netherlands, the yield on these will be slightly higher than the yield on German Bunds: even though one could argue that France or Netherlands have zero credit risk, you could conclude that the yield spread to German bunds is the priced-in credit risk.

As you move further down the credit curve, to for example Greece, the yield spread to German Bunds will be even higher. With Greece, you could argue that some credit risk exists: that's why you could argue that the price of Greek bonds fluctuates not just based on investors demand for bonds, but also with market's view on how likely Greece is to default.

As you move away from Government bonds to corporate bonds in the Eurozone, the yield spread to German Bunds will be even wider: this tells you the amount of credit risk that the market is pricing in.

In the US, the same structure works: start with 10y US Treasuries, and treat these as "credit-risk free bonds". Then work your way down to state-issued bonds (i.e. California), and the yield-spread to 10Y US Treasuries will give you an estimate of the credit risk priced in. Then as you move to corporate bonds, the yield spread will increase further in line with credit risk.

Ps: in this answer here, Expected Forward Volatility vs. Different Strikes, I explain why the yield-spread tends to exaggerate the real credit risk priced into the bonds.

• I agree with all of this. But we also talk about flight to quality during tough markets. If we go by this perspective, however, it follows that Treasuries are suddenly less risky then they were beforehand. That, of course, is not the case; people just want to park their money somewhere safe while things abate. If anything, the credit risk has increased, however marginally. Maybe my question relates more to the coupon than to the yield, but it seems to be more of a game question than a risk one, as it simply comes down to minimizing the interest rate at which the market will buy the issuance. Oct 21, 2020 at 22:20

what happens if such probability differs from those implied by CDS spreads? I would imagine there is an arbitrage opportunity there but don’t know what it would look like in practice

Bonds vs. CDS is known as 'basis'. If you think it's an arbitrage, I suggest you look at what happened to basis during the GFC. C.f. https://chairegestiondesrisques.hec.ca/wp-content/uploads/2019/11/19-04.pdf

Must it not be that any change in the price of a bond is the result of a change in the perceived risk of the bond — which in turn implies that the price of a bond must reflect probability of default and not just be the present value of future cash flows?

A default-free bond is priced using the sum of the PV of cash flows. In practice, yes, the bond prices reflect the credit-worthiness (spread) and liquidity.

Furthermore, I see some threads talk about implied probability of default from bonds, which leads me to the question: what happens if such probability differs from those implied by CDS spreads? I would imagine there is an arbitrage opportunity there but don’t know what it would look like in practice (not least because we don’t know which is “right”).

A CDS spread essentially reflects default probability as CDS on a company can be viewed as a portfolio of a long corporate bond on the same company and short a treasury (risk-free) bond.

Example: treasury bond and corporate bond have par value 100 and both have 1 year maturity. Portfolio A: long the risk-free-bond and short CDS (selling protection). Portfolio B: long corporate bond. Then the portfolios should provide the same return after the first year or else there's arbitrage. In the case of no default, Portfolio A's payoff at time 1 is 100 (payment from treasury) - 0 (no payment made to protection buyer) = 100 and Portfolio B's payoff is 100 (payment from corporate bond). In the case of a default with a recovery rate (RR) of 40%, then Portfolio A's payoff at time 1 is 100 for the risk-free bond and 1-RR=60% of notional of corporate bond which is 60 paid out which results in a total payoff of 40. Portfolio B's payment is 40 since that is the RR portion of the notional. If any of these numbers differed there is an opportunity for arbitrage.