Credit where credit is due

A banker's profile for an investment portfolio

It came up obliquely in the post about mismatch: the regulatory model only includes rates and equities and says nothing about other asset classes, particularly credit. The regulator has a case that any spread returns should be part of Rᴿ, and so not included in the matching portfolio. So we find ourselves in the common situation where corporate credits—which self-evidently add value to a portfolio of equities and bonds—are too risky for the matching portfolio, but not returny enough for the growth portfolio; so what to do with them? Hence the title.

This raises the following question: how to add a credit factor to the return portfolio, so that its overall risk remains the same? And, since this gives two conditions (same risk and weights sum to 100%) for 3 variables (equities, credits, cash), what is the optimal proportion? If you’ve taken any finance courses you may have encountered this kind of question. It could have been any asset class, but corporate credit is such a biggy.

Writing XE for the excess return on the Equity factor, and XC for the excess (spread) return of the Credit factor, this problem can be written as:

\(\max_{a.b} \left( aX_E + bX_C -\lambda(a^2 \sigma_E^2 + b^2 \sigma_C^2 +2ab\rho\sigma_E\sigma_C-\sigma_E^2)\right)\)

where λ is a Lagrange multiplier to ensure the level of risk. Taking derivatives:

\(\begin{split} \frac{\partial}{\partial a}: \qquad X_E - 2\lambda \sigma_E(a \sigma_E + b\rho\sigma_C)=0 \\ \frac{\partial}{\partial b}: \qquad X_C - 2\lambda \sigma_C(b \sigma_C + a\rho\sigma_E)=0 \\ \end{split}\)

so that, cancelling the 2λ:

\(X_C(a \sigma_E + b\rho\sigma_C) = X_E(b \sigma_C + a\rho\sigma_E) \implies b\sigma_C ( X_E - \rho X_C) = a\sigma_E (X_C-\rho X_E)\)

This tells us b should be a multiple of a, and we can use the risk condition (from setting ∂/∂λ to zero) to determine both!

Now for some illustrative figures¹, we default to the FTK risk parameters²: σE=30%, σC=10%, ρ=50%, and we furthermore assume equity risk premium XE = 4% and credit premium XC=2.5%³. Then b=a∙30/10∙(2.5-0.5∙4)/(4-0.5∙2.5)=a∙3∙(0.5/2.75) =a∙6/11. The risk relation can be used to find a, and so b:

\(a^2(1 + (6/11)^2 \frac{1}{3^2}+2\cdot (6/11)\cdot \frac{1}{3}\cdot 0.5)=1 \implies a^2 = 121/147 \implies a \approx 0.9 ,\; b \approx 0.5\)

So reducing the equity allocation by 10% allows us to invest 50% in credit risk, for the same total risk of the return portfolio but now with Rᴿ=4.85%, an uplift of ~0.85%.

Now, where to borrow that 40% cash? From the matching portfolio (V-VY) of course⁴! That should invest in corporate bonds and swap out the credit risk (for cash returns) with the return portfolio. The total portfolio is, with above parameters, then 0.9∙VY equity, 0.5∙VY credit, (V-1.4VY) cash⁵. Additional derivatives are used to match (V_XΔ - 0.5∙VY ∂CR/∂r): the liability sensitivity corrected for the rate sensitivity of the credits.

A final note on implications for markets. People who use the regulatory model usually aren’t professional investors, and so the thought process of the first paragraph suggests to not invest in credit at all. While Dutch pension funds and their advisors get their act together there will be selling pressure on credits, which will turn into buying pressure once they do.

1. As, to quote my hero Johan de Witt in our native Dutch: “de dagelijksche ondervindinghe openbaer maeckt, dat veele Menschen of niet genegen of niet bequaem zijn haer begrip op eenige aen een geschaeckelde, al-hoewel onfeylbare raisonnementen, soodanigh te appliceren, dat zy de kracht van de selve te rechte konnen vatten om daer door tot haer volkomen vergenoegen overtuycht te worden; en dat derhalve by haer de exempelen meer vermogen als alsulcke raisonnementen” … that is, people like examples.

2. Those risks were defined at the 97.5% confidence level, or roughly 2∙σ. That doesn’t matter for this example though.

3. First pass was XC=2%, but that gives b=a∙3∙(2-2)/(4-1)=0. That one’s too easy: no credit.

4. Again, refer to the post about mismatch for notation.

5. As long as VY/V < 1/1.4, this means no external leverage is needed. This is an internal trade.