The reward for risk in credit and option markets
Jan Ericsson
Overview of the talk



• Risk premia in corporate bond markets

• Risk premia in CDS and stock option markets
What’s in a credit spread?


• Consider a world without taxes and with perfectly liquid markets

• Suppose that default ri...
Systematic default risk
• So a positive spread over the risk free rate does not mean there is
  a premium for default risk...
Expected loss / Risk premia (EL / RP)

• So the total spread of 26.3% consists of

  11.67% compensation for expected loss...
Why is this important?

• Asset allocation (across products / over the cycle).


   • bonds with the same rating / default...
How we compute risk premia

             N
  P
 Bt,T   =         di · ci · (1 − Pt (τ < si )) + dN · p · (1 − Pt (τ < T ))...
How do we compute default probabilities (P)?

• As an illustration consider Merton (1974)


• The Q-probability of default...
Estimating the asset drift
we use historical returns for this - ideally one would use a
                  forward looking ...
Our default probabilities
     0.18


     0.16

                Model historical default probabilities
     0.14       Mo...
97% of the data (excluding AAA, CCC and less)

        0.14


        0.12       Model default probabilities
             ...
Our findings - preview

• risk premia are highly time-varying


      • they are similar to what has been found in CDS mark...
Risk premia measured in bond markets


    180
                                             Market bond risk premia (swap ...
300                                       300
                    Market spread (bps)              Expected loss component...
Expected losses



• Elton et al. (2001, JF) find using historical rating based default rates
  that expected losses consti...
The link between risk premia in equity and credit
markets is non-linear
        1500
                                     ...
Basic corporate finance tells us (MM prop II) that

                                                   E
                  ...
What drives expected losses?

     150                                                                   40
              ...
What drives risk premia?

     100                                                                    40
                 ...
Panel A: Spread components                             Panel B: Ratio of RP to total spread
4000                          ...
Summary so far

• spread components highly time varying both in absolute and relative
  terms.


• the link between risk p...
Derivatives markets - CDS and stock options

• We now move on to a slightly different question (based on current joint
  w...
Does knowledge of systematic risk help in pricing
derivatives relative to the underlying?

  • Duan & Wei (2008, RFS) have...
Stock options and systematic risk




   IVt,j − σt,j
                = ηA,t At,j + ηB,t BBBt,j + ηb,t bt,j + ηc,t · Contr...
CDS spreads and systematic risk




                                                                      ?




   st,j = ...
The RP spread




 st,j − ELt,j
              = ηA,t At,j + ηB,t BBBt,j + ηb,t bt,j + ηc,t · Controlst,j + εt,j
      st,j
Summary and implications.

• When pricing equity and credit derivatives relative to the
  underlying, systematic risk matt...
RiskLab Madrid meeting October 2009
RiskLab Madrid meeting October 2009
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RiskLab Madrid meeting October 2009

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A presentation on the compensation for risk in corporate bond, CDS and equity option markets.

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  • RiskLab Madrid meeting October 2009

    1. 1. The reward for risk in credit and option markets Jan Ericsson
    2. 2. Overview of the talk • Risk premia in corporate bond markets • Risk premia in CDS and stock option markets
    3. 3. What’s in a credit spread? • Consider a world without taxes and with perfectly liquid markets • Suppose that default risk is completely diversifiable: objective (P)= risk neutral (Q) survival rates • Assume P= 90%, zero recovery and r=5%. What is the bond yield (and spread)? 0.9 · 100 100 B= = 85.71 = 85.71 → y = 0.1667 1.05 1+y
    4. 4. Systematic default risk • So a positive spread over the risk free rate does not mean there is a premium for default risk - just compensation for expected losses. • Suppose now that default risk is systematic and as a result there is a default risk premium • This will translate into a lower risk-adjusted survival probability than the objective (Q<P) 0.9, say 0.8. So the bond price would be 0.8 · 100 100 B= = 76.16 = 76.16 → y = 0.3130 1.05 1+y
    5. 5. Expected loss / Risk premia (EL / RP) • So the total spread of 26.3% consists of 11.67% compensation for expected losses (EL spread) 14.63% default risk premium (RP spread). • Measuring these two components separately with and without corporate bond prices is part of what we do now (“Time varying risk premia in corporate bond markets” which is joint with Redouane Elkamhi).
    6. 6. Why is this important? • Asset allocation (across products / over the cycle). • bonds with the same rating / default rate can have very different spreads depending on the systematic nature of their default risk. • Bonds / CDS across rating categories appear to have different mixes of expected losses / risk premia. • the same is true for multi-name tranched products. Equity tranches may have more risk in an absolute sense but super senior tranches should compensate more for systematic risk than expected losses.
    7. 7. How we compute risk premia N P Bt,T = di · ci · (1 − Pt (τ < si )) + dN · p · (1 − Pt (τ < T )) i=1 T P +R · p · ds · dPt (s) →y t N Q Bt,T = di · ci · (1 − Qt (τ < si )) + dN · p · (1 − Qt (τ < T )) i=1 T +R · p · ds · dQt (s) → y Q,model , y Q,market t
    8. 8. How do we compute default probabilities (P)? • As an illustration consider Merton (1974) • The Q-probability of default is 1 − N (d2 ) vt ln F + (r − 2 σ )T 1 2 d2 = √ σ T • and the P-probability would be 1 − N dP 2 vt ln F + (µ − 2 σ )T 1 2 P d2 = √ σ T
    9. 9. Estimating the asset drift we use historical returns for this - ideally one would use a forward looking metric µv − r = (Rv (t) − r) = ∆E · (RE (t) − r) This is where the model comes in, we chose Leland & Toft (1996)
    10. 10. Our default probabilities 0.18 0.16 Model historical default probabilities 0.14 Moody’s default experience 1970−2004 0.12 0.1 0.08 0.06 0.04 0.02 0 2 4 6 8 10 12 14 16 18 20
    11. 11. 97% of the data (excluding AAA, CCC and less) 0.14 0.12 Model default probabilities Moody’s default experience 1970−2004 0.1 0.08 0.06 0.04 0.02 0 0 2 4 6 8 10 12 14 16 18 20 Horizon (years)
    12. 12. Our findings - preview • risk premia are highly time-varying • they are similar to what has been found in CDS markets during the 2001-2004 period. • Expected losses and risk premium spread components behave differently. • Risk premia in credit and equity markets look very different - • a link can only be established by accounting for the non-linearity and dependence on time varying parameters.
    13. 13. Risk premia measured in bond markets 180 Market bond risk premia (swap curve) 160 Market bond risk premia (CMT) 140 120 100 80 60 40 20 0 95 96 97 98 99 00 01 02 03 04
    14. 14. 300 300 Market spread (bps) Expected loss component (bps) 250 250 200 200 150 150 100 100 50 50 0 0 95 97 00 02 95 97 00 02 300 1 risk premium component (bps) EL ratio 250 RPratio 0.8 200 0.6 150 0.4 100 50 0.2 0 0 95 97 00 02 95 97 00 02
    15. 15. Expected losses • Elton et al. (2001, JF) find using historical rating based default rates that expected losses constitute about 25% of spreads on average. • Our average is very close to this - but there is significant time variation: near zero lows and highs in the 70-80% range. • using the average may not be sufficient in applications
    16. 16. The link between risk premia in equity and credit markets is non-linear 1500 Equity premium Bond risk premium (swap curve) 200 1000 100 500 0 95 95 96 96 97 97 98 98 99 99 00 00 01 01 02 02 03 03 04 04 and depends on time varying parameters.
    17. 17. Basic corporate finance tells us (MM prop II) that E RD = Rv + (Rv − RE ) D Augmenting this with a model such as Leland & Toft (1996, JF) highlights the 2nd point E(v; t, σ, N, T, r) RD (v; t, σ, N, T, r) = Rv + (Rv − RE (v; t, σ, N, T, r)) · D(v; t, σ, N, T, r)
    18. 18. What drives expected losses? 150 40 EL swap curve (bps) Mean asset volatility 100 30 50 20 0 10 95 96 97 98 99 00 01 02 03 04
    19. 19. What drives risk premia? 100 40 RP swap curve (bps) S&P 500 volatility in percentage 50 20 0 0 95 96 97 98 99 00 01 02 03 04
    20. 20. Panel A: Spread components Panel B: Ratio of RP to total spread 4000 1 spread 0.8 3000 RP EL 0.6 2000 0.4 1000 0.2 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Panel C: ratio of RN to historical DP Panel D: Diff. between RN and HIST DP 12 0.2 10 0.15 8 6 0.1 4 0.05 2 0 0 0 20 40 60 80 100 0 20 40 60 80 100
    21. 21. Summary so far • spread components highly time varying both in absolute and relative terms. • the link between risk premia in equity and credit markets is non-linear and time varying. • RP tends to be higher in a relative sense for higher grade credits and in times of relatively low default rates. • RP help in explaining bond spread data for high rated bonds in particular.
    22. 22. Derivatives markets - CDS and stock options • We now move on to a slightly different question (based on current joint work with Christian Dorion and Redouane Elkamhi) • so far we have discussed splitting the total bond spread into EL and RP components and shown • EL / RP depend on idiosyncratic / systematic risk. • But does knowing the split of volatility in its idiosyncratic and systematic components help to explain the total spread? • Does it help help in pricing derivatives (options, CDS) relative to the underlying?
    23. 23. Does knowledge of systematic risk help in pricing derivatives relative to the underlying? • Duan & Wei (2008, RFS) have shown that it matters for options on some 30 stocks during the early nineties. • We will now show that risk premia help in pricing CDS as well as stock options. • matched sample of CDS and equity options for about 130 firms in the CDX index between 2004 and 2007. 2 2 βj,k σM,k bj,k = systematic risk proportion 2 σj,k
    24. 24. Stock options and systematic risk IVt,j − σt,j = ηA,t At,j + ηB,t BBBt,j + ηb,t bt,j + ηc,t · Controlst,j + εt,j IVt,j
    25. 25. CDS spreads and systematic risk ? st,j = ηA,t At,j + ηB,t BBBt,j + ηb,t bt,j + ηc,t · Controlst,j + εt,j
    26. 26. The RP spread st,j − ELt,j = ηA,t At,j + ηB,t BBBt,j + ηb,t bt,j + ηc,t · Controlst,j + εt,j st,j
    27. 27. Summary and implications. • When pricing equity and credit derivatives relative to the underlying, systematic risk matters - prefence free pricing does not seem to hold. • The mix between compensation for expected losses and the compensation for risk in bond and CDS spreads behaves in a complex way. • Ratings alone do not give sufficient guidance for assessing a credit investment. Do we need risk-adjusted ratings?

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