#### RISK-ADJUSTED RETURN ON RISK-ADJUSTED CAPITAL (RARORAC) (Encyclopedia)

RARORAC (Risk-adjusted Return on Risk-adjusted Capital) is an indicator measuring efficiency in value creation as a function of risk. It belongs to the category of Risk-adjusted Performance Measures (RAPMs) together with the Return on Risk Adjusted Capital (RORAC) and the Risk-Adjusted Return on Capital (RAROC), among other indicators.

The RAPM approaches, aim to improve traditional valuation measures for business unit or portfolio profitability by quantifying the risk elements associated to uncertain factors. The terminology used for the definition of this methodological framework is wide and sometimes confusing (Saita, 2007). Frequently, this is just the result of using different names for the same indicator, although occasionally the same indicator is defined and implemented in different forms.

Generally speaking, the RORAC (Matten, 2000) and the RAROC are halfway measures of the risk profile, correcting for risk only the return or the economic capital, respectively (Wolfgang and Von Wendland, 2009). The former is commonly used to evaluate projects or investments involving a high risk element relative to the capital required. The latter measures risk-based profitability comparing risky financial returns over a range of investment alternatives.

The RARORAC combines RAROC and RORAC to propose a measure accounting for the risk dimension corresponding in the returns of a business line or on the profitability of investments (typically the RARORAC numerator) and in the economic capital allocated (typically the RARORAC denominator).

In the context of risk management the RARORAC supports the process of capital allocation among different business lines - or investment alternatives … allowing the achievement of the optimal proportion of equity to assets that minimizes the cost of funding. Similarly, in the context of performance measures it allows the evaluation of each project value (and therefore the comparison of mutually exclusive investment decisions) or the performance of each business line relative to the overall risk contribution and to the target performance to be reached. Therefore, this ratio is very useful as it offers a unified tool to compare any transaction to each other, on the same basis.

The formula used for the RARORAC calculation is the following:

In the numerator:

(*r _{p}* …

*r*) is the excess return given by the difference between the portfolio return and the risk-free return;

_{f} *? _{P}* is the systematic risk;

*r _{m}* is the market return;

*I _{0}* is the initial project investment at time t = 0, e.g., the

*Utilized Capital at Risk*.

Notice also that *r _{p}= CF/I_{0}-1, *where

*CF*is the expected cash flow at time t = 1 of the specific project.

For what concerns the denominator, the *Economic Capital* or *Allocated Capital at Risk* is normally calculated using - and referred to as - the Value at Risk (VaR). VaR represents another relevant measure in the context of investment projects evaluation. It became popular soon after the Basel Committee on Banking Supervision began requiring credit institutions to apply adjustments to performance measures by considering the risk underlying the amount of capital allocated for the business activity.^{1}

Specifically, the VaR (Jorion, 2007) is represented by the maximal expected loss (tail loss or ETL) for the line of business or portfolio. In the context of a bank, for example, by representing the unanticipated losses occurring in extreme situations or market conditions, VaR is the buffer required, above the average loss, for the credit institution to remain solvent in the case of extreme losses. This absolute measure of risk is calculated relatively to i) a defined time horizon; ii) a specific confidence interval consistent with the bank credit-rating target which assures the on-going survival of the business; iii) and a probability distribution for losses, generally obtained by Monte-Carlo simulations.

By construction the RARORAC increases as the value creation raises and decreases as assumed risk goes up. The risk dimension involves considerations on market risk, credit risk, operational risk within a unique comprehensive indicator. Hence, RARORAC “rewards” those enterprises operating prudent management and risk differentiation of their business, while it “penalizes” those showing a low asset quality due to the use of the leverage effect on returns in high-risk activities, or adverse selection situations.

Traditional accounting measures such as the Return on Investment (ROI) and the Return on Equity (ROE) calculating firms’ profitability and evaluating competing investment options present the shortfall of neglecting the overall risk connected with underlying projects or business activity. However, the omission of business risks, such as social, political, regulatory, reputational, environmental, and other “intangible” risks may translate in erroneous decisions. This lack in traditional performance measures has called therefore for the implementation of alternative indicators also in the light of the recent financial crisis where many financial and non financial intermediaries actually underestimated the risk they were running, especially in complex structured products or projects.

In this respect RAPMs measures represent a step forward in overcoming this limitation. Although organizations have been slow to adapt their risk measurement systems and to incorporate these “advanced” RAPM frameworks (mostly given to the lack of data on political, social, and other “intangible” hazards) the RARORAC is today widely used, especially for the evaluation of complex portfolios or operations. Indeed, it incorporates the benefits of portfolio diversification, and can be defined therefore as an approach which is consistent with the Markowitz Portfolio Theory (1952, 1959).

As other RAPM measures, the RARORAC evaluation has been recently attracting more and more attention becoming an important indicator especially in the field of corporate finance and in the broader context of investment decisions. On the one side its relevance has grown as a result of the Basel Committee on Banking Supervision regulation, given its use of the Basel II capital adequacy guidelines. On the other hand, creditors have begun requiring a proper risk management process to ensure debt financing at a fair cost.

Other RAPM frameworks based on returns which are worth to be mentioned are the following:

i) The Sharpe Ratio (Sharpe, 1966, 1975, 1994);

ii) The Treynor Ratio (Treynor, 1965);

iii) The Jensen’s Alpha (Jensen, 1967).

The former indicator can be interpreted as the risk premium for taking one unit of overall risk. It puts in a relation of proportionality the excess return and the overall risk, represented as the portfolio volatility, ?, which expresses both systematic and unsystematic risk. For this aspect, compared to the measures which follow, the Sharpe ratio presents the advantage that undiversified portfolios can be compared. Though, given that portfolios are normally well diversified this advantage is not actually relevant:

Also, the standard deviation is claimed, sometimes, not to be a good measure of risk, given that it accounts not only for downward movements (losses), but also for upward ones (gains).

The Treynor ratio substitutes ? with the ? factor, being equal to the ratio ?_{m,p}/?_{m}^{2} (where ?_{m,p} is the covariance between market and portfolio returns, while ?_{m}^{2} is the variance of the market return). In addition to still accounting for both directions movement (downward and upward) this measure is subject to the critique that only systematic risk is accounted for, meaning that portfolios comparison is generally not allowed:

In opposition to these first two measures which are relative, the Jensen’s Alpha is an absolute performance indicator. It assumes that profits can be gained out of a situation of market disequilibrium. J^{?} can be expressed as:

Where (r_{p}-r_{f}) can be seen as the realized excess return and ?_{p}(r_{m}-rf) as the theoretical (e.g., based on a factor model) or expected one.

Summarizing, while the Sharpe Ratio and the Treynor one standardize the excess return with some measures of risk, the Jensen’s Alpha also takes into account market risk in its calculation. In this regard, the latter indicator resembles the RARORAC measure.

Overall, these three measures normally generate as outcomes dimensionless figures which make difficult the overall risk control and management (Rachev et al., 2007). Also, the Treynor and the Jensen’s Alpha have been developed based on the market line of the Capital Asset Pricing Model or CAPM (Sharpe, 1964), and for this reason are subject to the same criticisms of the CAPM.

Comparing RAPMs with other similar performance indicators, such as the Net Present Value (NPV) which is based on the Discounted Cash Flows method (DCF), it can be noticed that whereas the NPV accounts for the systematic risk of a project or investment decision, RAPM indicators include the evaluation of both systematic and unsystematic risk. However, it has been claimed that while the traditional NPV calculation ensures the maximization of the shareholder’s value, the adoption of the RARORAC indicator does not, given that it maximises the excess return “conditional on an overall risk limit”. By using the Monte Carlo simulation technique Lampenius (2012) investigates whether the criterion for selecting alternative investment opportunities is consistent between RAPM-based approach and NPV calculations. He concludes that results from the two approaches are often inconsistent, producing different project rankings. Indeed, NPV cannot be substituted with the RARORAC framework in evaluating investment decisions. Also, he states that the definition of the RAPM-based denominator (VaR) produces significant differences in terms of maximization of shareholder’s value. Specifically, the existing literature distinguishes between two ways of calculating the VaR, which is either defined as the maximum expected loss relative to the expected value of the risky position or relative to the I_{0}. The first formulation is found to systematically outperform the latter one. As a result, the RARORAC which is based on the use of the first VaR calculation should be preferred whenever the aim is that of maximising shareholder’s value.

It should be said that for credit and financial institutions the RARORAC indicator is of great use given its ability to reduce the debt cost translating it into higher shareholder’s value. In this perspective, RARORAC should be seen as an efficient risk management tool. Additionally, by using this measure rather than traditional profit and loss calculations enables to award managers oriented towards risk minimization, a precautionary behaviour which is more consistent with long term decision-making than with a short term view which normally favours risky profits.

____________________________^{1}Find more on the Basel Banking Supervision regulations at:**http://www.bis.org/bcbs/index.htm**

Bibliography

Lampenius, N. (2012). Journal of Risk N. 15/2, pp. 77-101.

Jensen, M. C. (1967). The performance of mutual funds in the period 1945-1964. The Journal of Finance, 23(2), 389-416.

Jorion, P. (2007). Financial risk manager handbook (4 ed.). Hoboken, New York: John Wiley & Sons, Inc.

Markowitz, H.M. (1952). Portfolio Selection. The Journal of Finance 7 (1): 77…91.Markowitz, H.M. (1959). Portfolio Selection: Efficient Diversification of Investments. New York: John Wiley & Sons.

Matten, C. (2000). Managing bank capital: Capital allocation and performance measurement (2 ed.). New York: John Wiley & Sons, Ltd.

Rachev, S., Prokopczuk, M., Schindlmayr, G. and Trueck, S. (2007). Quantifying Risk in the ElectricityBusiness: A RAROC-based Approach. Energy Economics, Vol. 29, No. 5, 2007.

Saita, F. (2007). Value at Risk and Bank Capital Management. Risk Adjusted Performances, Capital Management, and Capital Allocation Decision Making. Academic Press Advances Financial Series, 2007, Elsevier Inc.

Sharpe, W.F., (1964). Capital Asset Prices: a Theory of Market Equilibrium Under Conditions of Risk. Journal of Finance 19(3), 425-442;

Sharpe, W. F. (1966). Mutual fund performance. Journal of Business, 39, 119-138.

Sharpe, W. F. (1975). Adjusting for risk in portfolio performance measurement. Journal of Portfolio Management, 29-34.

Sharpe, W. F. (1994). The Sharpe ratio. Journal of Portfolio Management, 49-58.

Treynor, J. (1965). How to rate management of investment funds. Harvard Business Review, 63-75.

Wolfgang S., M. Von Wendland (2009). Pricing, Risk, and Performance Measurement in Practice: The Building Block Approach to Modeling Instruments and Portfolios. Elsevier Science Publishing Co Inc. Academic Press Inc , 2009.

Editor: Melania MICHETTI

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