# Performance measures adjusted for the risk situation (PARS)

## Christoph Peters and Roland C. Seydel

#### Need to know

• Notion of risk depends on the situation of the institution (ie, cash flows and preferences)
• We introduce and advocate PARS (Performance measures Adjusted for the Risk Situation)
• PARS shall measure the financial performance in the actual quantity of interest
• Well-chosen PARS exhibit the right risk behavior instantaneously

#### Abstract

Nowadays it is widely recognized that, to assess the performance of a fund or trading business, the risk associated with the trading strategy has to be considered as well; frequently, so-called risk-adjusted performance measures are employed. But what is risk? In this paper, we argue that risk should not always be measured in terms of the variability of the portfolio present value; rather the notion of risk generally depends on the financial situation of the institution in question, in terms of the size and time of future cashflows including (financial) consumption preferences. To this end, we introduce performance measures adjusted for the risk situation (PARS), which will measure the financial performance in the actual quantity of interest. Due to the effected risk transformation, PARS have zero volatility under the investment strategy replicating these future cashflows. We give several examples of cashflow structures for individuals and companies, showing how their PARS could be defined. In the context of a debt manager, we show the results from applying PARS to the dynamic control of bond portfolios via sensitivities.

## 1 Introduction

It is widely recognized that, to assess the performance of a fund or trading business, the risk associated with the trading strategy also has to be considered; frequently, so-called risk-adjusted performance measures (RAPM) are employed. But what is risk? By measuring risk in terms of the variability of the portfolio present value (PV), RAPM-related papers implicitly assume that risk is any deviation from holding cash in a bank account.

In this paper, we argue that in general this is not the right notion of risk; rather the notion of risk depends on the situation of the institution in question in financial terms, ie, the size and time of future cashflows, including (financial) consumption preferences. To this end, we introduce “performance measures adjusted for the risk situation” (PARS), which will measure the financial performance in the actual quantity of interest. PARS are defined as the net PV of all future cashflows divided by the PV of one unit of preferred (financial) consumption; the resulting PARS key figure measures the level of such preferred consumption.

###### Example 1.1.

Consider the example of an asset manager at time $t$ with a limited-time mandate whose assets will be withdrawn at time $T>t$ (also known as a target date fund). What the investor is interested in is not the local variation or volatility of the fund’s PV, but the volatility of the fund’s PV forward at $T$. In other words, the investor wants to know the value of their assets at time $T$, assuming the fund manager is forced to sell all assets and put them into a default-risk-free zero-coupon bond with maturity $T$. Denoting the PV of the fund’s portfolio by $V_{t}$, and the value of such a zero-coupon bond by $P_{t,T}$, the key indicator of interest to the investor is therefore

 $\frac{V_{t}}{P_{t,T}}.$ (1.1)

Therefore, given the situation of the asset manager, the risk-free benchmark strategy is to put all assets into the zero-coupon bond with value $P_{t,T}$. In an RAPM-like assessment, any performance (or PV change) and its risk should be measured against this benchmark strategy. It turns out that the variations of (1.1) in essence already provide this information and thus it is not necessary to implement the risk-free benchmark strategy as (fictitious) trades.

### 1.1 Introducing PARS

In the example above, the changed investment horizon for $T>t$ already leads to a different perception of risk (compared with just considering the PV), which can be significant for a longer investment horizon of, say, 10 years. In general, PARS $\varPi$ are defined as follows:

 $\varPi_{t}:=\frac{V_{t}(P)-V_{t}(C)}{V_{t}(E)}.$ (1.2)

First, the current value of the institution’s portfolio $V_{t}=V_{t}(P)$ is reduced by the PV $V_{t}(C)$ of inevitable cashflows $C$ (“costs”), then the net PV $V_{t}(P)-V_{t}(C)$ is divided by the PV $V_{t}(E)$ of the preferred consumption or payout structure $E$ (“equity”). The resulting key figure $\varPi_{t}$ is the multiple (or quantity) of the equity cashflow structure $E$ which can be consumed or paid out to the owners. In more general terms, PARS will cast the portfolio PV into the quantity of interest to the owner. This can be the level of consumption or the payout to owners (ie, financial consumption).

PARS relate the success of the institution’s (investment) strategy (represented by $V_{t}(P)$) to its situation (represented by $V_{t}(C)$ and $V_{t}(E)$). Considering the time series in (1.2) makes visible the risk of this strategy with respect to future cashflows of the institution: replicating these future cashflows in the portfolio will lead to zero volatility in (1.2), while any deviation from the replicating strategy will have a positive volatility.

But what precisely is the situation of the institution? Answering this and thus determining appropriate cashflow structures for $V_{t}(C)$ and $V_{t}(E)$ is the practical challenge behind the seemingly simple definition in (1.2). These cashflow structures depend on what is taken to be the strategy and can thus be influenced, and what is exogenously given or even unknown. In the case of an individual, the choice of $C$ and $E$ can be intimately connected to the individual’s consumption preferences.

We shed some theoretical and practical light on this issue by elaborating on several examples in the paper.

### 1.2 Contributions and related literature

The concept of PARS is, to the best of our knowledge, new in this general cashflow-based setup. It is, however, not new in the special case of fund performance measurement, where overperformance relative to a benchmark (frequently a total return equity index) is computed. For instance, the tracking error of index funds is nothing else but the standard deviation of the corresponding PARS increments; in the risk-adjusted performance measurement of such funds, the information ratio is considered, which is defined as the mean of the relative increments of these PARS divided by their standard deviation (see, for example, Bacon 2013). If the bank account is used as a benchmark (and instantaneous increments are used), then this information ratio specializes to the well-known Sharpe ratio as introduced in Sharpe (1966). Indeed, the importance of using the differential return to the index (in our terminology, to use PARS with a benchmark) had already been pointed out in the original paper (Sharpe 1966), and again in Sharpe (1994) (see also Dowd 2000).11 1 There are a wealth of different indexes or ratios similar in fashion to the Sharpe ratio (see, for example, Sortino and van der Meer 1991; Young 1991; Sharpe 1994).

PARS can be interpreted as the price of the portfolio given as the quantity in the numéraire (or benchmark as Platen and Heath (2006) call it), where the numéraire can be any positive, even non-self-financing quantity. This interpretation is plainly evident in the introductory example when setting $T=t+2$ days: in this case, (1.1) is the portfolio price to be paid at the value on day $t+2$ (spot price). In other examples, such an interpretation may be less obvious and, as opposed to typical pricing applications, PARS are in general not martingales under typical probability measures. Under the real-world measure that will be of interest, the portfolio PV is typically only a martingale if benchmarked to the growth-optimal portfolio; see the monograph by Platen and Heath (2006) advocating this approach. In a similar spirit, PARS can also be seen as a generalization of the utility indifference price (applied to complete markets); see, for example, Henderson and Hobson (2005) for an introduction to the corresponding theory.

In some cases, where such a price interpretation is difficult, PARS are best interpreted as the level of consumption. Here, by choosing the preferred unit consumption $E$, the distribution of consumption over time and the allocation to different goods is fixed. Optimizing PARS in a dynamic setting will therefore typically yield solutions different from the classic dynamic optimal consumption and portfolio problem, where the distribution over time is not fixed ex ante (see, for example, Merton 1971; Framstad et al 1998). Further, optimizing a utility on the (one-dimensional) PARS outcome cannot change the allocation to different goods, unlike in economic theory, where the utility of consumption of different goods is optimized given a resource constraint.

Our model of a sovereign debt manager employs a special PARS where long-term interest expenses matter, which we call perpetuity cost. We present in this paper the results of Peters (2014), where, in an innovative model setup, the sensitivities with respect to the simple dynamic term-structure model from Henseler et al (2013) are controlled, borrowing ideas from Kraft (2003); see also the online appendix to this paper (Peters and Seydel 2018).

### 1.3 Background and applications

The concept of PARS arises naturally in the context of a (sovereign) debt manager (such as the German Finance Agency), where the situation is governed by long-term financing needs and therefore the PV is not an appropriate performance key figure.

Asset and liability management (ALM) is another important part of the background of our paper, which deals with managing risk and return of assets in the presence of liabilities; for a general overview of this topic, see, for example, Mitra and Schwaiger (2011) or Corlosquet-Habart et al (2015); and for an ALM perspective on performance measurement in the life insurance industry, see Braun and Schreiber (2019). Recent developments in finance regulation (such as IFRS fair value accounting, or Solvency II for insurance businesses) have further enforced the use of PV; yet using this key figure directly is frequently not in line with the traditional way of managing long-term liabilities.22 2 Another recent development confronting the traditional way of management with PV-based (economic value) key figures is the interest rate risk in the banking book (IRRBB) regulation (Basel Committee on Banking Supervision 2016), which requires banks to compute economic value risks as well as earnings risks on loans and deposits. On the impact of Solvency II for insurance businesses, there is a recent strand of literature (see, for example, Fischer and Schlütter 2015; Braun et al 2018; Kouwenberg 2018). PARS could provide a means to reconcile this traditional way of management with the PV key figure.

A further potential area of application of PARS is individual wealth management; see Rohner and Uhl (2017) for a holistic overview of this subject. For individuals, there are a wide variety of different situations and consumption preferences, ranging from ad hoc financing needs to long-term retirement planning. It is common wisdom that holistic individual wealth management should not just offer good risk-adjusted management, but also take into account the investor’s individual risk situation and goals, which is incorporated into PARS by design.

### 1.4 PARS: a new risk paradigm

The transformation of the PV in the PARS definition (1.2) also effects a risk transformation in the key figure. As PARS will measure the actual quantity of interest, their variability also exhibits the relevant risk. Therefore, it is our firm belief that risk should always be computed on appropriately defined PARS, where aversion to the relevant risk can be captured by applying a standard utility function, for instance.

As PARS already include the risk situation, the relevant risk can be computed in an instantaneous evaluation of PARS (eg, using a PARS sensitivity together with a covariance matrix), as detailed in our paper. This implies that a simulation of future values is often not necessary, and hence bespoke market evolution models can be replaced by market-standard valuation models.

### 1.5 Overview of the paper

Section 2 introduces PARS and their theory. Section 3 discusses how a risk situation can be incorporated into suitable PARS, applying the concept to a broad range of different situations. If you are more interested in applications, you are encouraged to skip the precise definition in Section 2 and directly jump to the examples in Section 3. Section 4 carries out an extended analysis of the risk situation and PARS of a sovereign debt manager, including the illustrative results of a dynamic control of the PARS quantity. We draw some conclusions in Section 5. Our paper is complemented by the online appendix (Peters and Seydel 2018), which generalizes the PARS theory (for example, to the case of incomplete markets), and contains further details on the dynamic control of the sovereign debt manager.

## 2 Performance measures adjusted for the risk situation

### 2.1 Definition

#### 2.1.1 Notation and PARS

Given a complete filtered probability space $(\varOmega,\mathcal{F},(\mathcal{F}_{t})_{t\in\mathcal{T}},\mathbb{P})$, we assume a discrete time set $\mathcal{T}\subset\mathbb{R}$ (we use a discrete-time formulation for simplicity – see Appendix A online for a continuous-time formulation).33 3 For an introduction to stochastic processes, see, for example, Jacod and Protter (2003) or Øksendal (2000). Let $\mathcal{C}$ be the space of all real-valued adapted stochastic processes on $\mathcal{T}$. We also call $\mathcal{C}$ the cashflow space, as for $C\in\mathcal{C}$ and $t\in\mathcal{T}$ the value $C_{t}$ will have the meaning of a cashflow at time $t$ (measured in a chosen, fixed currency). Let $V\colon\mathcal{C}_{0}\to\mathcal{C}$ be the valuation function such that $V_{t}(C)$ is the adapted PV at time $t\in\mathcal{T}$ of the cashflow structure $(C_{s})_{s>t}$; we require $V_{t}$ to be additive, ie, $V_{t}(A+xB)=V_{t}(A)+xV_{t}(B)$ for any $A,B\in\mathcal{C}_{0}$ and $x\in\mathbb{R}$. The subspace $\mathcal{C}_{0}\leq\mathcal{C}$ is chosen such that the valuation function $V$ is well-defined (typically subject to integrability conditions).

Let the cost cashflows be denoted by $C\in\mathcal{C}_{0}$, let the equity (unit) cashflows be denoted by $E\in\mathcal{C}_{0}$ and let the portfolio (endogenous or internal) cashflows implied by the holdings at $t$ (after rebalancing at $t$) be denoted by $P^{t}\in\mathcal{C}_{0}$. We assume that $V_{t}(E)>0$ for all $t$ up to some terminal stopping time $T\in\mathcal{T}$, and we can then define PARS $\varPi_{t}$ at time $t by

 $\varPi_{t}:=\varPi_{t}(P^{t}):=\frac{V_{t}(P^{t})-V_{t}(C)}{V_{t}(E)}.$ (2.1)

Another (standard) notation we frequently use is the price $P(t,T)$ (or $P_{t,T}$) of a nondefaultable zero-coupon (or discount) bond at time $t$ with a cashflow of 1 at $T$. Under the standard risk-neutral valuation,

 $P(t,T)=\mathbb{E}^{\mathbb{Q}}\bigg{[}\exp\bigg{(}{-}\int_{t}^{T}r(s)\mathrm{d% }s\bigg{)}\biggm{|}\mathcal{F}_{t}\bigg{]}=\mathbb{E}^{\mathbb{Q}}\bigg{[}% \frac{B(t)}{B(T)}\biggm{|}\mathcal{F}_{t}\bigg{]},$

for the risk-neutral measure $\mathbb{Q}$ associated with the bank account $B$, for the filtration $(\mathcal{F}_{t})_{t\in\mathcal{T}}$ and for the short rate $r(t)$ at time $t$. For more background on risk-neutral valuation and interest rates, see Brigo and Mercurio (2006) or Zagst (2002).

#### 2.1.2 Interpretation

PARS relate the success of the institution’s (investment) strategy44 4 We always use the general term “institution” in the paper, which may refer to any entity with an investment strategy, ie, an individual, a company, a fund, etc. (represented by $V_{t}(P^{t})$) to its risk situation (represented by the two types of future cashflows, $C\in\mathcal{C}_{0}$ and $E\in\mathcal{C}_{0}$). $C$ represents the cashflows of the first, inevitable type, which are not directly dependent on the PV time series; they are often interpreted as costs or fees. Cashflows of the second type are assumed to scale with the (net) PV; they are often interpreted as payments to owners or stock holders (if any), or as excess consumption. One unit of such “equity” cashflows is denoted by $E$.

The definition of PARS (2.1) can be interpreted as follows: first the current value of the institution’s portfolio $V_{t}=V_{t}(P^{t})$ is reduced by the PV of costs $V_{t}(C)$; then the net PV $V_{t}(P^{t})-V_{t}(C)$ is distributed among the owners by dividing by $V_{t}(E)$. The resulting key figure is the multiple (or quantity) of the equity cashflow structure that can be consumed or paid out to the owners, after deducting the costs.

This amounts to selling the portfolio as a whole, replacing it by a combination of $C$ and $E$ with the same PV,

 $V_{t}(P^{t})\lx@stackrel{{\scriptstyle!}}{{=}}V_{t}(C)+xV_{t}(E)=V_{t}(C+xE).$ (2.2)

The solution $x=x(t)\in\mathbb{R}$ of (2.2) is equal to $\varPi_{t}$. We also call the portfolio $C+\varPi_{t}E$ in the second equality of (2.2) the benchmark strategy (at time $t$); we say that the benchmark strategy is implemented at $t$ if $P^{t}$ is replaced by $C+\varPi_{t}E$ (assuming no transaction costs) and there is no further portfolio rebalancing.

Choosing the benchmark strategy fixes the future (external) cashflows due to $E$ of the portfolio; as $\varPi_{t}$ measures the quantity of those future cashflows, it is then constant.

###### Proposition 2.1.

Assume there are no transaction costs. Then $\varPi_{t+\Delta}=\varPi_{t}$ for all $\Delta>0$ with $t+\Delta if the benchmark strategy is implemented in $t$, ie, the benchmark strategy is risk-free in $\varPi$.

###### Proof.

In the absence of transaction costs, the benchmark strategy $C+\varPi_{t}E$ can result from rebalancing. Now insert

 $V_{t+\Delta}(P^{t+\Delta})=V_{t+\Delta}(C+\varPi_{t}E)=V_{t+\Delta}(C)+\varPi_% {t}V_{t+\Delta}(E)$

into the PARS definition (2.1) for $t+\Delta$. ∎

###### Remark 2.2.

From our way of defining the portfolio PV, it follows that we need to rebalance the portfolio to make it self-financing (ie, $P^{t_{1}}\not=P^{t_{2}}$ for some $t_{1}\not=t_{2}\in\mathcal{T}$). In the absence of such rebalancing (such as for the risk-free portfolio $C+\varPi_{t_{0}}E$), by definition there will be external cashflows as implied by $P^{t_{0}}$. More on external cashflows and a self-financing version of Proposition 2.1 can be found in can be found in the online appendix to this paper (Peters and Seydel 2018).

#### 2.1.3 Risk versus risk-free benchmark strategy

Proposition 2.1 suggests measuring risk in terms of the standard deviation of PARS, for example, of a historical time series of PARS increments. Instead of computing risk on PARS, we can obtain practically the same result by considering the increments (eg, day by day) of the PV compared with the benchmark strategy $C+\varPi_{t}E$: to simplify notation, we use the time-difference operator $\Delta_{t}(f):=f_{t+\Delta}-f_{t}$ for a function $f\colon\mathbb{R}\to\mathbb{R}$ and some $\Delta>0$. Assume further for the moment that there are no transaction costs, then

 $\displaystyle\Delta_{t}(V_{\cdot}(P^{t})-(V_{\cdot}(C)+\varPi_{t}V_{\cdot}(E)))$ $\displaystyle\qquad\qquad=V_{t+\Delta}(P^{t})-V_{t}(P^{t})-(V_{t+\Delta}(C)-V_% {t}(C))$ $\displaystyle\qquad\qquad-\frac{V_{t}(P^{t})-V_{t}(C)}{V_{t}(E)}(V_{t+\Delta}(% E)-V_{t}(E))$ $\displaystyle\qquad\qquad=V_{t+\Delta}(P^{t})-V_{t+\Delta}(C)-\frac{V_{t}(P^{t% })-V_{t}(C)}{V_{t}(E)}V_{t+\Delta}(E)$ $\displaystyle\qquad\qquad=V_{t+\Delta}(E)\bigg{(}\frac{V_{t+\Delta}(P^{t})-V_{% t+\Delta}(C)}{V_{t+\Delta}(E)}-\frac{V_{t}(P^{t})-V_{t}(C)}{V_{t}(E)}\bigg{)}$ $\displaystyle\qquad\qquad=V_{t+\Delta}(E)\Delta_{t}\bigg{(}\frac{V_{\cdot}(P^{% t})-V_{\cdot}(C)}{V_{\cdot}(E)}\bigg{)}=V_{t+\Delta}(E)\Delta_{t}(\varPi_{% \cdot}),$ (2.3)

where in the last step we have used the portfolio rebalancing property $V_{t+\Delta}(P^{t})=V_{t+\Delta}(P^{t+\Delta})$ (under absence of transaction costs).

Thus, considering the increments of PARS (2.1) amounts to considering the increments of the portfolio PV compared with the freshly set up benchmark strategy from (2.2), normalized by $V_{t+\Delta}(E)$. The risk (standard deviation) of the increments (2.3) vanishes if the benchmark strategy is followed; therefore, the term risk-free benchmark strategy is justified once more. We suggest that the risk in an RAPM-like assessment should be measured against this risk-free benchmark strategy, normalized by $V_{t+\Delta}(E)$, ie, the risk should be computed on the time series of PARS (2.1).

###### Example 2.3.

Consider Example 1.1. Here, the increments due to risk taking are

 $\displaystyle\Delta_{t}\bigg{(}V_{\cdot}-\frac{V_{t}}{P_{t,T}}P_{\cdot,T}\bigg% {)}=P_{t+\Delta,T}\Delta_{t}\bigg{(}\frac{V_{\cdot}}{P_{\cdot,T}}\bigg{)}.$ (2.4)

(Recall that $P_{t,T}$ denotes here the price of a zero-coupon bond with maturity $T$, which is not to be confused with the portfolio endogenous cashflows $P^{t}\in\mathcal{C}_{0}$.) Choosing one step $\Delta=T-t$ in (2.4) yields the difference $V_{T}-\varPi_{t}$ directly, but in this case it is not possible to measure the volatility in (2.4) on a historical (realized) time series. For $\Delta\ll T-t$, measuring historical volatility in (2.4) is possible, and it will yield similar results to measuring it on the increments of (2.1): the deltas versus the benchmark strategy (2.4), normalized by $P_{t,T}$ (to get the same order of magnitude), differ from the deltas of the key indicator (2.1) only in the factor $P_{t+\Delta,T}/P_{t,T}$, covering the carry cost of implementing the risk-free benchmark strategy.

#### 2.1.4 Risk preferences and utility functions

PARS account for the risk situation of the institution, casting the portfolio’s PV into the quantity of interest (eg, the quantity which can be paid out to owners in the preferred equity cashflow structure). Investment decisions should be based on this quantity of interest and the owners’ risk preferences. These risk preferences can be incorporated into the decision process in the standard way, for example, by weighing PARS expectation and standard deviation (or any other risk measure) of different alternatives, or by considering the average utility of PARS. Using a concave increasing utility function on PARS allows us to capture aversion to the relevant risk; PARS and utility function concatenated code a utility which can be stochastic in the portfolio PV.

###### Remark 2.4 (PARS and change of numéraire).

Assume that $V_{t}(E)$ is self-financing on the interval $[0,T]$ for $T>0$ bounded. Then $V_{t}(E)$ can serve as numéraire (with the corresponding risk-neutral measure $\mathbb{P}^{E}$). Hence, we can apply the change-of-numéraire technique (see, for example, Brigo and Mercurio 2006) for any $t\in[0,T]$ to obtain

 $\displaystyle\mathbb{E}^{E}\bigg{[}\frac{V_{t}(E)}{V_{T}(E)}V_{T}(P^{T})\biggm% {|}\mathcal{F}_{t}\bigg{]}$ $\displaystyle=\mathbb{E}^{T}\bigg{[}\frac{P(t,T)}{P(T,T)}V_{T}(P^{T})\biggm{|}% \mathcal{F}_{t}\bigg{]}$ $\displaystyle=\mathbb{E}^{T}[P(t,T)V_{T}(P^{T})\mid\mathcal{F}_{t}].$ (2.5)

While the left-hand side of (2.5) represents the expectation of a (scaled) future PARS (under $\mathbb{P}^{E}$), the right-hand side of (2.5) represents the expectation of the (scaled) future PV (under the $T$-terminal measure $\mathbb{P}^{T}$). However, as soon as we introduce a strictly concave utility function under the expectation in both sides of (2.5), the equality does not hold any more, implying that changing the numéraire actually matters in terms of risk. In the same way, a risk transformation is effected if we switch from the PV to the PARS key figure under the – unchanged – physical probability measure.

### 2.2 PARS and simulation

PARS incorporate the risk situation of the institution (represented by future cashflows) in today’s performance key figure. Hence, it is not necessary to simulate future actions and/or market evolutions; instead the forward-looking part is implicitly incorporated into the PV calculations.

We illustrate this in the following by considering the risk horizon: the risk horizon is the time after which the portfolio’s PV is of interest to the owner or investor, eg, the time after which the investor might withdraw their money. Given the risk horizon $T>t$, everything will be withdrawn at time $T$; therefore, $E=1_{T}$ and $\varPi_{t}=V_{t}(P^{t})/P(t,T)$ (Example 2.3).

Without PARS, to get an idea about the relevant key properties of the portfolio current at time $t$, we would have to simulate market and portfolio evolutions until $T$. This would mean in turn that we would need a market model until $T$, and we would have to devise a strategy to follow until $T$, although any choice for the latter is hypothetical.

PARS make this burdensome decision for you: the (benchmark) strategy implicitly chosen in PARS is to reduce the risk (with respect to the horizon) to zero. Because the risk is reduced to zero, the implicit portfolio value at $T$ is already known at time $t$ and is given by $\varPi_{t}$ (here equal to the forward). Actually, for the “expectation” part of the PARS performance, no market model at all is needed, and only market-standard (objective) valuation for the zero-coupon bond is needed. Even for computing PARS portfolio risk, an instantaneous market model (eg, an instantaneous covariance, combined with a sensitivity) is sufficient.

To summarize, PARS allow us to use an instantaneous evaluation, making a simulation with a positive simulation horizon redundant. This allows us to concentrate on the current portfolio properties instead of specifying arbitrary market and/or portfolio evolutions. Also the use of market-standard valuation formulas improves transparency.

Now consider the slightly more general risk horizon interval $[T_{1},T_{2}]$ for $t: money will be withdrawn constantly from time $T_{1}$ to $T_{2}$; therefore, $E=1_{[T_{1},T_{2}]}$ and

 $\varPi_{t}=V_{t}(P^{t})\bigg{(}\int_{T_{1}}^{T_{2}}P(t,s)\mathrm{d}s\bigg{)}^{% \!-1}.$

Without PARS, we would have to simulate until $T_{2}$, but any simulation result from $T_{1}$ onward could be of interest – collapsing this high-dimensional result space is circumvented by using PARS.

Of course, there may be cases where a time-dependent market model is employed and therefore a true simulation of future market and portfolio outcomes is desired; in such cases the full simulation is not made redundant by PARS (although they can still be helpful in that they display the performance in the actual quantity of interest).

## 3 Risk situations and performance measures adjusted for the risk situation

This section discusses how to condense the risk situation into the cashflow processes $C$ and $E$ and how appropriate PARS can be defined. To shed some light on this issue, we elaborate on several instructive examples in the following.

### 3.1 Determining the risk situation

We have seen in the previous sections how PARS, representing a holistic performance measure, can be computed, given the risk situation of the institution.

But what precisely is the situation of the institution? Answering this and thus determining appropriate cashflow structures for $C,E\in\mathcal{C}_{0}$ is the practical challenge behind the seemingly simple definition (2.1). We address this question by considering a number of practical examples as listed below.

What is appropriate depends on what is taken to be the strategy and can thus be influenced, and what is exogenously given or even unknown. Anything which cannot be controlled by the strategy goes into $C$ and $E$. Of the two, $C$ denotes the inevitable cashflows, which are not directly dependent on the strategy’s success. (Note that any cashflow in $C$ could be included in $P^{t}$ instead, the choice being irrelevant due to the additive pricing rule $V_{t}$.) By contrast, $E$ denotes the cashflow structure in which the level is of interest to the institution (respectively, to its owners); see the schematic view in Figure 1. This means the (consumption) preferences of the institution typically show up in $E$, often expressed as very simple cashflow structures.

##### Financial versus nonfinancial use.

Here, preferred consumption can mean financial consumption (configured mainly by the preferred times of payout), or consumption of real goods including nonmaterial ones (anything which has a use in itself and has costs or can be given a price). Whether the former or the latter is the appropriate modeling choice depends on the use case in question.

##### Nonfinancial use.

An in-depth analysis of real consumption preferences is carried out, where the consumption is in real or nonmaterial goods. The resulting cashflow structures $C$ and $E$ typically include inflation, and their PVs are then used to define PARS. This approach is the most fundamental one and is particularly useful for individuals, as shown in Section 3.3.1.

##### Financial use.

$E$ does not represent a consumption but a payout, configured by the preferred time and currency distribution of cashflows. This is a natural approach for companies or financial intermediaries where it is barely possible to carry out an analysis of underlying (real) consumption preferences, as typically the ultimate owners or investors are not known. Several examples for financial use can be found in Section 3.3.2 and Section 3.3.3; an extended example of financial use for a sovereign debt manager is presented in Section 4.

##### Shortcut PARS definition.

A special case of financial use is the shortcut definition of PARS (see also Figure 1). It makes use of the one-to-one correspondence of “What is risk-free?” and the consumption preferences of the institution: while the standard way of deriving $\varPi$ is to first specify $C$ and $E$, and then to deduce by Proposition 2.1 what risk-free is, in some cases it may be more natural to follow the reverse path. If the risk-free instrument is already known, then its PV can directly be used as $V_{t}(E)$ in (2.1), making a detailed analysis of preferences and corresponding cashflows redundant. This approach is demonstrated in Section 3.2 for a mutual fund benchmarked to an equity index.

### 3.2 Mutual fund benchmarking and index cashflow structures

Now let us consider a special case of PARS, where

 $\varPi_{t}=\frac{V_{t}(P^{t})}{I_{t}},$ (3.1)

with respect to a (self-financing) index $I$.55 5 Formally, (3.1) can be considered as a performance measure adjusted for the risk situation, as under risk-neutral valuation (with standard assumptions), by a change of numéraire $\smash{V_{t}(1_{\tau}I_{\tau})=\mathbb{E}^{\mathbb{Q}}[I_{\tau}\exp(-\int_{t}^% {\tau}r(s)\mathrm{d}s)]=\mathbb{E}\mathrm{{}^{I}}[I_{\tau}I_{t}/I_{\tau}]=I_{t}}$ for any bounded stopping time $\tau\geq t$. The (nonannualized) return of PARS (3.1) (assuming no inflows/outflows) from $t=0$ to $t=T$ is equal to the difference of returns:

 $\log\bigg{(}\frac{\varPi_{T}}{\varPi_{0}}\bigg{)}=\log\bigg{(}\frac{V_{T}(P^{T% })}{V_{0}(P^{0})}\bigg{)}-\log\bigg{(}\frac{I_{T}}{I_{0}}\bigg{)}.$ (3.2)

The prototypical example of PARS (3.1) is performance measurement of mutual funds benchmarked to a total return equity index $I$. Here, the overperformance (3.2) compared with the benchmark is an important key figure of the fund. Yet often the risk of such a benchmarked fund is computed not on PARS, but directly on the fund’s price or PV time series.

We strongly believe that PARS (3.1) should be used consistently for return and risk measurement, as the excess return generated by deviating from the benchmark should be held against the risk incurred by deviating from the benchmark. Such an approach is already followed in the analysis of the tracking error of index funds (see, for example, Bacon 2013). Here, the information ratio key figure is merely the return of (3.1) divided by the risk of this return.

If the bank account is used as benchmark $I$ (and instantaneous increments are used), then this information ratio specializes to the well-known Sharpe ratio as introduced in Sharpe (1966) (under the name of reward-to-variability ratio). In the Sharpe ratio, the return minus the risk-free rate (the differential return) is divided by the standard deviation of the rate of return; this standard deviation is equal to the standard deviation of the differential return only if instantaneous increments are used in the measurement (as the bank account is only locally deterministic; see also Section 3.3.2).

In a later paper, Sharpe (1994) emphasizes the importance of using the differential return to the index (in our terminology, to use PARS with a benchmark), but this time switching (in the denominator of the Sharpe ratio) to the standard deviation of the differential return.

PARS (3.1) for mutual funds is a good example for the shortcut definition of PARS as described in Section 3.1. It is probably difficult to derive (3.1) from real consumption preferences, as the equity index itself cannot be consumed; the same holds for the bank account as index $I$. The cashflow-based derivation of (3.1) is more convincing for commodity or real property indexes as they represent the price of real needs.

Nevertheless, a more fundamental analysis of investor consumption preferences could be beneficial; examples for this will be discussed in Section 3.3.

###### Remark 3.1.

Performance benchmarking against a (self-financing) index can also result from a real cashflow structure using so-called anticipated PARS (aPARS) as introduced in the online appendix to this paper (Peters and Seydel 2018). Consider for example a stock whose current price can also be seen as the PV of a stream of future dividends. If $E$ is set as the stream of future dividends, then reinvesting the dividends in the aPARS framework leads to a self-financing stock as the denominator in the ratio $\varPi$, or (3.1).

### 3.3 Further examples

Our aim in the following is to show the full breadth of potential applications, by presenting instructive examples of PARS in diverse use cases. The presented exemplary PARS choices are certainly not the only possible ones and may be regarded as a starting point for a more detailed analysis of the respective risk situations.

#### 3.3.1 Personal finance

##### Real property investor.

Consider an individual who wants to buy real property (eg, a family home), although the precise time for buying is not yet clear. Therefore, the consumption cashflow structure is $E=1_{\tau}I_{\tau}$ for a suitable bounded stopping time $\tau$ and the real property index $I$, and the quantity of interest to this individual is the multiple of the current real property index,

 $\varPi_{t}=\frac{V_{t}(P^{t})}{V_{t}(E)}=\frac{V_{t}(P^{t})}{I_{t}},$

where the second equality holds as in Section 3.2.

##### Individual saving for retirement.

Consider an individual saving for their own retirement. The main uncertainties or risks they face are unemployment, illness, death and inflation. The individual earns a constant (non-inflation-adjusted) yearly salary $s$ until their (stopping) time of retirement $\tau_{\mathrm{R}}$ (or time of final unemployment). To fulfill their basic needs, they need to spend $cI_{t}$ per year until their (stopping) time of death $\tau_{\mathrm{D}}$, where $I_{t}$ is the inflation index of their country at time $t$. For simplicity, we assume that any expenses due to illness are covered by a health insurance whose premium is contained in $cI_{t}$. The individual wants to spend any excess money on luxury consumption, ideally distributed evenly over the rest of their life.66 6 Note that this wish for uniform distribution of consumption will not come true in reality unless the individual pursues a risk-free strategy. The PV of costs is

 $V_{t}(C)=\mathbb{E}^{\mathbb{Q}}\bigg{[}\sum_{r\colon t

for the bank account $B$, while the value of one unit of equity cashflows is

 $V_{t}(E)=\mathbb{E}^{\mathbb{Q}}\bigg{[}\sum_{r\colon t

#### 3.3.2 Asset management

##### Asset manager with limited-time mandate.

We take up once more the example from Section 1 (or Example 2.3) with $E=1_{T}$ and $C\equiv 0$ (also known as a target date fund). We use this example to show that in general it is a misconception to take the bank account to be risk-free: indeed, for a positive risk horizon $T>t_{0}+1$ day, the zero-coupon bond maturing at $T$ (and not the bank account) is free of risk. The (present-value) risk over the time of investing in the (instantaneously risk-free) bank account is illustrated in Figure 2 for $T=t_{0}+10$ years: after approximately six years, the PV of the bank account turns out to be more risky than that of a zero-coupon bond with initial maturity of 10 years.

We note that Figure 2, in terms of PARS risk instead of present-value risk, would show a flat line at 0 for the 10Y zero-coupon bond.

##### Asset manager with regular review dates.

Consider an asset manager whose performance is evaluated at regular review dates $(T_{i})_{i\geq 1}>t_{0}\in\mathcal{T}$, but without any external cashflow. As the value to the owner is simply achieved by looking at the quantity of interest, we are in a so-called aPARS (anticipated PARS) setting as introduced in the online appendix to this paper (Peters and Seydel 2018). Iterating the example above (or Example 2.3), with the next review date always being the only relevant one, the aPARS are

 $\varPi^{\mathrm{a}}_{t}:=\frac{V_{t}(P^{t})}{P(t,T_{j(t)})\prod_{i=2}^{j(t)}(1% /P(T_{i-1},T_{i}))},$

where $j(t)=\min\{i\colon T_{i}>t\}$ is the index of the next $T_{i}$.

##### Mutual fund.

A mutual fund has capital inflows and capital outflows, yet often a limited lifetime. We model this by introducing the stopping time of “fund death” $\tau_{\mathrm{D}}$, where the remaining assets will be liquidated and paid out; further small-size capital inflows and outflows take place up until $\tau_{\mathrm{D}}$. We model capital inflows as part of $C$ (in the numerator of the PARS definition), as they are not directly dependent on the fund’s PV. Let $P\mathrm{{}^{I}}$ and $P\mathrm{{}^{O}}$ be increasing compound Poisson processes on $\mathcal{T}$ (ie, processes with random jump times and positive jump distributions). Then using the jumps of these processes, $\Delta P\mathrm{{}^{I}}_{t}$ and $\Delta P\mathrm{{}^{O}}_{t}$, we define

 $C_{t}:=-\Delta P\mathrm{{}^{I}}_{t}1_{t<\tau_{\mathrm{D}}}\quad\text{and}\quad E% _{t}:=\Delta P\mathrm{{}^{O}}_{t}1_{t<\tau_{\mathrm{D}}}+1_{t=\tau_{\mathrm{D}}}$

to be used in the PARS definition. Both $P\mathrm{{}^{I}}$ and $P\mathrm{{}^{O}}$ should be calibrated to yield cashflows $C_{t}$ and $E_{t}$ of a small size compared with the final liquidation cash outflow. In the stochastic processes of fund inflows and outflows as well as in the definition of the time of fund death $\tau_{\mathrm{D}}$, market observables such as stock market performance could of course enter.

##### From mutual funds to individual wealth management.

In the previous example, we modeled the asset manager’s cashflows in a reduced form, simply incorporating the observed timing patterns in $C$ and $E$. The ultimate objective of an asset manager however should be to pursue a more structural approach and align the strategy with the investor’s risk situation, in particular with their consumption plans. This is by definition perfectly possible in individual wealth management, where the systematic application of PARS could bring substantial benefit.

For mutual funds, investors’ consumption plans depend on the current investor basis and therefore are difficult to predict. However, if funds publish the PARS they use for risk management, then an investor can at least choose the fund that best matches their individual risk situation.

##### Pension fund.

The risk situation of a pension fund is characterized by long-term liabilities (created by the pensions to be paid), regular cash inflows (by current contributors) and of course the need to invest to achieve a satisfactory return.77 7 For more background on the pension fund risk situation, see, for example, Kouwenberg (2001), Bogentoft et al (2001), Braun et al (2011) and Chen and Delong (2015).

A pension fund with a defined-contribution scheme could be modeled with the following PARS: let $C$ contain the future contributions and fixed pension payments, and choose the preferred payout structure $E:=1_{(0,\infty)}$, reflecting the long-term nature of pensions. An uneven distribution of pension payments over time could additionally be incorporated into the preferred payout structure $E$ by making $E$ time-dependent.

If the pension fund has a defined-benefit scheme, the PARS modeling would get more involved, as the contribution amount can also be changed in this case, and hence it becomes part of the strategy; we leave this for future research.

##### Money market fund.

A money market fund is a mutual fund investing in short-term securities, with the (published) objective of minimal price volatility. Here, liquidity aspects clearly dominate the risk situation, and so the PV is an appropriate performance measure. The PV (which is also one of the PARS) can either be derived from the choice of the bank account as risk-free (shortcut definition of PARS, see Section 3.1) or it can follow as a limiting case from the mutual fund example by setting the stopping time of “fund death” $\tau_{\mathrm{D}}$ appropriately.

#### 3.3.3 Corporate finance

##### Publicly listed company with long-term investors.

Consider a publicly listed company whose investors are interested in long-term dividend payments. In this case we set $E:=1_{(0,\infty)}$, and let $C$ be the cashflows from the company’s core business. $P^{t}$ is the financial portfolio of the company that will be managed in an optimal way; here typically the financial portfolio will have a negative PV (debt portfolio).

###### Remark 3.2.

The PARS $\varPi_{t}$ theoretically could become negative (if $V_{t}(P^{t})), although owners of corporates in general do not have to inject capital (they own a call option on the asset value of the company). The negative PARS can be mostly avoided by modeling the impact of own default in $C$ or $P^{t}$ (eg, by reducing debt repayments); if in doubt, additionally the positive part of PARS can be considered.

##### Bank.

As in the previous example, for a typical bank, $C$ would be the cashflows from the bank’s core business of lending (eg, mortgage loans), while the cashflow structure of interest could again be $E:=1_{(0,\infty)}$. The financial portfolio $P^{t}$ consists of the remaining activities, including bank deposits, liquidity reserves and derivatives. It is the task of the bank’s asset–liability management unit to structure $P^{t}$ in an optimal way, taking into account the risk appetite of the bank (or its investors); this may also include hedging parts of the loan portfolio using derivatives.

For a nonlife insurance company, the core business is often modeled using a compensated compound Poisson process where the insurance claims represent the jumps and the insurance premiums represent the drift; let $C$ represent the increments of such a process. Choosing again the long-term financial consumption $E:=1_{(0,\infty)}$, we obtain the quantity of interest to the owners of the insurance company.

The situation is more complex for a life insurance company, as the customers (policyholders) frequently participate in portfolio management success, while being insured against downside risks (cliquet-style capital guarantee).88 8 Such policyholder participation is actually a regulatory rule in countries such as Switzerland and Germany (see, for example, Albrecher et al 2017). Several papers have shown that those guarantees exhibit a substantial influence on the asset allocation of a life insurer (see, for example, Kling et al 2007; Braun et al 2019; Ruß and Schelling 2017). We assume for simplicity that the insurance customer and the insurance company have the same preferred cashflow structure $E$. Further, let the cashflows as currently guaranteed to the customer be contained in $C$. If $d\in(0,1)$ is the ratio of the portfolio management success due to the customer, then the quantity of interest to the owners of the insurance company is

 $\varPi_{t}:=(1-d)\frac{V_{t}(P^{t})-V_{t}(C)}{V_{t}(E)}.$ (3.3)

This standard form of PARS (the multiplier $1-d$ could also be included in $E$) can be ensured to be nonnegative as explained in Remark 3.2. Note, however, that the representation (3.3) does not take into account guarantees that might be issued in the future.

Braun and Schreiber (2019) introduce the concept of the asset–liability Sharpe ratio for a life insurance company, which is a Sharpe ratio on the return on equity of the insurance company. In PARS notation, their approach amounts to taking the institution’s liabilities (represented approximately by their first-order sensitivity to a parallel shift of the yield curve) as $C$, and taking the bank account as denominator $V_{t}(E)$ (see Section 3.2); the asset–liability Sharpe ratio is then given as the Sharpe ratio on these PARS.

## 4 Case study: performance measures adjusted for the risk situation for a sovereign debt manager

We present the following case study on how PARS could be used in the situation of a sovereign debt manager, such as the German Finance Agency.

### 4.1 Risk situation and PARS of sovereign debt manager

Which cashflows constitute the risk situation of a central government’s debt management, ie, which values should $V_{t}(E)$ and $V_{t}(C)$ in (2.1) take? First of all, a country has no fixed life span, and it is most likely that the government will always be a net debtor. In any case, it is safe to assume that the government cannot and will not reduce its net debt by a significant proportion in a short period of time – rather, repaying debt will be a slow and tedious process driven by relatively stable tax earnings. Yet we will assume that the government always pays back its debt in the long run.

All in all, the components of a realistic government cashflow structure could have the following components:

• fixed, known cashflows for the first few years, where tax earnings and expenses can be relatively well predicted $\leadsto V_{t}(C)$; and

• long-term constant cashflows thereafter, maybe GDP-linked (just like tax earnings) $\leadsto V_{t}(E)$.

For simplicity, in the following we will constrain ourselves to nominally constant cashflows (in euros, say) starting right away. This also has the advantage that it is closer to typical government debt instruments.

##### Quantity of interest: long-term cashflows.

Set $C:\equiv 0$ and $E:=1_{(0,\infty)}$ (a perpetuity, ie, an instrument with a continuous cashflow stream of 1 until $\infty$), to get for $t>0$ the PARS

 $\displaystyle\varPi(t)=\frac{V_{t}(P^{t})-V_{t}(C)}{V_{t}(E)}=\frac{V_{t}(P^{t% })}{p^{\infty}(t)},$ (4.1)

with the value of the perpetuity

 $V_{t}(E)=p^{\infty}(t):=\int_{0}^{\infty}P(t,t+s)\mathrm{d}s,$

the integrated discount factors (provided the integral exists). As the value $\varPi(t)$ can be interpreted as the coupon of the current portfolio when refinanced by a perpetuity, we also term (4.1) the perpetuity cost.

###### Example 4.1.

If we assume a constant continuously compounded rate of $r\equiv 2\%$, then the value of the perpetuity is

 $p^{\infty}(t)=\int_{0}^{\infty}\exp(-rs)\mathrm{d}s=\frac{1}{r}=50.$

Inspired by Example 4.1, we can interpret (4.1) also as the PV multiplied by the perpetuity rate, where the perpetuity rate is defined by $1/p^{\infty}(t)$.

It can be seen from (4.1) that in the given situation of the sovereign debt manager, the risk-free benchmark strategy is to refinance the complete debt by a perpetuity, thereby fixing debt servicing costs up to $\infty$. By contrast, choosing GDP-linked cashflows instead of constant nominal ones in $E$ would make GDP-linked perpetuities risk-free in $\varPi$ (and would make inflation-linked securities less risky due to the positive correlation of inflation and GDP).

### 4.2 Optimal solution

An optimal stochastic control problem can be set up and solved to maximize the expected utility of terminal PARS $\varPi(T)$ for some $T>t$, as detailed the online appendix of this paper (Peters and Seydel 2018). In the following, we show the structure of the (approximate) optimal solution in a risk–return graph, which is illustrative even without knowledge about the precise problem setup and solution. We mention that the optimal solution consists of keeping the portfolio sensitivities at a certain (stochastic) level, similar to the continuous-time setting in Appendix A online; for details we refer the reader to Peters and Seydel (2018) or Peters (2014).

#### 4.2.1 Risk–return graphs

In Figure 3, the optimal approximate solution for the sovereign debt manager is shown in a risk–return graph for different levels of risk aversion, compared with other strategies. Figure 3 illustrates the risk transformation effected by the chosen PARS: while part (a) shows mean and standard deviation of the (negative) PV change over one year (with its optimal solution shown by the blue squares), part (b) displays the same in $\varPi$ (with its optimal solution shown by the green triangles). The red circles in Figure 3 represent stable portfolios resulting from a constant monthly issuance of coupon bonds with the maturities 6M, 1Y, 2Y, 5Y, 10Y and 30Y (rectangular nominal shape in time to maturity); for example, the 30Y stable portfolio has an average time to maturity of about 15Y.

As illustrated in Figure 3, the key advantage of choosing the perpetuity cost key figure (or a similar PARS) is that it appropriately reflects the risk situation of the sovereign debt manager: less risky assets (in the debt manager’s situation) show a lower risk in the perpetuity cost measure (Figure 3(b)). A portfolio optimization should therefore be carried out based on risk and return in the perpetuity cost measure; using the PV-optimal solution instead leads to riskier portfolios for the same return. This observation carries over from the sovereign debt manager to any institution: any investment decision should be based on appropriately chosen PARS, as already argued several times in this paper.

## 5 Conclusion

In this paper we proposed the use of a new class of performance measures adjusted for the risk situation (PARS), as the perception of risk depends on the individual situation including risk preferences. PARS take into account future cashflows of the institution including (financial) consumption preferences (the risk situation) and will measure the financial performance in the actual quantity of interest.

The definition of PARS is closely related to that of the risk-free strategy: if you know one, then you also know the other. The risk-free strategy is to sell your portfolio with PV $V_{t}(P^{t})$ and eliminate risks by offsetting the inevitable cashflows and by fixing the multiple $\varPi_{t}$ of your preferred consumption cashflows:

 $V_{t}(P^{t})=V_{t}(C)+\varPi_{t}V_{t}(E).$ (5.1)

PARS are future-oriented key figures, as they transform the PV (ie, the value of future cashflows, collapsed to one time point) into a series of future cashflows, distributed over time. Concerning the choice of this distribution (or risk situation), we have seen a number of examples ranging from individuals to fund managers or sovereign debt managers; for the latter, we saw how risk–return-optimal long-term debt financing costs could look.

This cashflow transformation of PARS also transforms the risks. Well-chosen PARS exhibit the right risk behavior instantaneously: as a consequence, taking the standard deviation of daily PARS increments gives a meaningful estimate of the risks incurred; further, a meaningful instantaneous risk evaluation is possible using just sensitivities and a covariance matrix. With only the PV, such behavior can be difficult and time-consuming to obtain (eg, using a simulation). Also, this advantage of PARS comes at practically no cost, as PARS are easy to compute using market-standard valuation techniques.

We believe that PARS is a useful concept for both academia and industry. Industry professionals are invited to work out examples of appropriate cashflow structures for their companies, while academics may want to further investigate the case of individuals or to extend the theory in several directions.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper. The views expressed in this paper are personal views and do not represent those of the German Finance Agency.

## Acknowledgements

The research was carried out while the second author was at the German Finance Agency. We want to thank all our colleagues at the German Finance Agency involved in the development and discussion of new key figures, in particular Andreas Ricker and Gerrit Handrich for encouraging research and for their valuable feedback. Thanks also to the participants of the 2017 Winter School in Lunteren for the lively discussions, in particular to Joachim Paulusch, Thorsten Schmidt and Jérôme Spielmann. Further, we want to thank an anonymous referee for valuable comments and for suggesting a wealth of additional citations, mainly in the area of insurance.

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