Value-at-risk capital requirement regulation, risk taking and asset allocation: a mean–variance analysis

Abstract

In this study, the mean–variance framework is employed to analyze the impact of the Basel value-at-risk (VaR) market risk regulation on the institution's optimal investment policy, the stockholders’ welfare, as well as the tendency of the institution to change the risk profile of the held portfolio. It is shown that with the VaR regulation, the institution faces a new regulated capital market line, which induces resource allocation distortion in the economy. Surprisingly, only when a riskless asset is available does VaR regulation induce the institution to reduce risk. Otherwise, the regulation may induce higher risk, accompanied by asset allocation distortion. On the positive side, the regulation implies an upper bound on the risk the institution takes and it never induces the firm to select an inefficient portfolio. Moreover, when the riskless asset is available, tightening the regulation always increases the amount of maintained eligible capital and decreases risk.

Keywords:

• risk management
• capital requirement regulation
• value-at-risk
• Basel regulations
• regulated capital market line

JEL Classification::

• G11
• G18
• G28
• G38
• E58

Acknowledgements

We are grateful to two anonymous Referees and the Associate Editor for their helpful suggestions that helped improved the paper. We acknowledge financial support from the Krueger Center of Finance.

Notes

1. In practice, these marketable assets also include foreign currencies and commodities throughout the institution. Yet, this is only a semantic differentiation, which does not change the analysis.

2. For extensive empirical evidence of regulation costs on financial intermediation, see Demirgüç-Kunt, Laeven, and Levine (2004). They study 1400 banks in 72 countries showing that a tighter regulation on banks boosts their costs. Following their empirical finding, our model assumes that regulation is involved with costs.

3. Of course, the magnitude of asset allocation distortion is a function of the total amount of marketable assets held by all regulated institutions relative to the total market value of the assets in the economy.

4. For a review on bank regulation, see Bhattacharya, Boot, and Thakor (1998).

5. An introduction and overview of VaR can be found in Duffie and Pan (1997), Kupiec (2004a) and Jorion (2007), which has become the industry standard.

6. Santos et al. (2012) explore how to minimize capital requirement under this revised scheme, while taking into account imposed penalties due to poor estimation of VaR. Alexander, Baptista, and Yan (2013) find that while the Basel (2011) revised scheme increases minimum capital requirements, it also tends to be less effective in preventing banks from taking substantive tail risk. This suggests that the analysis presented in this study also remains relevant under the revised scheme. Similar to other studies, in the mathematical analysis, we consider only a single VaR. However, this does not affect the results, as we assume elliptical distribution, which implies that both VaRs are linear functions of the portfolio's standard deviation. Namely, this single VaR may be the sum of the two VaRs, where the difference is only a constant multiplier.

7. For the difficulties of international convergence of bank capital requirements in the presence of divergent policies of different central banks, see Acharya (2003).

8. The Tier 1 capital may consist of the following items: (i) common stockholders’ equity; (ii) qualifying noncumulative perpetual preferred stock (including related surplus); and (iii) minority interest in the equity accounts of consolidated subsidiaries. The Tier 2 capital may consist of the following items: (i) allowance for loan and lease losses; (ii) perpetual preferred stock and related surplus; (iii) hybrid capital instruments and mandatory convertible debt securities; (iv) term subordinated debt and intermediate-term preferred stock, including related surplus; (v) unrealized holding gains on equity securities.

9. In Kim and Santomero's (1988) analysis, for example, short sales are restricted as they analyze credit regulation where short sales are irrelevant. This is because in their classic case, a bank borrows money from numerous depositors and lends this money together with equity capital. Hence, all assets are long positions in credit loans. In contrast, in our case regulation is over market risk; namely, on the market portfolio, which may include short-sale positions. For further discussion of this subject, see Kupiec (2004b).

10. For the definition of VaR, see Duffie and Pan (1997), Alexander (1998), Kupiek (2004a) and Jorion (2007).

11. Kupiek (2004a) correctly argues that while the latter definition estimates the so-called ‘unexpected losses’, it will not produce accurate estimates of buffer stock capital requirements.

12. This bound from below is the local minimum; the global minimum is even lower. This is because from a certain pre-regulated portfolio variance, the regulated portfolio variance decreases such that the regulated firm's M–V frontier bounces backward and turns left in (μ, σ) space (Figure 1 and imagine a continuation of segment until the variance is lower than that of portfolio m^′). Hence, for certain portfolios, the regulated portfolio variance is lower than this bound. However, only the local minimum is economically relevant.

13. Substituting from Equation (10) in Equation (9) yields . The expression can also be written as or . Substituting this expression in yields Equation (11).

14. With corporate taxes the firm will not invest in the riskless asset, but will rather let the stockholder do it directly.

15. In this case of no riskless asset, we use the notion optimal regulated risky portfolio somewhat differently as it is not on the RCML, but rather, the portfolio that permits the representative stockholder to obtain the highest feasible indifference curve.

16. In Merton (1972), the frontier is formulated somewhat differently such that the goal is minimizing the variance, , for a given expected return, μ_P. We use this formulation for consistency with the next steps. Naturally, this does not change the results.

17. Substituting μ_P from Equation (14) in Equation (6) and adding and subtracting r⁻ yields, . Next, the two solutions of Equation (7), which is quadratic in σ_P, are given by . However, as we focus only on the range where or equivalently, , then only the solution with the negative sign is relevant. Namely, where . Substituting in from above yields Equation (16). Note that for , Equation (7) yields . Thus, is consistent with the assumption that and therefore does not add any new constraint.

18. As was previously mentioned, Basel (2011) acknowledges that the 10-day VaR methodology may not cover possible future extraordinary market conditions or shocks and therefore also requires a separate stress-testing scheme accompanied by capital requirements against the stressed VaR. Thus, it is only reasonable to assume, consistent with Basel's stress-testing approach, that there are elements of liquidity risk that might not be covered by the current VaR methodology. Yet, these elements are also reduced by the amount of maintained eligible capital.

19. To see that the second expression on the right-hand side of Equation (20) is always positive, recall Footnote 5 in Merton (1972, p. 1853): D>0. Obviously, in the numerator and, by assumption, (r−r⁻)>0. By definition, ; hence, in the denominator .

In addition, . Finally, as , (A−Cr)+2>0 and G^e>0, then, H^e>0.

20. Note that our analysis only covers market risk regulation, as the financial risk is covered by a different credit regulation scheme. Thus, one may wonder whether the desirable reduction in the risk of the risky portfolio is accompanied by an increase in leverage and, correspondingly, an increase in financial risk. Fortunately, there is no reason to assume such a response. Recall that according to the Separation Theorem, the firm's decision about the degree of leverage does not depend on the composition of its risky portfolio. Thus, the firm should employ the same degree of leverage, regardless of the selected risky portfolio. In addition, the financial institution is separately regulated for the financial risk by credit capital adequacy regulations and for market risk by VaR regulation. Thus, the decision on leverage is regulated separately from market risk.

21. Recall from Equation (22) that the increase in n in the denominator of Equation (23) is always larger than the implied decrease in ; hence, the denominator in Equation (23) always decreases with n while the nominator is unchanged.

22. This decreasing rate is due to both a decrease of in the numerator and an increase of n and 1/M in the denominator with the increase in n.