3.1 Time discretised net asset value models (no jumps)

If we consider the case that ( $\lambda _F=\lambda _I = 0)$ , then we have a pure diffusion model. For a small time interval $\Delta t$ , given the NAV process values, $X^{(F)}_t$ and $X^{(I)}_t$ , at time $t$ , we discretise the fund NAV $X^{(F)}_{t+\Delta t}$ and the benchmark NAV $X^{(I)}_{t+\Delta t}$ at time $t+\Delta t$ by applying the lognormal model given by

(3) \begin{align} \begin{split} X^{(F)}_{t+\Delta t} &= X^{(F)}_t\,\exp \!\left(\left(\widetilde{\mu }_F-0.5\sigma _{F}^2 \right)\Delta t+\sigma _{F} \sqrt{\Delta t} Z_{X^{(F)},t+\Delta t} \right),\\ X^{(I)}_{t+\Delta t} &= X^{(I)}_t\,\exp \!\left(\left(\widetilde{\mu }_I-0.5\sigma _{I}^2 \right)\Delta t+\sigma _{I} \sqrt{\Delta t} Z_{X^{(I)},t+\Delta t}\right), \end{split} \end{align}

where $\widetilde{\mu }_F$ and $\widetilde{\mu }_I$ are the generic drift functions. For instance, in the model in Equation (1) these would be given by $\widetilde{\mu }_F = r-d_F-\mu _F$ and $\widetilde{\mu }_I = r-d_I-\mu _I$ , respectively. Furthermore, under this discretised lognormal distribution form, the random vector $\left(Z_{X^{(F)},t+\Delta t}, Z_{X^{(I)},t+\Delta t}\right)$ is distributed as a standard bivariate normal distribution with correlation $\sigma _{FI}$ . We also denote the tracking error standard deviation of the difference between fund and benchmark returns by $\sigma _{F-I}$ .

3.2 Time discretised net asset value models (with jumps)

If we consider the case that ( $\lambda _F=\lambda _I = 1)$ , then we have a jump diffusion model where we consider the Lévy Process structures which incorporate the Poisson Process jump diffusion components in addition to the geometric random walk components. It was shown in Kou (Reference Kou2002) that the solution to this class of NAV jump diffusion models is given by

(4) \begin{align} \begin{split} X^{(F)}_t &= X^{(F)}_0\exp \!\left \{\left (\widetilde{\mu }_F - \frac{1}{2}\sigma _F^2\right )t + \sigma _F W^{(F)}_t \right \}\prod _{i=1}^{N^{(F)}_t}V_i^{(F)},\\ X^{(I)}_t &= X^{(I)}_0\exp \!\left \{\left (\widetilde{\mu }_I - \frac{1}{2}\sigma _I^2\right )t + \sigma _I W^{(I)}_t \right \}\prod _{i=1}^{N^{(I)}_t}V_i^{(I)}, \end{split} \end{align}

where in each model one has (dropping temporarily index F or I)

(5) \begin{equation} \mathbb{E}[V] = p\frac{\eta _1}{\eta _1-1} + (1-p)\frac{\eta _2}{\eta _2+1}, \;\; \eta _1 \gt 0, \eta _2 \gt 0 \end{equation}

with $\eta _1 \gt 1$ ensuring that $\mathbb{E}[V] \leq \infty$ and $\mathbb{E}[X_t] \leq \infty$ . From these solutions to the diffusion equations, one can then obtain a discretised return over an interval of time $\Delta t$ as follows:

(6) \begin{align} \begin{split} \frac{X^{(F)}_{t+\Delta t} - X^{(F)}_{t}}{{X^{(F)}_{t}}} &= \exp \!\left \{\left (\widetilde{\mu }_F - \frac{1}{2}\sigma _F^2\right )\Delta t + \sigma _F \!\left (W^{(F)}_{t+\Delta t} - W^{(F)}_{t} \right ) + \sum _{i=N^{(F)}_t+1}^{N^{(F)}_{t+\Delta t}}Y^{(F)}_i \right \},\\ \frac{X^{(I)}_{t+\Delta t} - X^{(I)}_{t} }{X^{(I)}_{t}} &= \exp \!\left \{\left (\widetilde{\mu }_I - \frac{1}{2}\sigma _I^2\right )\Delta t + \sigma _I \!\left (W^{(I)}_{t+\Delta t} - W^{(I)}_{t} \right ) + \sum _{i=N^{(I)}_t+1}^{N^{(I)}_{t+\Delta t}}Y^{(I)}_i \right \}. \end{split} \end{align}

where the summation over an empty set is taken to be zero. Of course, one can then make a small time scale approximation for the exponential which would expand as approximately $\exp\! (x) \approx 1 + x + x^2/2$ by

(7) \begin{equation} \frac{X^{({\cdot})}_{t+\Delta t} - X^{({\cdot})}_{t} }{X^{({\cdot})}_{t}} = \widetilde{\mu }_F\Delta t + \sigma _F \sqrt{\Delta t} Z + B Y \end{equation}

where $Z$ and $B$ are standard normal and Bernoulli random variables, respectively, with $\mathbb{P}\text{r} (B=1 ) = \lambda \Delta t$ and $\mathbb{P}\text{r} (B=0 ) = 1 - \lambda \Delta t$ and $Y$ is given by the following

(8) \begin{equation} Y=\begin{cases} E^+, & w.p. \;\; p\\[4pt] E^-, & w.p. \;\; 1-p,\\ \end{cases} \end{equation}

where $E^+$ and $E^-$ are exponential random variables with means $1/\eta _1$ and $1/\eta _2$ respectively.

Having defined the real-world processes and their time discretisations under consideration for describing the NAV of the fund and the reference index fund, we next introduce the fees models.

4.1 Model-specific Performance Fee (PF) structures

A daily accrual of a fee will take place constantly, for each fee model type, whenever the following fee model structure conditions are satisfied:

• $\mathcal{C}_1$ (Claw-Back condition) $\sum _{s=\widetilde{\tau }}^{s=t}\,FR_{s,s-1} \gt \sum _{s=\widetilde{\tau }}^{s=t}\,IR_{s,s-1}$ ,

• $\mathcal{C}_2$ (HWM condition) $\sum _{s=t_Y}^{s=t}\,FR_{s,s-1} \gt HWMR_t$ ,

• $\mathcal{C}_3$ (CAP condition) $\sum _{s=t_Y}^{s=t}\,FR_{s,s-1} -\sum _{s=t_Y}^{s=t}\,IR_{s,s-1} \gt CAP$ ,

• $\mathcal{C}_4$ (RHWM condition) $\sum _{s=t_Y}^{s=t}\,FR_{s,s-1} \gt RHWM_t$ ,

where the threshold $CAP$ is a percentage set by the fund manager charging the fees. Using these conditions, one may define several versions of Performance Fee Model (PPF). We will distinguish the $i$ -th model via a model index $\mathcal{M}_i$ . For each model, there will be a family of PPF’s for that model such that each member of the family of models is determined by the parameter $a_2 \in \mathbb{R}^+$ , which set by the fund manager.

Definition 1. (Reference Mechanism (Model $\mathcal{M}_{0}$ )) The (daily) accruing payoff of performance fee $PPF_t^{(\mathcal{M}_0)}$ at day $t$ is given by

(13) \begin{equation} PPF_t^{(\mathcal{M}_0)}=\max \!\left (a_2 ER_{t,t-1} AX_{t-1},0\right ). \end{equation}

Definition 2. (Claw-Back Mechanism (Model $\mathcal{M}_{1}$ )) The (daily) accruing payoff of performance fee $PPF_t^{(\mathcal{M}_1)}$ at day $t$ is given by

(14) \begin{equation} PPF_t^{(\mathcal{M}_1)} = \begin{cases} a_2 ER_{t,t-1} AX_{t-1}, \quad \, \text{if $\mathcal{C}_1$ is true,} \\[4pt] 0, \qquad \qquad \qquad\quad \text{otherwise.} \end{cases} \end{equation}

In the $\mathcal{M}_1$ fee model family, the indicator function of the event that $\sum _{s=\widetilde{\tau }}^{s=t}\,IR_{s,s-1}$ acts as a hurdle rate, and the word “Claw-Back” reflects the characteristic of the mechanism that the payoff will be accrued in case of claw-back performance but not a jump-back performance (i.e. the under-performance needs to recover before any accrual). Although $PPF_t^{(\mathcal{M}_1)}$ can be negative, to penalise the manager in case of under-performance, the cumulative $PPF_t^{(\mathcal{M}_1)}$ will never be negative since we will not accrue negative $PPF_t^{(\mathcal{M}_1)}$ if it makes the cumulative amount drop below zero. $PPF_t^{(\mathcal{M}_1)}$ is also called the “Periodic Performance Fee Accrual” in typical fund manager parlance.

Definition 3. (High Watermark Mechanism (Model $\mathcal{M}_{2}$ )) The (daily) accruing payoff of performance fee $PPF_t^{(\mathcal{M}_2)}$ at day $t$ is given by

(15) \begin{equation} PPF_t^{(\mathcal{M}_2)} = \begin{cases} a_2 ER_{t,t-1} AX_{t-1}, \quad \, \text{if $\mathcal{C}_1$ and $\mathcal{C}_2$ are true,} \\[4pt] 0, \qquad \qquad \qquad \quad \text{otherwise,} \end{cases} \end{equation}

In the case of the $\mathcal{M}_2$ fee model family, it builds upon the claw-back condition and adds an additional High Water Mark condition to be exceed in order to satisfy the condition to begin accruing fees.

Definition 4. (High Watermark with Cap Mechanism (Model $\mathcal{M}_{3}$ )) The (daily) accruing payoff of performance fee $PPF_t^{(\mathcal{M}_3)}$ at day $t$ is given by

(16) \begin{equation} PPF_t^{(\mathcal{M}_3)} = \begin{cases} a_2 \!\left (CAP-\sum _{s=t_Y}^{s=t-1}\,ER_{s,s-1}\right ) AX_{t-1}, \, \text{if $\mathcal{C}_1$, $\mathcal{C}_2$, and $\mathcal{C}_3$ are true,} \\[4pt] a_2 ER_{t,t-1} AX_{t-1}, \qquad \qquad \qquad \quad \quad \,\, \text{if $\mathcal{C}_1$ and $\mathcal{C}_2$ are true but $\mathcal{C}_3$ is false,} \\[4pt] 0, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \, \text{otherwise,} \end{cases} \end{equation}

In the case of the $\mathcal{M}_3$ fee model family, it builds upon the claw-back condition, the High Watermark condition and adds a third condition based on a performance fee cap in case the fund does exceptionally well relative to the reference index. This has the effect of protecting the investors gains should exceptional gains be achieved in a given period by the fund manager, where $CAP-\sum _{s=t_Y}^{s=t-1}\,ER_{s,s-1}$ implies there will be no accrual of performance fee above $CAP$ .

Definition 5. (Replicate High Watermark Mechanism (Model $\mathcal{M}_{4}$ )) The (daily) accruing payoff of performance fee $PPF_t^{(\mathcal{M}_4)}$ at day $t$ is given by

(17) \begin{equation} PPF_t^{(\mathcal{M}_4)} = \begin{cases} a_2 ER_{t,t-1} AX_{t-1}, \quad \, \text{if $\mathcal{C}_1$ and $\mathcal{C}_4$ are true,} \\[4pt] 0, \qquad \qquad \qquad \qquad \text{otherwise,} \end{cases} \end{equation}

Having now defined the different fee model structures, it is now important to ask about the valuation of these different fee structures under the specified fund NAV stochastic models and the reference index NAV stochastic models. We will do so by considering two basic questions:

• Does their valuation warrant their application with regard to performance gains for investors? and

• Which of these fee structures is most reasonably valued from the investors perspective?

To achieve the required analysis to address these questions, we will reformulate the valuation question for these fee structures as an option pricing question. We will then consider the option pricing framework in both a classical risk-neutral and also an actuarial incomplete distortion pricing contexts.

This will involve consideration of the following pricing challenge. We would like to determine, at any time $t \in [0,T_j]$ in the $j$ -th year, the values of performance fees at the end of the year which are paid annually, denoted generically by $\mathcal{P}_{j} \left(\mathcal{F}^{X^{(F)}}_{t},\mathcal{F}^{X^{(I)}}_{t};\, \mathcal{M}_s \right)$ . This is the model $\mathcal{M}_s$ performance fee structures discounted present valuation, which can be obtained by taking the pricing kernel discounted valuation as follows

(18) \begin{equation} \mathcal{P}_{j}\!\left (\mathcal{F}^{X^{(F)}}_{t},\mathcal{F}^{X^{(I)}}_{t};\ \mathcal{M}_s\right ) = \mathbb{E}_{Q}\!\left [\exp \!\left ({-}r(T_j-t)\right )H_T \right ] \end{equation}

for an appropriately selected pricing framework, to be discussed in the following section and an option-like discounted end-of-year payoff denoted generically by $H_t$ which is selected from one of the fee models $ \left\{PPF_t^{(\mathcal{M}_1)},PPF_t^{(\mathcal{M}_2)},PPF_t^{(\mathcal{M}_3)},PPF_t^{(\mathcal{M}_4)} \right\}$ .

4.2 Valuation incorporating Performance Fees (PF), Operation Fees (OF) and Management Fees (MF)

We denote the (daily) accruing payoff of OF $OF_t$ at day $t$ and the (daily) accruing payoff of management and advisory fee $MF_t$ at day $t$ , respectively, by:

(19) \begin{align} \begin{split} OF_t &= \frac{a_0}{252} X_t,\\ MF_t &= \frac{a_1}{252} X_t, \end{split} \end{align}

for pre-defined constants $a_0, a_1 \in \mathbb{R}^+$ where the gross asset value per share $X_t$ is given by

(20) \begin{equation} X_t=\frac{X^{(F)}_t+\mathcal{P}_{j}\!\left (\mathcal{F}^{X^{(F)}}_{t},\mathcal{F}^{X^{(I)}}_{t};\, \mathcal{M}_s\right )}{1-\frac{a_0}{252}-\frac{a_1}{252}}. \end{equation}

which is derived from the relation $X^{(F)}_t=X_t-OF_t-MF_t-\mathcal{P}_{j} \left(\mathcal{F}^{X^{(F)}}_{t},\mathcal{F}^{X^{(I)}}_{t};\ \mathcal{M}_s \right)$ .

5.1 Scenario-based analysis versus stochastic optimal control approaches

There are two approaches one could adopt to study the valuation of performance fees, the first assumes that the fund manager desires to fix apriori the fee structures based on their target level of risk-return over a range of various investment strategies that they have a mandate to consider to apply when deploying capital from investors. The objective in this context is then to study which fee structures will achieve the desired outcomes for the fund manager profit expectations as well as satisfy the risk averse investors with regard to returns that keep the fund competitive with regard to competitors for a tolerable level of risk dictated by the investors at initial date of investment.

The second approach assumes that the fund manager may dynamically modify any combination of their fee structures, strategies or investment practices over time and that the optimal sequence of such decisions is then obtained by solving a stochastic optimal control problem, see examples in Soner (Reference Soner2004) and Guasoni & Obłój (Reference Guasoni and Obłój2016).

We have not adopted this latter case, as we believe that the scenario-based framework aligns more realistically with a framework that would be practically useful for industry practitioners and investors when compared to the assumptions required to be made for the second case in the context of the studies undertaken in this work. In the first approach, one develops a framework based on scenario analysis valuation that prices the different fee structures in various economic scenarios and under different fund strategy simulations which may seek to assess for each fee structure the expected best case outcomes, the worst case outcomes and typical outcomes under various economic scenarios. This may also include assessment of the sensitivity and stability of the fee structures to changes in various model components.

From the fund manager perspective, such an approach would provide the fund manager with a model-based assessment of the fee dynamic behaviour. In this context, different strategies for the investment practice of the fund combined with various fee structures considered can then be assessed to determine which were optimal for a given risk-return profile of the investors and fund manager cost basis.

From the investors perspective, such a framework is also valuable as it would allow them to compare for a given class of investment strategies characterised by the fund NAV model and a reference benchmark, how different fee structures may impact on their profit in various economic scenario contexts.

Furthermore, fees are not paid out every day but rather accumulate daily over some fixed period, say annually. Then they are paid out at the end of each period. So, if a fund manager was accumulating fees throughout the year, it is unlikely they would be altering their strategies drastically throughout that period too often to try to maximise fees. Rather we believe that they would explore a scenario-based approach to understanding the likely accumulation of fees over time, based on different economic scenarios and investment strategies adopted.

If one were to adopt the second approach based on the stochastic optimal control perspective to study the valuation of performance fees, this would require modifications to the fund NAV jump diffusion model. Ideally, one would adopt a class of models that incorporates a feedback mechanism between the fund NAV and the historical dynamic of the fees accrued by the fund manager making investment decisions. Such an approach would consequently result in a non-linear path-dependent diffusion model, with an integrated fee functional incorporated into the drift, volatility or jump diffusion components of the fund NAV jump diffusion model. This would add significant complexity to the model and the problem formulation would be rewritten as a type of optimal stochastic control problem, which we do not believe reflects the approach that practitioners in the industry would adopt in practice. It is not the tendency of fund managers to dynamically change their fee structures regularly over time, rather these are typically fixed apriori and rarely get changed. A more realistic approach would be one in which the fund manager could alternatively vary their investment strategy for a given fee structure. We have not approached this class of solution in this manuscript, see approaches to formulate such a framework mathematically in Soner (Reference Soner2004)

5.2 Risk-neutral versus actuarial valuation

In this section, we set out the mathematical framework under which we seek to study the pricing of performance fees in fund management under the scenario-based framework described in context in section 5.1. It will be based upon the fact that one may identify fees attributed to a fund performance as a form of option type payoff. Take for example the case of a hedge fund, where incentive plans are primarily bonus plans. In such asset management structures, fund managers typically receive a fraction of the funds return each year in excess of the high watermark. The high watermark for each investor is the maximum share value since the time of his or her investment in the fund. We refer to this fraction of the funds return as a performance fee. This performance fee can be considered as a call option on the profits associated with managing other people money, since the fee structure gives the managers the positive fees with profits but no negative fees with losses.

We make a few further remarks on the treatment of fee structures as a type of optional like payoff and pricing problem formulation. Firstly, it is apparent that the fee structures act as an incentive mechanism for fund managers to attempt to manage funds to generate greater fees. When this activity aligns with improved PnL with managed risk profiles acceptable to investors, then it becomes an effective mechanism. This raises a few interesting considerations. Are there market contexts in which the fund manager can manipulate this outcome to their benefit without a tangible outcome for investors and secondly which fee structures may be more likely for this to occur?

We first remark on these points as follows. We first discuss the ideal setting of efficient markets. In such markets, there should be seldom opportunities to realise an arbitrage and any short-term or instantaneous arbitrage opportunities between an underlying asset market and derivatives such as options written on that asset would be rapidly closed. Furthermore, traders who hold in their portfolio positions in traded derivative contracts would seldom have sufficient capital to significantly influence or alter the price substantially in the underlying asset, simply by trading the underlying in a manner that would benefit their option positions and generate PnL, perhaps to the detriment of investors holding the underlying, such as the fund investors. This is because such efficient markets would have significant liquidity in the order books to prevent significant and permanent price impacts to be effectively enacted to drive price in any given direction. Therefore, in the context of this paper, it would be analogous to saying the fund managers would find it difficult to alter the state of the markets in which they manage positions for the fund in order to generate higher fees from particular fee mechanisms.

However, one can also argue that in practice it is certainly the case that derivative contracts can be used to influence the underlying asset price upon which they were written or that positions in such instruments can result in incentives for market participants to make attempts to influence the underlying assets price behaviour. This is regularly observed when there are large option positions that may expire out of the money, and there is a vested interest for market participants that may have positions expire well out of the money to incentivise shocks or changes in price momentum and volatility of the underlying asset, through either active trading in the underlying or motivating others to do similar actions.

This is clearly more effective in illiquid or incomplete markets more so than very liquid markets as the amount of capital would indeed make it prohibitive for most participants in deeply liquid efficient markets. We note that much of the paper is focussed on the actuarial pricing context of relevance to incomplete market pricing contexts and compared to risk-neutral approach in these settings.

Furthermore, the ability to influence the fund performance will depend on the type of fee structure designed, some fee structures would be harder to manipulate than others as they are referenced to a benchmark index that is not traded by the fund using the investors capital.

Hence, the natural question arises on how best to price such fee-based call options. We will explore the application of well-known methods for complete and incomplete actuarial pricing model frameworks in this context. We do not innovate in this regard but rather we apply these methods in an interesting problem domain in asset management, which in and of itself is a practically meaningful contribution.

Here we will distinguish between efficient and inefficient market valuation. This is of relevance to the context of the study of fees in asset management since some funds are highly liquid and possibly exchange traded and therefore more likely to be efficient in their price discovery while other funds are much less liquid and either only traded at end of day market close or just OTC traded.

One would typically perform valuation of a financial asset under an actuarial framework via a “deflator” or “pricing kernel” for a measurable payoff $H_t$ , which is given by

(21) \begin{equation} H_t = \mathbb{E}\!\left [\frac{\xi _T}{\xi _t} H_T\right ], \end{equation}

where the deflator $\xi _t$ is concerned with achieving market-consistent valuations of assets and liabilities, see detailed discussion in Bühlmann & Hansjörg (Reference Wüthrich, Bühlmann and Hansjörg2010). In addition to using deflators, the arbitrage-free market-consistent value can also be obtained in the mathematical finance audience by considering the notion of risk-neutral pricing according to a change of measure. These methods involve using probabilistic expectations of discounted present values of future cash flows, but in a world where all investors are risk-neutral. In the ideal complete market setting, the deflator and risk-neutral settings can yield the same unique result in general; however, if one moves to incomplete markets, this starts to differ.

5.3 Risk-neutral pricing for option-like fund performance fees

Under a risk-neutral pricing framework, one typically assumes that an investor would have the ability to trade the fund position as often as desired in a frictionless manner and this market participant behaviour would be reflected by the NAV of the fund and the benchmark index at all times. The pricing of options in the risk-neutral formulation has been developed for several decades now, based on early seminar works by Black & Scholes (Reference Black and Scholes1973), Merton (Reference Merton1973, Reference Merton1976) and Cox et al. (Reference Cox, Ross and Rubinstein1979). See the review articles of Smith (Reference Smith1976) and Broadie & Detemple (Reference Broadie and Detemple2004). In Merton’s aforementioned work, “ideal conditions” in the market for the underlying stock and the option are discussed. These ideal conditions were based around the concepts of frictionless market characteristics, see further discussion in Rogers (Reference Rogers1998), and may not lead to practically realistic frameworks upon which to price the fee structures outlined in this manuscript. As such we will compare the classical risk-neutral option pricing framework to alternative approaches developed in the actuarial pricing literature, see coverage of such topics in the actuarial literature in Gerber & Shiu (Reference Gerber and Shiu1996, Reference Gerber and Shiu1995), Embrechts (Reference Embrechts2000), Goovaerts & Laeven (Reference Goovaerts and Laeven2008) and a book length treatment in Bühlmann & Hansjörg (Reference Wüthrich, Bühlmann and Hansjörg2010) .

Consider a maturity (future time) $T$ that the derivative on the asset price process $ (X_t )_{0 \leq t \leq T}$ pays an amount (payoff function for the fee from one of the models $\mathcal{M}_0$ to $\mathcal{M}_4$ ) denoted by $H_T$ , which is a $\mathcal{F}_T$ measurable random variable. Under the fundamental theorem of asset pricing, it is implied that in a complete market with no-arbitrage opportunities a derivative’s price is the discounted expected value of the future payoff under the unique risk-neutral measure. This leads to the standard risk-neutral pricing formulation producing the present fair value given by a discounted expected value

(22) \begin{equation} H_t =\mathbb{E}_{\mathbb{Q}}\!\left [\exp\! ({-}r(T-t)) H_T\right ] = \mathbb{E}_{\mathbb{P}}\!\left [\exp\! ({-}r(T-t))H_T\frac{d\mathbb{Q}}{d\mathbb{P}}\right ], \end{equation}

where $r$ is the risk-free interest rate and $\mathbb{Q}$ denotes the risk-neutral pricing measure and $\mathbb{P}$ the real-world measure that characterises the observed probabilities of price dynamics, where these are connected via the standard change of measure formula based on the Radon–Nikodym derivative $\frac{d\mathbb{Q}}{d\mathbb{P}}$ .

This framework was relaxed and extended in works by Merton (Reference Merton1973) to allow for stochastic interest rates and other features. Furthermore, if the asset price process follows a continuous time stochastic jump process such as a Lévy process examples of works treating option valuation include Duffie et al. (Reference Duffie, Pan and Singleton2000) and Sepp & Skachkov (Reference Sepp and Skachkov2003).

In the pricing undertaken for the performance fee valuation in this manuscript, one already sees from the specification of the payoff structures under fee models $\mathcal{M}_0$ to $\mathcal{M}_4$ that the payoff functions will be less trivial and generally path-dependent; therefore, they will not admit closed form solutions and are therefore alternatively evaluated numerically via Monte Carlo methods.

In the setting of no jump structure, $\lambda _F=\lambda _I=0$ , we will be able to develop an efficient Monte Carlo sampling strategy based on the discretisation schemes previously presented with $\widetilde{\mu }_F$ and $\widetilde{\mu }_I$ being specified under the Girsanov transform drift adjustment and with the added exception that now $\tilde{W}_t$ is a $Q_{F}$ -Brownian motion and $\tilde{V}_t$ is a $Q_{I}$ -Brownian motion representing the risk-neutral drivers. Under the Black–Scholes model, there exists a unique probability measure $Q_{F}$ and a unique probability measure $Q_{I}$ such that they are equivalent martingale measures. In this framework, one can then use the martingale measures so that the models for the fund and benchmark NAVs are arbitrage-free and complete.

5.4 Actuarial incomplete market distortion pricing for option-like fund performance fees

One may question the veracity of the assumptions regarding the existence of a complete arbitrage-free market in the context of the funds dynamics studied in this manuscript. This is especially the case when one recalls that market completeness assumes that the trading of the fund shares has negligible transaction costs and that the complete set of possible bets on future states of the world, valuations of the funds NAV incorporating fee structures, can be constructed with existing assets without friction. However, as discussed in Birge & Linetsky (2007, Chapter 12), one is sometimes best served to recognise explicitly that in reality, markets such as those under consideration in this manuscript are incomplete, meaning that some payoffs cannot be replicated by trading in security markets. The classic no-arbitrage theory of valuation in a complete market, based on the unique price of a self-financing replicating portfolio, is not adequate for non-replicable payoffs in incomplete markets. For a guide to surveys on incomplete market pricing settings in financial mathematics contexts, see Jouini & Napp (Reference Jouini and Napp2001), Tankov (2003, Chapter 10). There are several ways in which market incompleteness may manifest, as discussed in Birge & Linetsky (Reference Birge and Linetsky2007), for example, insufficiency of marketed assets relative to the class of risks that one wishes to hedge, which may involve jumps or volatility of asset prices, or variables that are not derived from market prices; market frictions related to transaction costs and portfolio constraints; or ambiguity in the appropriate model for the market prices.

In order to develop alternative approaches to pricing derivatives, when markets are incomplete, the actuarial literature has developed numerous solutions approaches. In this section, we will recall some basic details of actuarial pricing methods that can be made consistent with risk-neutral arbitrage-free efficient market pricing methods described above. These are namely the Esscher transform approach based on early actuarial pricing methods from Swedish actuary (Escher, Reference Escher1932), and the Wang distortion measure approach based on Wang (Reference Wang2000). An excellent coverage of market-consistent actuarial pricing methods is provided in Bühlmann & Hansjörg (Reference Wüthrich, Bühlmann and Hansjörg2010).

Numerous authors have explored aspects of actuarial approaches to option pricing that arose after the works of Gerber & Shiu (Reference Gerber and Shiu1994, Reference Gerber and Shiu1996), Bühlmann et al. (Reference Bühlmann, Delbaen, Embrechts and Shiryaev1996), Wang (Reference Wang2000), Embrechts (Reference Embrechts2000) and Labuschagne & Offwood (Reference Labuschagne and Offwood2013). In the paper by Gerber & Shiu (Reference Gerber and Shiu1994), they consider the family of dispersion measures known as the Esscher transform, that they utilise for option pricing (as the deflator methodology equivalent of risk-neutral pricing in financial mathematics). Under the Esscher transform, they were able to show that one may obtain an efficient technique for valuing derivative securities if the logarithms of the prices of the underlying security come from a particular class that follows a stochastic processes with independent and stationary increments. Therefore, under this family of dispersion measures, one may select the parameter of this transform such that when it is applied to a security price process this would produce an equivalent probability measure. This resulting equivalent probability measure for the specially selected Esscher transform parameter (corresponding to the market price of risk) will produce a martingale for the discounted price of any underlying security with respect to the new transformed measure. Hence, in terms of valuation, one may calculate the value of any derivative or contingent claim future cash flows as the expectation, with respect to the equivalent martingale measure, of the discounted payoffs.

The Esscher transform is given in Definition 6 and is also commonly known in statistics as an “exponential tilting” and used widely in developing asymptotic series expansions for distributions and densities.

Definition 6. (Esscher Transform or Exponential Tilting) Consider a continuous random variable $X$ defined with respect to a probability measure $\mathbb{P}$ and a non-zero constant real number $h$ such that $\mathbb{E}_{\mathbb{P}} [\exp\! (hX) ]$ exists. One can then define the Esscher transform, denoted as $\mathcal{E}_h[{\cdot}]$ , of the original probability measure $\mathbb{P}$ for $X$ in terms of an equivalent new probability measure $\mathbb{Q} \sim \mathbb{P}$ (same null sets) with the following properties:

1. 1. $\mathcal{E}_{h_1}\mathcal{E}_{h_2}[\mathbb{P}] = \mathcal{E}_{h_1+h_2}[\mathbb{P}]$ ;

2. 2. $\mathcal{E}^{-1}_h[\mathbb{P}] = \mathcal{E}_{-h}[\mathbb{P}]$ .

If the measure $\mathbb{P}$ that characterises random variable $X$ admits a Radon–Nikodym derivative with respect to a suitable measure $\nu$ given by a density $f_X(x)$ , then we see that the Esscher transform of the density, denoted by $\mathcal{E}_h [f_X(x) ]$ , will be a new density given by

(23) \begin{equation} f\!\left (x;\,h\right ) \,:\!=\, \mathcal{E}_h\!\left [f_X(x)\right ] = \frac{\exp\! (hx)f_X(x)}{\int _{-\infty }^{\infty }\exp\! (hx)f_X(x)dx}. \end{equation}

Now, for $t \geq 0$ , consider a price process (e.g. contingent claim such as a particular fee payoff structure linked to a reference index) denoted by process $ (H_t )_{t\geq 0}$ on a market characterised probabilistically by $ (\Omega,\mathcal{F}, (\mathcal{F}_t )_{0\geq t \geq T},\mathbb{P} )$ . One can then apply the Esscher transform change of measure to obtain an Esscher pricing measure. The Esscher principle of pricing allows one obtain a unique martingale measure for the pricing of future contingent claims that, for the right continuous price process denoted $ (H_t )_{t\geq 0}$ , can be used to perform the pricing under discounting with regard to the new measure $\mathbb{Q}$ according to the expectation $H_0 = \mathbb{E}_{\mathbb{Q}} [H_t |\mathcal{F}_t ]$ for some unique $h = h^*$ such that the risk-neutral Esscher measure is given by

(24) \begin{equation} \frac{d\mathbb{Q}}{d\mathbb{P}} = \frac{\exp \!\left (hH_t\right )}{\mathbb{E}_{\mathbb{P}}\!\left [\exp \!\left (hH_t\right )\right ]}, \end{equation}

and the process $ (H_t )$ is a $\mathbb{Q}$ -martingale. The resulting measure $\mathbb{Q}$ is the Esscher transform with parameter $h$ , known in actuarial science as the “risk-neutral Esscher measure.” Then, one can select a unique parameter $h^*$ such that the process $ (\exp\! ({-}rt )H_t )_{t \geq 0}$ is a martingale.

An alternative widely used actuarial pricing formulation is based on the Choquet integral and is termed the Wang transform after the universal pricing paper for actuarial valuation presented by Shaun Wang, see Wang (Reference Wang2002) and the double parameter Wang transform (Wang, Reference Wang2004). Furthermore, extensions that generalise the standard Wang transform have been developed in Kijima & Muromachi (Reference Kijima and Muromachi2008). The key to this transform is the introduction of a parameter known as the “market price of risk,” which is utilised to obtain a “risk-adjusted” fair valuation price. It is assumed that the market price of risk is a continuously increasing function of duration. This pricing method can be applied to any contingent claim or payoff so long as it is co-monotone with its underlying assets or liabilities; see an excellent review of properties of comonotonicity of functions in economic and risk applications in Chateauneuf et al. (Reference Chateauneuf, Cohen and Kast1997).

Definition 7. (Wang Transform) Consider a financial asset or liability, with value denoted by $X_t$ , over time horizon $[0,T]$ . Assume $X_t$ has distribution $F(x)$ , then the Wang transformation of this distribution is given by the “risk-adjusted distribution”

(25) \begin{equation} F^*(x) \,:\!=\, \mathcal{W}_{\lambda }\!\left [F(x)\right ] = \Phi \!\left (\Phi ^{-1}\!\left ( F(x) \right ) + \lambda \right ) \end{equation}

with $\Phi ({\cdot})$ representing the standard Gaussian distribution and $\lambda$ representing the market price of risk that indicates the associated amount of systematic risk. Under this transform, the expected value of $\mathbb{E}^* [X ]$ under $F^*(x)$ will correspond to the risk-adjusted actuarial fair value of the asset or liability at time $T$ , which can then be discounted to any time via the risk-free rate.

The Wang transform and the Esscher transform are the only two known distortion measures that are able to recover the risk-neutral Black–Scholes option pricing formula as a special case.

Given an underlying risk $X$ and a function $h$ that maps this risk to a payoff $Y = h(X)$ , that is, a derivative or contingent payoff, then Wang provides two methods that, though they involve different mathematical steps at each stage, will produce equivalent fair value prices; see Wang (2002, p. 218).

1. 1. First take the Wang transform $F_Y^*(y;\,\lambda )=\mathcal{W}_{\lambda } [F_X(x) ]$ of the underlying risk random variable (processes) $X$ with respect to its distribution. Then derive the distribution of the risk-adjusted derivative or contingent payoff $F_Y^*(y;\,\lambda )$ as a function of $F^*_X(x)$ using change of variable or probability transforms with the relationship $Y^* = h (X^* )$ to obtain

   (26) \begin{equation} F_Y^*(y;\,\lambda ) = \mathbb{P}\text{r}\!\left [Y^* \leq y\right ] = \mathbb{P}\text{r}\!\left [h\!\left (X^*\right ) \leq y\right ] =\mathbb{P}\text{r}\!\left [X^* \leq h^{-1}(y)\right ] = F_X^*(h^{-1}(y)). \end{equation}

2. 2. Alternatively, one can first derive the distribution of $F_Y$ using the relationship $Y=h(x)$ ,

   (27) \begin{equation} F_Y(y) = \mathbb{P}\text{r}\!\left [Y \leq y\right ] = \mathbb{P}\text{r}\!\left [h\!\left (X\right ) \leq y\right ] =\mathbb{P}\text{r}\!\left [X \leq h^{-1}(y)\right ] = F_X(h^{-1}(y)). \end{equation}
   Then apply the Wang transformation $F_Y^*(y;\,\lambda ) = \mathcal{W}_{\lambda } [F_Y(y) ]$ .

This equivalence in relationship will hold if the transformation $h$ is monotone.

In the continuous time setting where one considers the change of measure for a diffusion process, there have also been studies of the associated relationships and consistency of the Wang transform and risk-neutral arbitrage-free pricing, see discussion in Pelsser (Reference Pelsser2008, section 4). There have been studies performed in Goovaerts & Laeven (Reference Goovaerts and Laeven2008) that consider the stochastic process generalisation of the Esscher transform for pricing, termed the Esscher-Girsanov distortion measures. This is the distortion measure analog of the change of measure achieved by the Girsanov theorem in classical risk-neutral option pricing.

5.5 Applying distortion measure pricing via Monte Carlo simulation

A performance fee can be considered as an option due to the fact that the accumulated payoff for the manager will never go below zero (Xiao, Reference Xiao2006). Monte Carlo simulation is a numerical technique used to approximate the expected values based on generating random samplings. The method is useful in pricing an option with unavailable closed form solution.

In this section, we outline a generic Monte Carlo Pricing framework as shown in Algorithm 1 adopted to price the different fee option payoff functions. Under the Monte Carlo procedure, this is relatively straightforward to achieve via a sequence of simulation steps of the real-world price process for the NAV of each fund, followed by known transformations and Monte Carlo integral approximation. We note that the algorithm is intended as a pseudo code and will outline the basic steps one can follow to implement these frameworks.

Algorithm 1. Monte Carlo Pricing

6.1 Case study fund selections

The case study undertaken will explore an application based on the three JP Morgan funds run in Luxemburg which are Europe Research Enhanced Index Equity and Income Opportunity Plus appearing in the 2017 Prospectus (J.P.Morgan Asset Management, 2017) and Income Opportunity appearing on 2021 Prospectus (J.P.Morgan Asset Management, 2021). The details of each fund are shown in Table 2. We note that the selection of these illustrative funds provides an anchoring for the scenario case studies explored for various fee mechanisms, to be based in a realistic practical framework.

Table 2. Fund details used in the case studies.

According to the 2017 (J.P.Morgan Asset Management, 2017) and 2021 (J.P.Morgan Asset Management, 2021) prospectuses, model $\mathcal{M}_1$ is applied to Europe Research Enhanced Index Equity (share class I) while model $\mathcal{M}_2$ is applied to Income Opportunity Plus (2017) as well as Income Opportunity (2021).

For Europe Research Enhanced Index Equity (share class I), the fee rates are $MF=0.16\%$ , $OF=0.19\%$ and $PF=10\%$ . For both Income Opportunity Plus and Income Opportunity, the values are $MF=0.55\%$ , $OF=0.11\%$ and $PF=20\%$ . The summary of fund fees together with their fee mechanism is shown in Table 3. All Fund NAVs including MSCI Europe Index NAVs are displayed in Figure 1.

Table 3. The selected funds together with their share class and fee structures.

Note: (acc) means the share class of interest does not pay dividends. Earned income is reinvested in the NAV (J.P.Morgan Asset Management, 2017).

Figure 1 Fund and benchmark NAVs.

Although model $\mathcal{M}_3$ is defined in the 2017 Prospectus (J.P.Morgan Asset Management, 2017), the mechanism is not actually applied to Income Opportunity Plus, but it is applied to US opportunistic Long-Short Equity which is not included in this study as the fund and its benchmark data is not sufficiently available to undertake a complete study. The CAP at $11.5\%$ is indicated in the 2017 Prospectus (J.P.Morgan Asset Management, 2017) to apply to the US opportunistic Long-Short Equity. Instead, to study model $\mathcal{M}_3$ in comparison with model $\mathcal{M}_2$ , we apply the same CAP at $11.5\%$ , shown in Table 3, to Income Opportunity Plus and Income Opportunity. Model $\mathcal{M}_4$ is also developed in order to analyse model $\mathcal{M}_2$ .

In this work, the period of study is set to be three years. We assume the period is sufficiently long to accurately measure the long-term performance of a mutual fund so that the assessment of performance is not affected solely by the short-term fluctuations in market cycles over a single business cycle in the equity markets.

6.2 Discounting interest rates

We assume the investor can deposit money in a secure “bank account” and earn a risk-free interest rate, when not investing. In this section, we discuss the bank account yields and we will differentiate between the Eurozone and the US investor.

For Europe Research Enhanced Index which invested in the European region, its risk-free rate is set to equal the yield of AAA-rated Euro-area central government bonds with one-year term to maturity and is assumed to be fixed throughout the period of our study. The rate on 25th July 2017 is $-0.70\%$ (European Central Bank, 2020). We use the rate quoted in year 2017 since the fund with performance fee structure was available in year 2017 and the fee structure was cancelled in the following year.

During recent years, all yields in this region are negative for various reasons related to the current macroeconomy of the region and geopolitical factors that may have caused uncertainty in this economic region. Negative yielding rates on fixed income products are a recent phenomenon that would typically discourage both fund manager and investor to invest in such instruments in cash-based bank accounts, however for large capital investments, they are seen as somewhat of a safer option for investing even with this negative yield than keeping money in more volatile assets such as equities or commodities. They can still offer a long-term safe haven for large amounts of total assets that are required to be deployed for investment purposes even during high volatility periods of the markets. Therefore, we still assume the investor and manager choose to keep riskless assets in a bank account with risk-free rate although the risk-free rate itself is negative.

For Income Opportunity Plus and Income Opportunity which invested most in the American region, the United States Treasury bond yield curve rates with maturity time of one year are used as its risk-free rate and are assumed to be fix throughout three years of our study period. The rate for Income Opportunity Plus is at $1.24\%$ quoted on 30th June 2017 (Daily Treasury Yield Curve Rates). We use the rate quoted in year 2017 since the fund was available during the year, and later, it was merged with Income Opportunity in year 2018. As for Income Opportunity, the risk-free rate is $0.09\%$ on 1st July 2021 (Daily Treasury Yield Curve Rates). The rate quoted in year 2021 is adopted since the fund is available in J.P.Morgan Prospectus of year 2021 (J.P.Morgan Asset Management, 2021). All parameter values of risk-free rates are summarised in Table 4.

Table 4. Fund and fee parameter values.

Note 1: Estimated model parameters for $\mu _F$ are obtained over the 3-year data history. The estimated $\sigma _F$ low and high values were estimated based on a 1-year sliding window and the lowest historical estimate and highest historical estimates were used in the scenarios studied.

Note 2: While the spot NAVs 196.77 (EUR) and 111.1 (USD) were the NAVs on 25 July 2017, 119.17 USD was the NAV on 11 August 2021 (Morgan).

Note 3: setting the extreme scenario volatility at $10^*\%$ was not estimated from observed historical NAVs but set up in order to study the effect of CAP on model $\mathcal{M}_3$ in an extreme scenario.

7.1 Estimation of the diffusion components and construction of scenarios

For the development of the stochastic models used in the NAV dynamics, we will require to estimate certain model parameters. In Equation (1), $\mu _F$ is the annual mean return of fund, while $\mu _I$ is the annual mean return of the benchmark index fund. The two parameters were calibrated based on the fund and benchmark data (Reference MorganMorgan; MSCI Europe Net EUR Historical Rates (MIEU00000NEU)) and their values are shown in Tables 4 and 5, respectively.

Table 5. Benchmark parameter values.

Note: The NAV 211.60 (EUR) is the benchmark NAV on 8 August 2017 (MSCI Europe Net EUR Historical Rates (MIEU00000NEU)).

Calibration of the pure diffusion models, which involved setting $\lambda _I=\lambda _F=0$ , was based on a moment-matching method of estimation of the discretised Lognormal models from Equation (3) for the fund and benchmark NAV processes. The estimated drift and volatility parameters in Table 4 are estimated based on the simple moment-matching approach from historical data from NAV data.

Given $Z$ a standard normal variable with $\mu$ and $\sigma \gt 0$ parameters. Then, the distribution of the random variable $X=e^{\mu +\sigma Z}$ is the lognormal distribution with parameters $\mu$ and $\sigma$ and the moment-matching equations solved are given by:

(28) \begin{align} \mu =\ln \!\left ({\frac{\mu _{X}^{2}}{\sqrt{\mu _{X}^{2}+\sigma _{X}^{2}}}}\right ), \;\;\;\; \sigma ^{2} =\ln \!\left (1+{\frac{\sigma _{X}^{2}}{\mu _{X}^{2}}}\right ). \end{align}

Historical volatility and correlation are also used as parameters in Equation (3) to simulate NAVs of fund and benchmark. We estimate the annualised historical volatility rather than the implied volatility based on the justification that options are generally priced more fairly based on historical norms. The historical volatilities of each fund NAV, $\sigma _{F}$ , and each benchmark NAV, $\sigma _{I}$ , and their correlation, $\sigma _{FI}$ , as well as tracking error $\sigma _{F-I}$ , are estimated based on available NAV data. The historical volatility period is set to be three years, and one year is assumed to consist of 252 trading days. The following steps are used to estimate $\sigma _F$ , $\sigma _I$ , and $\sigma _{FI}$ :

1. 1. Calculate day-to-day returns $DR_F$ and $DR_I$ of the fund and benchmark, respectively.

2. 2. Calculate the standard deviation of the fund return $\sigma _{DRF}$ , benchmark return $\sigma _{DRI}$ , and their correlation $\sigma _{DRFI}$ during the historical volatility period.

3. 3. Annualise the standard deviations based on the selected number of trading days which is 252 days, that is, $\sigma _{F}=\sigma _{DRF}\sqrt{252}$ , $\sigma _{I}=\sigma _{DRI}\sqrt{252}$ , and $\sigma _{FI}=\sigma _{DRFI} (252)$

For $\sigma _{F-I}$ , we adopt the formula below: $\sigma _{F-I}=\sqrt{\sigma ^2_F+\sigma ^2_I-2\sigma _F\sigma _I\sigma _{FI}}$

For models $\mathcal{M}_0$ and $\mathcal{M}_1$ , we provide two scenarios (low and high estimates) of volatilities and correlations to analyse the sensitivity of fees under the different financial market conditions. For models $\mathcal{M}_2$ , $\mathcal{M}_3$ , and $\mathcal{M}_4$ , one more extreme scenario is added to analyse the effect of CAP on model $\mathcal{M}_3$ .

The values of $\sigma _{F}$ are provided in Table 4, while values for $\sigma _{I}$ , $\sigma _{FI}$ , and $\sigma _{F-I}$ are provided in Table 5.

7.2 Jump component parameter estimation and scenario specifications

In this section, the jump diffusion model calibration and scenario specifications are discussed. The focus will be on the discretised version in Equation (7), and the procedure adopted for model estimation. There are numerous approaches proposed in the literature to calibrate a jump diffusion model, see Cont & Tankov (Reference Cont and Tankov2004) and Schoutens et al. (Reference Schoutens, Simons and Tistaert2003). In this work, we have opted to use a framework developed in Matlab version “9.12.0.1884302 (R2022a)” framework (https://www.mathworks.com/company/newsletters/articles/estimating-market-implied-value-with-jump-diffusion-models.html) with adaption to the small time scale approximation considered in Equation (7).

For the jump diffusion models, we have based the diffusion components of these models on the method of moments results discussed previously. Below we will discuss how to estimate the parameters for the jump components. Under this estimation procedure, one first segments the return series into change point segmentations, see the optimal selections of the estimated segmentation’s obtained in each time series in Figure 2. One has a choice to segment the price process according to the trend changes induced by a jump or to alternatively work with the detrended log-returns process via a segmentation of volatility changes (std. dev.) or Root-Mean-Square error changes. We utilised a detrending of the price process, but explored the log-returns segmentation also and found very similar results in the location of detected jumps. The number of optimal change points detected were explored between 1 and 50 changes, and the optimal choices are presented in the figure for each NAV data set. The criterion used for selecting optimal change point counts was an AIC criterion on the fitted local trend models as depicted and the optimal segmentation order was selected to minimised the AIC.

Figure 2 Jumps detected in fund and benchmark NAVs.

For the jump diffusion models, $\lambda _F\gt 0$ , $\lambda _I\gt 0$ were estimated based on the number of detected jumps annually which produced and average annualised jump intensity for each fund estimated to be: Europe Research Enhanced $\lambda _F = 0.003$ ; Income Opportunity $\lambda _F = 0.0033$ ; Income Opportunity Plus $\lambda _F = 0.008$ ; and benchmark MSCI Europe Index $\lambda _F = 0.012$ . These values were then used to set the probability of $B=1$ in small time scale representation Equation (7). Furthermore, the distribution of the jump sizes over time is plotted in the following Figure 3. The probability of an upward move ( $p$ ) was estimated to be: Europe Research Enhanced $p=0.2$ ; Income Opportunity $p=0.7$ ; Income Opportunity Plus $p=0.3$ ; and benchmark MSCI Europe Index $p=0.5$ . The intensities for the jumps were estimated to be: Europe Research Enhanced $\eta _1=6,\eta _2=1.2$ ; Income Opportunity $\eta _1=1.4,\eta _2=3.3$ ; Income Opportunity Plus $\eta _1=3,\eta _2=1.5$ ; and benchmark MSCI Europe Index $\eta _1=2.2,\eta _2=1.8$ .

Figure 3 Distribution of jump sizes. Top left panel: Europe Research Enhanced; Top right panel: Income Opportunity; Bottom left panel: Income Opportunity Plus; Bottom right panel: MSCI Europe Index Benchmark.

One can observe from this that for some NAV processes there was a greater negative support and for others a greater positive support in jumps, so there is no clear consensus in this case study as to a strictly greater magnitude of downward moves and chance of downward moves in these markets in general. We note that compared to other studies that have been undertaken in equity markets that tend to indicate a greater support for larger negative jumps as compared to positive jumps this is a difference for these markets under study in this work, see discussions in Kou (Reference Kou2007) and (Reference Kou2002). We suspect such findings will highly depend on the window of time studied and assets studied, and there is no definitive consensus that $p \gt 0.5$ in general or that the size of negative jumps should be greater than positive jumps, that is $\frac{1}{\eta _1} \lt \frac{1}{\eta _2}$ for all periods and market cycles.

We now point out that the analysis performed using these estimated parameter settings produced results that were very closely aligned with those obtained from the pure diffusion setting, due to the low chance of having any simulated jumps in the price process and the fact that when a jump was simulated, they did not lead to a significant change in the outcomes of the findings relative to the diffusion setting. Due to space, these results were omitted as they did not change the perspective learnt on fee fund structure and pricing behaviour compared to those already discovered from the diffusion model analysis.

We have therefore just focused on what we define as an extreme scenario for jump diffusion models which represents a case where jumps were more consequential in affecting the outcomes. This will better provide an indication of the influence of jumps in a market environment in which jumps had greater impact. For instance, this may be consistent with time periods like those in March 2020 at the start of the COVID-19 pandemic or consistent with those in May-June 2022 as inflation is rising and the Federal Reserve has significantly increased interest rates compared to historical changes, leading to either very large declines in asset prices or increases in asset prices as was the case when stimulus support was announced after the initial COVID-19 lockdowns occurred.

Consequently, in the scenarios developed we decided to simulate the chance of an upward or downward move as equal in magnitude for upward or downward moves, setting $p=0.5$ .

8.1 Results: Pricing method $1$ (risk-neutral & no jumps)

We conduct three-year studies for the models with the selected fund. For each study, we price performance fee PF $\%$ , management and advisory fee MF $\%$ (shown in square brackets), operating and administrative fee OF $\%$ (shown in parentheses) and tracking error $\%$ (shown in angle brackets). Furthermore, we also undertake an analysis of how the fee performance relates to profit performance. In this regard, we provide results that determine the return on investment ROI $\%$ (shown in curly brackets).

8.1.1 Models $\mathcal{M}_0$ and $\mathcal{M}_1$ on Europe Research Enhanced Index

Under pricing method 1, Figure 4 shows an example of simulated realisations of discounted cumulative performance fee for model $\mathcal{M}_0$ (left) and $\mathcal{M}_1$ (right) calculated by assuming risky interest rate, low fund volatility, low benchmark volatility, and low correlation as specified according to settings in section 6. Interestingly, the cumulative performance fee under model $\mathcal{M}_1$ (right) is not monotonically increasing during the year like in case of model $\mathcal{M}_0$ (left) before being reset after 252 days as the performance fee is paid. This may be caused by the symmetric-penalised structure of the claw-back mechanism of model $\mathcal{M}_1$ . Furthermore, the results of all scenarios for model $\mathcal{M}_0$ are provided in Table 9 in the first set of columns, and those for model $\mathcal{M}_1$ are provided in Table 9 in the second set of columns. For both models $\mathcal{M}_0$ and $\mathcal{M}_1$ , annual MFs and OFs do not vary significantly and are close to 0.19 $\%$ and 0.16 $\%$ which is consistent with what is observed in practice for these funds. Note that, from the tables, the PFs under the same scenario of the two examples are $5.97\%$ and $0.79\%$ , respectively, which is consistent with the daily cumulative fee average of the two models coloured in blue in Figure 4.

Figure 4 Present value of cumulative performance fee (red) of Europe Research Enhanced Index Equity simulated via pricing method 1 together with the daily cumulative fee average (blue) plotted over 756 trading days (3 years).

As for PFs for $\mathcal{M}_0$ model, the values can vary in range $5-7\%$ . The highest performance fee at $6.87\%$ falls into the scenario of high volatility (for both fund and benchmark), but low correlation under a negative discount rate. Nonetheless, the lowest performance fee at $5.71\%$ falls into the scenario of low volatility (for both fund and benchmark), but high correlation under high positive discount rate. By assuming ceteris paribus and varying only one variable at a time: either lowering correlations; increasing fund volatility; or increasing benchmark volatility, all possible individual changes result in higher PFs.

Regarding the returns of investment ROIs for $\mathcal{M}_0$ model, the values fall in range $8-9\%$ . The highest ROI at $8.72\%$ given by the scenario of low fund volatility and low correlation, but high benchmark volatility under either positive or negative discount rates. On the contrary, the lowest ROI at $8.64\%$ is given by the scenario of high fund volatility and high correlation, but low benchmark volatility under either positive or negative discount rates. By assuming ceteris paribus and varying only one variable at a time: either lowering fund volatility, lowering correlation, or increasing benchmark volatility, all possible individual changes result in higher ROIs. When considering the cost of investment relative to profit (PF per ROI), we found that the relative cost follows the PF trend, that is, highest at $79.31\%$ when PF reaches maximum at $6.87\%$ with ROI $8.66\%$ and vice versa. Interestingly neither the highest PF nor the lowest one lead to the highest ROI.

This set of results therefore demonstrates that in the risk-neutral setting in which one models the NAV processes as pure diffusion models then the following takeaway considerations are important to the investor, regarding the impact of performance fees. Firstly, performance fees in the range of 5–7% are very substantial in nature, compared to just a simple buy and hold strategy which would yield around 10% returns in one simply bought an ETF on say the S&P500. In particular, by considering the plausible range of ROIs between $8-9\%$ , the ratios of PF per ROI fall between $65-80\%$ which is indeed extremely high for investors. This means most of investment profit is paid to a fund manager in terms of performance fee. Furthermore, the results show that when the fund NAV is increasingly uncorrelated with the reference index fund NAV performance, this will result in greater expense to the investor over the investment horizon. This indicates that investors would be best suited to select funds for which the reference index was suitably related and therefore correlated with the assets comprised in the investment fund. The subsequent findings, that increasing periods of volatility lead to higher performance fees, should only be acceptable to investors if the fund manager is able to ensure that either funds grow sufficiently in value during heightened volatility periods due to active management investment decisions or alternatively that invested capital is secured against large potential draw downs in highly volatile periods, where the investment fund manager may adopt a risk off investment attitude and therefore the higher performance fees indicate a reward for such fund stewardship.

As for PFs and ROIs, although the values of PFs for $\mathcal{M}_1$ model varied from $0.8-0.9\%$ , which are drastically lower than the PFs for model $\mathcal{M}_0$ (factor of 10 times lower), the ROIs for $\mathcal{M}_1$ model varied from $8.4-8.5\%$ which were only $3\%$ lower than the ROIs for model $\mathcal{M}_0$ . This leads to a huge drop in the relative cost of investment (PF per ROI) for model $\mathcal{M}_1$ to the range $8.9-10.8\%$ (factor of 10 times lower than one for model $\mathcal{M}_0$ ). The same scenario which generated the highest PF for model $\mathcal{M}_0$ also yields both the highest PF and PF/ROI at $0.9\%$ and $10.75\%$ , respectively, for model $\mathcal{M}_1$ . That also holds true in case of the lowest PF and PF/ROI at $0.76\%$ and $8.94\%$ respectively.

8.1.2 Models $\mathcal{M}_2$ , $\mathcal{M}_3$ , and $\mathcal{M}_4$ on Income Opportunity Plus

An example of the movement of discounted cumulative performance fee over 3 years for model $\mathcal{M}_2$ (left), $\mathcal{M}_3$ (middle), and $\mathcal{M}_4$ (right) when interest rate is high and fund volatility is extremely high is shown in Figure 5. Similar to model $\mathcal{M}_1$ , the cumulative performance fee under models $\mathcal{M}_2$ , $\mathcal{M}_3$ , and $\mathcal{M}_4$ are all not monotonically increasing during each year before being reset to zero after 252 days. This results from symmetric-penalised structure of the claw-back mechanism which is used as the baseline mechanism of the three models. Interestingly, in Figure 7(c), the first-year PF of model $\mathcal{M}_4$ paid at the end of the year is above $1.5\%$ which is significantly higher than PFs of the later two years which are clearly lower than $0.5\%$ . This may be due to the effect of HWM condition on the first year which may set the watermark to be too high to catch in the following years.

Figure 5 Present value of cumulative $\%$ performance fees (red) of Income Opportunity Plus simulated via pricing method 1 together with the daily cumulative $\%$ fee average (blue) plotted over 756 trading days (3 years).

In addition, Table 10 reveals the results for model $\mathcal{M}_2$ , $\mathcal{M}_3$ , and $\mathcal{M}_4$ when applied to the Income opportunity Plus fund, under pricing method 1. It is obvious that MFs and OFs remain nearly stable across all scenarios around $0.55\%$ and $0.11\%$ , respectively, which are consistent with the rates the company promised in the fund investor information.

By considering results under different estimated values of volatility (low and high), PFs are varied from $0.1-0.3\%$ for models $\mathcal{M}_2$ , $\mathcal{M}_3$ , and $\mathcal{M}_4$ . In fact, $\mathcal{M}_2$ , $\mathcal{M}_3$ give exactly the same results under these two values (low and high) volatility. $\mathcal{M}_4$ yields a bit higher PFs than $\mathcal{M}_2$ in all scenarios. Again, by assuming Ceteris Paribus and varying only one variable, either an increase in fund volatility or a decrease in discount rate lead to an increase in PFs for all scenarios. Under volatility values estimated from the NAV data, we see no effect of CAP on model $\mathcal{M}_3$ ; however, once the volatility was increased to reach $10\%$ , under the same scenario, PF for model $\mathcal{M}_3$ are much lower than one for model $\mathcal{M}_2$ and $\mathcal{M}_4$ .

As for ROIs and PF/ROIs for models $\mathcal{M}_2$ , $\mathcal{M}_3$ and $\mathcal{M}_4$ , the ROIs and PF/ROIs varied from $1.23-1.24\%$ and $12.5-21.2\%$ , respectively, under these two values (low and high) volatility. In fact, the ROI results of models $\mathcal{M}_2$ – $\mathcal{M}_4$ are very close to each other (different by only $0.01-0.02\%$ ). Again once the volatility was increased to reach $10\%$ , under the same scenario, by considering the relative cost of investment (PF/ROI), it reveals that the effect of CAP does reduce the relative cost of model $\mathcal{M}_3$ significantly when we compare the model relative cost against ones for model $\mathcal{M}_2$ and $\mathcal{M}_4$ .

8.2 Results: Pricing method 2 (actuarial distortion pricing & jumps)

As in the previous section, we again conduct three-year studies for the models with the selected fund. For each study case, we price performance fee PF $\%$ , management and advisory fee MF $\%$ (shown in square brackets), operating and administrative fee OF $\%$ (shown in parentheses), and also provide return on investment ROI $\%$ (shown in curly brackets) under pricing method 2. Results for these case studies are developed in the following section.

8.2.1 Models $\mathcal{M}_0$ and $\mathcal{M}_1$ on Europe Research Enhanced Index

By adopting the pricing method 2, an example of realisations of discounted cumulative performance fee for model $\mathcal{M}_0$ (left) and $\mathcal{M}_1$ (right), calculated by assuming risky interest rate, low fund volatility, low benchmark volatility, and low correlation, is provided in Figure 6. Similar to the example of model $\mathcal{M}_1$ under pricing method 1, the cumulative performance fee shows non-monotonically increasing behaviour during each year which is directly a result of the penalty structure of the claw-back mechanism. The results for all scenarios are given in Table 11. For both models $\mathcal{M}_0$ and $\mathcal{M}_1$ , annual MFs and OFs are both larger than the rates indicated in the prospectus at 0.19 $\%$ and 0.16 $\%$ , respectively. MF values were in the range $0.22-0.24\%$ , and OF values were in the range $0.19-0.20\%$ which are quite stable regardless of any changes in the scenarios explored or fee models studied.

Figure 6 Present value of cumulative % performance fee (red) of Europe Research Enhanced Index Equity simulated via pricing method 2 together with the daily cumulative % fee average (blue) plotted over 756 trading days (3 years).

Figure 7 Present value of cumulative % performance fee (red) of Income Opportunity Plus simulated via pricing method 2 together with the daily cumulative % fee average (blue) plotted over 756 trading days (3 years).

Under pricing method 2, the values of PFs for $\mathcal{M}_0$ model were found to be between $7\%$ and $9\%$ , while those for $\mathcal{M}_1$ model were between $1.3\%$ and $1.5\%$ . All PFs derived from the two models under pricing method 2 are significantly larger than ones under pricing method 1. As for model $\mathcal{M}_0$ , the highest performance fee at $8.58\%$ falls into the scenario of high volatility (of both fund and benchmark), but low correlation under a negative discount rate. The lowest performance fee is at $7.25\%$ derived from the scenario of low volatility (for both fund and benchmark), but high correlation under high positive discount rate. Ceteris Paribus the assumptions adopted, if we vary only one variable which is either lowering correlation, increasing fund volatility, or increasing benchmark volatility, all yield higher PFs. This pattern of PFs is consistent with what we found in the results of model $\mathcal{M}_0$ under pricing method 1.

For PFs of $\mathcal{M}_1$ model under pricing method 2, the values of PFs varied between $1.3-1.5\%$ which was twice as large as those under pricing method 1. Under the same pricing method 2, we found a similar pattern in the PFs, namely that $\mathcal{M}_1$ was much lower than those obtained for $\mathcal{M}_0$ (7 times lower). Indeed, the behaviour of PFs of $\mathcal{M}_1$ is the same as the pattern we found in the case of pricing method 1. Note that the highest PF is at $1.49\%$ and the lowest was at $1.34\%$ . Both extremes appeared in the same scenarios as in the case of model $\mathcal{M}_0$ .

As for the ROIs of $\mathcal{M}_0$ and $\mathcal{M}_1$ models, the ROI values are slightly varied between $17.30-17.33\%$ and $17.02-17.04\%$ , respectively. However, when considering the cost of investment relative to profit (PF per ROI), PF/ROI for $\mathcal{M}_0$ model is at least five times higher than one for $\mathcal{M}_1$ model given the same scenario.