# Risk Appetite and the Virtues of the Board

This article consists of two parts: Risk Appetite, and The Virtues of the Board. (Upcoming) This first part can be read as a standalone article, the second will be based on concepts developed in this part.

### Risk Appetite

Multiple sources of risk are a fact of life. Only rarely will decisions concerning various risks be neatly separable. Intuitively, even when risks are statistically independent, bearing one risk should make an agent less willing to bear another. (Kimball, 1993)

Risk appetite – the board’s willingness to bear risk – will depend both on the degree to which it dislikes uncertainty and to the level of that uncertainty. It is also likely to shift as the board respond to emerging market and macroeconomic uncertainty and events of financial distress.

The following graph of the “price of risk[1]” index developed at the Bank of England shows this. (Gai & Vause, 2005)[2] The estimated series fluctuates close to the average “price of risk” most of the time, but has sharp downward spikes in times of financial crises. Risk appetite is apparently highly affected by exogenous shocks:

In adverse circumstances, it follows that the board and the investors will require higher expected equity value of the firm to hold shares – an enhanced risk premium – and that their appetite for increased risk will be low.

### Risk Management and Risk Appetite

Despite widespread use in risk management[3] and corporate governance literature, the term ‘risk appetite’[i] lacks clarity in how it is defined and understood:

- The degree of uncertainty that an investor is willing to accept in respect of negative changes to its business or assets. (Generic)
- Risk appetite is the degree of risk, on a broad-based level, that a company or other entity is willing to accept in the pursuit of its goals. (COSO)
- Risk Appetite the amount of risk that an organisation is prepared to accept, tolerate, or be exposed to at any point in time (The Orange Book October 2004)

The same applies to a number of other terms describing risk and the board’s attitudes to risk, as for the term “risk tolerance”:

- The degree of uncertainty that an investor can handle in regard to a negative change in the value of his or her portfolio.
- An investor’s ability to handle declines in the value of his/her portfolio.
- Capacity to accept or absorb risk.
- The willingness of an investor to tolerate risk in making investments, etc.

It thus comes as no surprise that risk appetite and other terms describing risk are not well understood to a level of clarity that can provide a reference point for decision making[4]. Some takes the position that risk appetite never can be reduced to a sole figure or ratio, or to a single sentence statement. However to be able to move forward we have to try to operationalize the term in such a way that it can be:

- Used to commensurate risk with reward or to decide what level of risk that is commensurate with a particular reward and
- Measured and used to sett risk level(s) that, in the board’s view, is appropriate for the firm.

It thus defines the boundaries of the activities the board intends for the firm, both to the management and the rest of the organization, by setting limits to risk taking and defining what acceptable risk means. This can again be augmented by a formal ‘risk appetite statement’ defining the types and levels of risk the organization is prepared to accept in pursuit of increased value.

*However, in view of the “price of risk” series above, such formal statements cannot be carved in stone or they have to contain rules for how they are to be applied in adverse circumstances, since they have to be subject to change as the business and macroeconomic climate changes.*

Deloitte’s Global Risk Management Survey 6. ed. (Deloitte, 2009) found that sixty-three percent of the institutions had a formal, approved statement of their risk appetite. (See Exhibit 4. below) Roughly one quarter of the institutions said they relied on quantitatively defined statements, while about one third used both quantitative and qualitative approaches:

Using a formal ‘risk appetite statement’ is the best way for the board to communicate its visions, and the level and nature of the risks the board will consider as acceptable to the firm. This has to be quantitatively defined and be based on some opinion of the board’s utility function and use metrics that can fully capture all risks facing the company.

We will in the following use the firm’s Equity Value as metric as this will capture all risks – those impacting the balance sheet, income statement, required capital and WACC etc.

We will assume that the board’s utility function[5] have diminishing marginal utility for an increase in the company’s equity value. From this it follows that the board’s utility will decrease more with a loss of 1 $ than it will increase with a gain of 1 $. Thus the board is risk averse[ii].

### The upside-potential ratio

To do this we will use the upside-potential ratio[6] (UPR), a measure developed as a measure of risk-adjusted returns (Sortino et al., 1999). The UPR is a measure of the potential return on an asset relative to a preset return, per unit of downside risk. This ratio is a special case of the more general one-sided variability ratio Phi_{b}

Phi_{b}^{ p,q} (X) := E^{1}^{/p}[{(X – b)^{+}}^{p}] / E^{1/}^{q}[{(X- b)^{–}}^{q}],

Where X is total return, (X-b) is excess return over the benchmark b[7] and the minus and plus sign denotes the left-sided moment (lower partial moment) and the right sided moment (upper partial moment) – of order p and q.

The lower partial moment[8] is a measure of the “distance[9]” between risky situations and the corresponding benchmark when only unfavorably differences contribute to the “risk”. The upper partial moment on the other hand measures the “distance” between favorable situations and the benchmark.

The Phi ratio is thus the ratio of “distances” between favorable and unfavorable events – when properly weighted (Tibiletti & Farinelli, 2002).

For a fixed benchmark b, the higher Phi the more ‘profitable’ is the risky asset. Phi can therefore be used to rank risky assets. For a given asset, Phi will be a decreasing function of the benchmark b.

The choice of values for p and q depends on the relevance given to the magnitude of the deviations from the benchmark b. The higher the values, the more emphasis are put on that tail. For p=q=1 we have the Omega index (Shadwick & Keating, 2002).

The choice of p=1 and q=2, is assumed to fit a conservative investor while a value of p>>1 and q<<1 will be more in line with an aggressive investor (Caporin & Lisi, 2009).

We will in the following use p=1 and q=2 for calculation of the upside-potential ratio (UPR) thus assuming that the board consists of conservative investors. For very aggressive boards other choices of p and q should be considered.

The UPR for the firm can thus be expressed as a ratio of partial moments; that is as the ratio of the first order upper partial moment (UPM_{1})[10] and the second order lower partial moment (LPM_{2}) (Nawrocki, 1999) and ( Breitmeyer, Hakenes & Pfingsten, 2001), or the over-performance divided by the root-mean-square of under-performance, both calculated at successive points on the probability distribution for the firm’s equity value.

As we successively calculates the UPR starting at the left tail will the lower partial moment (LPM_{2}) increase and the upper partial moment (UPM_{1}) decrease: The upside potential ratio will consequently decrease as we move from the lower left tail to the upper right tail – as shown in the figure below: The upside potential ratio have many interesting uses, one is shown in the table below. This table gives the upside potential ratio at budgeted value, that is the expected return above budget value per unit of downside risk – given the uncertainty the management for the individual subsidiaries have expressed. Most of the countries have budget values above expected value exposing downward risk. Only Turkey and Denmark have a ratio larger than one – all others have lager downward risk than upward potential. The extremes are Poland and Bulgaria.

Country/ Subsidiary | Upside Potential Ratio |
---|---|

Turkey | 2.38 |

Denmark | 1.58 |

Italy | 0.77 |

Serbia | 0.58 |

Switzerland | 0.23 |

Norway | 0.22 |

UK | 0.17 |

Bulgaria | 0.08 |

We will in the following use five different equity distributions, each representing a different strategy for the firm. The distributions (strategies) have approximately the same mean, but exhibits increasing variance as we move to successive darker curves. That is; an increase in the upside also will increase the possibility of a downside:

By calculating the UPR for successive points (benchmarks) on the different probability distribution for the firm’s equity value (strategies) we, can find the accompanying curves described by the UPR’s in the UPR and LPM_{2}/UPM_{1} space[12], (Cumova & Nawrocki, 2003):

The colors of the curves give the corresponding equity value distributions shown above. We can see that the equity distribution with the longest upper and lower tails corresponds to the right curve for the UPR, and that the equity distribution with the shortest tails corresponds to the left (lowest upside-potential) curve.

In the graph below, in the LPM_{2}/UPM_{1} space, the curves for the UPR’s are shown for each of the different equity value distributions (or strategies). Each will give the rate the firm will have to exchange downside risk for upside potential as we move along the curve, given the selected strategy. The circles on the curves represent points with the same value of the UPR, as we move from one distribution to another:

By connecting the points with equal value of the UPR we find the iso-UPR curves; the curves that give the same value for the UPR, across the strategies in the LPM_{2}/UPM_{1} space:

We have limited the number of UPR values to eight, but could of course have selected a larger number both inside and outside the limits we have set.

The board now have the option of selecting the strategy they find most opportune, or the one that fits best to their “disposition” to risk by deciding the appropriate value of LPM_{2} and UPM_{1} or of the upside-potential ratio, and this what we will pursue further in the next part: “The Virtues of the Board”.

### References

Breitmeyer, C., Hakenes, H. and Pfingsten, A., (2001). The Properties of Downside Risk Measures. Available at SSRN: http://ssrn.com/abstract=812850 or http://dx.doi.org/10.2139/ssrn.812850.

Caporin, M. & Lisi,F. (2009). Comparing and Selecting Performance Measures for Ranking Assets. Available at SSRN: http://ssrn.com/abstract=1393163 or http://dx.doi.org/10.2139/ssrn.1393163

CRMPG III. (2008). The Report of the CRMPG III – Containing Systemic Risk: The Road to Reform. Counterparty Risk Management Policy Group. Available at: http://www.crmpolicygroup.org/index.html

Cumova, D. & Nawrocki, D. (2003). Portfolio Optimization in an Upside Potential and Downside Risk Framework. Available at: http://www90.homepage.villanova.edu/michael.pagano/DN%20upm%20lpm%20measures.pdf

Deloitte. (2009). Global Risk Management Survey: Risk management in the spotlight. Deloitte, Item #9067. Available at: http://www.deloitte.com/assets/Dcom-UnitedStates/Local%20Assets/Documents/us_fsi_GlobalRskMgmtSrvy_June09.pdf

Ekern, S. (1980). Increasing N-th degree risk. Economics Letters, 6: 329-333.

Gai, P. & Vause, N. (2004), Risk appetite: concept and measurement. Financial Stability Review, Bank of England. Available at: http://www.bankofengland.co.uk/publications/Documents/fsr/2004/fsr17art12.pdf

Illing, M., & Aaron, M. (2005). A brief survey of risk-appetite indexes. Bank of Canada, Financial System Review, 37-43.

Kimball, M.S. (1993). Standard risk aversion. Econometrica 61, 589-611.

Menezes, C., Geiss, C., & Tressler, J. (1980). Increasing downside risk. American Economic Review 70: 921-932.

Nawrocki, D. N. (1999), A Brief History of Downside Risk Measures, The Journal of Investing, Vol. 8, No. 3: pp. 9-

Sortino, F. A., van der Meer, R., & Plantinga, A. (1999). The upside potential ratio. *, The Journal of Performance Measurement*, *4*(1), 10-15.

Shadwick, W. and Keating, C., (2002). A universal performance measure, J. Performance Measurement. pp. 59–84.

Tibiletti, L. & Farinelli, S.,(2002). Sharpe Thinking with Asymmetrical Preferences. Available at SSRN: http://ssrn.com/abstract=338380 or http://dx.doi.org/10.2139/ssrn.338380

Unser, M., (2000), Lower partial moments as measures of perceived risk: An experimental study, Journal of Economic Psychology, Elsevier, vol. 21(3): 253-280.

Viole, F & Nawrocki, D. N., (2010), The Utility of Wealth in an Upper and Lower Partial Moment Fabric). Forthcoming, Journal of Investing 2011. Available at SSRN: http://ssrn.com/abstract=1543603

### Notes

[1] In the graph risk appetite is found as the inverse of the markets price of risk, estimated by the two probability density functions over future returns – one risk-neutral distribution and one subjective distribution – on the S&P 500 index.

[2] For a good overview of risk appetite indexes, see “A brief survey of risk-appetite indexes”. (Illing & Aaron, 2005)

[3] Risk Management all the processes involved in identifying, assessing and judging risks, assigning ownership, taking actions to mitigate or anticipate them, and monitoring and reviewing progress.

[4] The Policy Group recommends that each institution ensure that the risk tolerance of the firm is established or approved by the highest levels of management and shared with the board. The Policy Group further recommends that each institution ensure that periodic exercises aimed at estimation of risk tolerance should be shared with the highest levels of management, the board of directors and the institution’s primary supervisor in line with Core Precept III. Recommendation IV-2b (CRMPG III, 2008).

For an extensive list of Risk Tolerance articles, see: http://www.planipedia.org/index.php/Risk_Tolerance_(Research_Category)

[5] See: http://en.wikipedia.org/wiki/Utility, http://en.wikipedia.org/wiki/Ordinal_utility and http://en.wikipedia.org/wiki/Expected_utility_theory.

[6] The ratio was created by Brian M. Rom in 1986 as an element of Investment Technologies’ Post-Modern Portfolio theory portfolio optimization software.

[7] ‘b’ is usually the target or required rate of return for the strategy under consideration, (‘b’ was originally known as the minimum acceptable return, or MAR). We will in the following calculate the UPR for successive benchmarks (points) covering the complete probability distribution for the firm’s equity value.

[8] The Lower partial moments will uniquely determine the probability distribution.

[9] The use of the term distance is not unwarranted; the Phi ratio is very similar to the ratio of two Minkowski distances of order p and q.

[10] The upper partial-moment is equivalent to the full moment minus the lower partial-moment.

[11] Since we don’t know the closed form for the equity distributions (strategies), the figure above have been calculated from a limited, but large number of partial moments.

### Endnotes

[i] Even if they are not the same, the terms ‘‘risk appetite’’ and ‘‘risk aversion’’ are often used interchangeably. Note that the statement: “increasing risk appetite means declining risk aversion; decreasing risk appetite indicates increasing risk aversion” is not necessarily true.

[ii] In the following we assume that the board is non-satiated and risk-averse, and have a non-decreasing and concave utility function – U(C) – with derivatives at least of degrees five and of alternating signs – i.e. having all odd derivatives positive and all even derivatives negative. This is satisfied by most utility functions commonly used in mathematical economics including all completely monotone utility functions, as the logarithmic, exponential and power utility functions.

More generally, a decision maker can be said as being n^{th}-degree risk averse if sign (u^{n}) = (−1)^{n+1 }(Ekern,1980).