# The price of liquidity in the reinsurance of fund returns

## David Saunders, Luis Seco and Markus Senn

#### Abstract

We consider a new type of contract for insuring the returns of hedge funds and aim to extend downside protection to an investment portfolio beyond the first tranche of losses insured by first-loss fee structures, which have become increasingly popular in the market. By considering a second tranche, we suggest an up-front premium to a reinsurance party, in exchange for which the investor gains full protection against all losses, not just those occurring in the first tranche. We identify a fund’s underlying liquidity as a key parameter in deriving the price for the additional reinsurance and provide a method for computing the premium using two approaches: an analytic closed-form solution based on the Black–Scholes framework, and a numerical simulation using a Markov-switching model. In addition, a simplified backtesting method is implemented to evaluate the practical application and performance of the concept for both mathematical models.

## 1 Introduction

The low interest rate environment, an uncertain and fragile political climate and cost pressures are the main drivers of today’s challenging market environment. It is increasingly difficult for hedge fund managers to justify high fees for return performances that tend not to be superior to alternative low-cost investment opportunities, eg, exchange-traded funds. A hedge fund’s capability to deliberately incorporate more risk by either adding new risk sources or including riskier positions is an attractive alternative for investors wishing to diversify their existing investment portfolio. Nonetheless, weak performance, more difficult and volatile markets, and pressure to raise assets – combined with investors’ calls for better alignment of interest – have resulted in new fee structures in the hedge fund industry. Managers offer downside protection by insuring hedge fund investments in return for a bigger share in generated profits. These fee structures are commonly referred to as “first loss”, “shared loss” or “shared profit”.

In traditional fee structures, the manager receives a flat 2% management fee and a performance fee of 20% of net profits (known as “2 and 20”). Even though the “2 and 20” concept is commonly applied, other fixed and variable fee combinations are possible, eg, Escobar et al (2018) suggest an optimal equilibrium in management and performance fees by optimizing the Sharpe ratio to maximize the utility of both manager and investor alike. Nonetheless, new fee structures in the form of first-loss schemes are emerging. There are several versions of the first-loss principle, but the basic framework has the following feature: the manager provides downside protection by covering the first tranche of portfolio losses in return for a higher participation rate in the case of a positive fund performance. In contrast to traditional fee structures, a first-loss structure might allow a manager to raise the performance fee to 50% by insuring losses of up to 10% of the initial investment, ie, providing protection for the first tranche of portfolio losses. He and Kou (2018) derive trading strategies with a first-loss structure based on cumulative prospect theory and conclude that certain parameter combinations result in increasing utility for both the manager and the investor while reducing the risk of an investment portfolio. However, in other cases, the traditional fee structure produces higher investor utility. Djerroud et al (2016) price the contract using risk-neutral valuation, thereby obtaining the “fair” level of the performance fee. Bhaduri et al (2018) extend the traditional first-loss principle by introducing a minimum return guarantee in addition to covering the first tranche of losses in a portfolio. The investment resembles a coupon bond, with a guaranteed minimum return (fixed part) and performance-dependent return net of accrued fees (variable part).

Independent of applied fee structures, hedge funds are often subject to so-called liquidation barriers. These barriers can be introduced exogenously, ie, they are investor demanded, or endogenously, ie, the manager sets an internal barrier. They are a commonly used option to protect investors from negative fund performances. Once poorly performing funds breach the liquidation barrier, their operations shut down to prevent further losses for investors. Hodder and Jackwerth (2007) analyze the effects of a shutdown or liquidation barrier on the risk-taking behavior of the manager as another option to provide downside protection for the investor. They find a significant change in the willingness to take on riskier positions when the fund performance is close to the barrier.

Our approach aims to extend the protection beyond the first tranche of losses by considering a second one that insures the investor against all losses, not just the first level. The interest in this second layer of protection is manifold. On the one hand, investors may have a different appetite for the second layer of losses from a third-party reinsurer, thus creating a financial transaction between investor and reinsurer. On the other hand, even if investors’ risk appetites lead them to accept this second tranche of losses, or if reinsurance cost is deemed too high, there may be regulatory reasons for this tranche to be outsourced to a third party, eg, if the investor is an insurance company subject to capital ratios linked to a loss profile that regulators may penalize in excess of the external reinsurance cost. In any case, while most related research focuses on optimizing fee structures in a first-loss framework, little work has been done on protecting losses greater than the first tranche. We address the subject of providing additional downside protection that goes beyond the insurance of the first tranche of losses.11 1 A similar payoff in a different financial contract is afforded to investors in variable annuities, who may receive a minimum guaranteed rate of return, together with fractional participation in the upside of an investment portfolio managed by an insurance company.

Theoretically, by introducing a liquidation barrier at the lower end of the first tranche, the investor gains full portfolio protection. In this setting, the liquidation is triggered as soon as the portfolio value breaches the barrier, in which case the manager compensates for all resulting losses. In practice, however, the investor is exposed to gap risk arising from the inability of the manager of the investment structure to liquidate the portfolio before losses exceed the protection level of the first tranche. We should point out that the aim of this paper is to model not the gap risk itself, but rather its impact on portfolio insurance, where the effects of gap risk and other liquidity restrictions are similar. We will still refer to it as gap risk, and discuss the associated risk premiums. The risk premium that is relevant in the valuation of the second loss tranche, ie, pricing further downside protection, should reflect this exposure to gap risk. Two related market events can give rise to this exposure.

1. (1)

A situation where loss momentum is higher than the manager’s ability to liquidate the portfolio in a timely manner: the fund is exposed to price fluctuations from the moment the fund value first hits the barrier (at which point the manager is aware of the situation and starts the liquidation of all assets) up to the end of the liquidation process.

2. (2)

A situation where the liquidity of the underlying assets is low, to the point that an orderly liquidation of the portfolio extends in time beyond the liquidation barrier and assets can continue to lose value: whereas highly liquid assets (eg, most stocks, futures or certain bonds) can be liquidated within seconds or minutes to minimize the gap risk, illiquid assets (eg, real estate or certain over-the-counter products) take time to liquidate and are exposed to market risk over a longer period of time. As the manager only covers losses up to the barrier in both cases, the investor is exposed to the gap risk and losses can exceed the first tranche.

We analyze a contract in which an investor pays an up-front premium to a reinsurance party and in exchange gains full portfolio protection net of fees. The contribution of this paper is a method of computing the premium for the reinsurance as a function of the liquidity of the underlying portfolio. Assuming risk-neutral valuation and a geometric Brownian motion (GBM) for the value of the underlying portfolio, we derive a closed-form expression for the premium. In addition to the analytical approach, we conduct a numerical simulation using a Markov-switching model as an alternative. Further, we provide some sensitivity analysis with respect to certain market parameters and variations of the initial setting.

The remainder of the paper is structured as follows. Section 2 elaborates on established first-loss concepts and introduces our modifications. In Section 3, we consider the analytical model using a GBM and derive a closed-form solution. Section 4 contains a simulation in a Markov-switching model. In Section 5, we present our numerical results. Section 6 is dedicated to backtesting the introduced models. Section 7 states our conclusions.

## 2 Hedge fund fee structures

### 2.1 Established first-loss framework

First-loss schemes were established around a decade ago but have more recently found rising interest.22 2 See Barr (2011) about rising interest in first-loss schemes in recent years and Weiss (2018) for a Bloomberg article on recent first-loss related news. Particularly in the aftermath of the financial crisis, managers lacked investor confidence. The first-loss principles not only signal a manager’s dedication, as their own funds are at stake, but also allow smaller funds and those without a long track record to attract new investors and quickly grow assets in the early seeding phase. However, the first-loss scheme comes with a price. The manager provides downside protection by covering the first tranche of portfolio losses in return for a higher participation rate in the event of a positive fund performance. Whereas in traditional fee structures the manager receives a flat 2% management fee and a 20% performance-dependent fee on net profits (known as “2 and 20”), a first-loss structure might allow a manager to raise the performance fee to 50% by insuring losses of up to 10% of the initial investment.

There exists a variety of first-loss variations. In this paper, we follow the framework of Djerroud et al (2016) closely. The fund value $X_{t}$ consists of two parts: the manager’s part $V_{\mathrm{M}}(t)$, and the investor’s part $V_{\mathrm{I}}(t)$, where $X_{t}=V_{\mathrm{I}}(t)+V_{\mathrm{M}}(t)$. The payoffs at the terminal time $T$ are defined as

 $\displaystyle V_{\mathrm{I}}(T)$ $\displaystyle\!=\!\begin{cases}X_{T}-m_{\mathrm{M}}X_{0}-\alpha_{\mathrm{M}}(X% _{T}-m_{\mathrm{M}}X_{0}-X_{0})^{+},&X_{T}\geq X_{0},\\ (1-m_{\mathrm{M}})X_{0},&(1-c_{\mathrm{M}})X_{0}\leq X_{T}\leq X_{0},\\ X_{T}+(c_{\mathrm{M}}-m_{\mathrm{M}})X_{0},&X_{T}\leq(1-c_{\mathrm{M}})X_{0},% \end{cases}$ $\displaystyle V_{\mathrm{M}}(T)$ $\displaystyle\!=\!\begin{cases}m_{\mathrm{M}}X_{0}+\alpha_{\mathrm{M}}(X_{T}-m% _{\mathrm{M}}X_{0}-X_{0})^{+},&X_{T}\geq X_{0},\\ m_{\mathrm{M}}X_{0}-(X_{0}-X_{T}),&(1-c_{\mathrm{M}})X_{0}\leq X_{T}\leq X_{0}% ,\\ m_{\mathrm{M}}X_{0}-c_{\mathrm{M}}X_{0},&X_{T}\leq(1-c_{\mathrm{M}})X_{0},\end% {cases}$

or, in a more condensed form, as

 $\displaystyle V_{\mathrm{I}}(T)$ $\displaystyle=X_{T}-m_{\mathrm{M}}X_{0}-\alpha_{\mathrm{M}}[X_{T}-m_{\mathrm{M% }}X_{0}-X_{0}]^{+}$ $\displaystyle\qquad+[X_{0}-X_{T}]^{+}-[(1-c_{\mathrm{M}})X_{0}-X_{T}]^{+},$ (2.1) $\displaystyle V_{\mathrm{M}}(T)$ $\displaystyle=m_{\mathrm{M}}X_{0}+\alpha_{\mathrm{M}}[X_{T}-m_{\mathrm{M}}X_{0% }-X_{0}]^{+}$ $\displaystyle\qquad-[X_{0}-X_{T}]^{+}+[(1-c_{\mathrm{M}})X_{0}-X_{T}]^{+},$ (2.2)

where $X_{0}$ is the initial investment, $\alpha_{\mathrm{M}}$ is the performance-dependent fee, $m_{\mathrm{M}}$ is the flat management fee and $c_{\mathrm{M}}$ is the managerial deposit, ie, $(1-c_{\mathrm{M}})X_{0}$ is the lower level of the first tranche insurance. In other words, in the event of negative performance the manager compensates the investor with up to $c_{\mathrm{M}}$ of the initial investment.

The payoff functions form the base case of our derivative-pricing framework. The investor holds a long position representing their investment, ie, the fund’s assets net of fees ($X_{T}-m_{\mathrm{M}}X_{0}$), a short position in a call option on the fund’s assets with strike $m_{\mathrm{M}}X_{0}+X_{0}$ representing potential performance-dependent fees, a long position in a put option on the fund’s assets with strike $X_{0}$ representing first-loss insurance by the manager and a short position in a put option on the fund’s assets with strike $(1-c_{\mathrm{M}})X_{0}$ representing the cap on the manager’s deposit, ie, the manager’s limited liability in the event of a negative fund performance beyond the first-loss protection. The manager holds the respective counterparts, ie, a long call option, a long put option and a short put option.

### 2.2 Modified framework: reinsurance

Theoretically, by introducing a liquidation barrier at the level of the lower end of the managerial deposit (or first tranche of losses), ie, at $K=(1-c_{\mathrm{M}})X_{0}$, the investor gains full portfolio protection. In this setting, the liquidation is triggered as soon as the portfolio value breaches the barrier, in which case the manager compensates for all resulting losses. As noted above, however, in practice the investor is exposed to gap risk owing to potential delays when liquidating the portfolio, eg, due to market conditions or the illiquid nature of the instruments in the investment portfolio.

Consider the situation in which the investor pays a premium to a third party reinsurer in return for protection from losses not covered by the manager. Denote the value of the reinsurer’s position by $V_{\mathrm{R}}(t)$. The fund value now follows $X_{t}=V_{\mathrm{I}}(t)+V_{\mathrm{M}}(t)+V_{\mathrm{R}}(t)$, where the payoff functions are defined as follows:

 $\displaystyle V_{\mathrm{I}}(T)$ $\displaystyle=\begin{cases}X_{T}-(m_{\mathrm{M}}+m_{\mathrm{R}}\mathrm{e}^{rT}% )X_{0}-\alpha_{\mathrm{M}}(X_{T}-m_{\mathrm{M}}X_{0}-X_{0})^{+},&X_{T}\geq X_{% 0},\\ (1-m_{\mathrm{M}}-m_{\mathrm{R}}\mathrm{e}^{rT})X_{0},&X_{T}\leq X_{0},\end{cases}$ $\displaystyle V_{\mathrm{M}}(T)$ $\displaystyle=\begin{cases}m_{\mathrm{M}}X_{0}+\alpha_{\mathrm{M}}(X_{T}-m_{% \mathrm{M}}X_{0}-X_{0})^{+},&X_{T}\geq X_{0},\\ m_{\mathrm{M}}X_{0}-(X_{0}-X_{T}),&(1-c_{\mathrm{M}})X_{0}\leq X_{T}\leq X_{0}% ,\\ m_{\mathrm{M}}X_{0}-c_{\mathrm{M}}X_{0},&X_{T}\leq(1-c_{\mathrm{M}})X_{0},\end% {cases}$ $\displaystyle V_{\mathrm{R}}(T)$ $\displaystyle=\begin{cases}m_{\mathrm{R}}\mathrm{e}^{rT}X_{0},&X_{T}\geq(1-c_{% \mathrm{M}})X_{0},\\ m_{\mathrm{R}}\mathrm{e}^{rT}X_{0}-((1-c_{\mathrm{M}})X_{0}-X_{T}),&X_{T}\leq(% 1-c_{\mathrm{M}})X_{0},\end{cases}$

where $X_{0}$ is the initial investment; $\alpha_{\mathrm{M}}$ is the performance-dependent fee; $m_{\mathrm{M}}$ is the flat management fee; $c_{\mathrm{M}}$ is the managerial deposit, ie, $(1-c_{\mathrm{M}})X_{0}$ is the lower level of the first tranche insurance; $m_{\mathrm{R}}$ is the up-front premium for the full portfolio insurance; and $r$ is the risk-free interest rate. Written more compactly,

 $\displaystyle V_{\mathrm{I}}(T)$ $\displaystyle=X_{T}-(m_{\mathrm{M}}+m_{\mathrm{R}}\mathrm{e}^{rT})X_{0},$ $\displaystyle\qquad-\alpha_{\mathrm{M}}[X_{T}-m_{\mathrm{M}}X_{0}-X_{0}]^{+}+[% X_{0}-X_{T}]^{+},$ (2.3) $\displaystyle V_{\mathrm{M}}(T)$ $\displaystyle=m_{\mathrm{M}}X_{0}+\alpha_{\mathrm{M}}[X_{T}-m_{\mathrm{M}}X_{0% }-X_{0}]^{+}$ $\displaystyle\qquad-[X_{0}-X_{T}]^{+}+[(1-c_{\mathrm{M}})X_{0}-X_{T}]^{+},$ (2.4) $\displaystyle V_{\mathrm{R}}(T)$ $\displaystyle=m_{\mathrm{R}}\mathrm{e}^{rT}X_{0}-[(1-c_{\mathrm{M}})X_{0}-X_{T% }]^{+}.$ (2.5)

Note that we assume the investor cannot redeem their investment during the investment horizon, eg, due to an imposed lockup period, even though it might be optimal to prematurely withdraw from the arrangement from the investor’s point of view (see Chen et al (2016) and Meng and Saunders (2021), who analyze optimal stopping times). The payoff functions (2.3)–(2.5) extend the functions in (2.1) and (2.2). The short position representing the cap on the manager’s deposit is transferred from the investor to the reinsurer, ie, the investor gains full portfolio protection net of fees. However, the investor is subject to another future liability representing the premium ($m_{\mathrm{R}}\mathrm{e}^{rT}X_{0}$). Despite the full portfolio protection, the investor now faces a certain credit risk, as they now rely on the financial solvency of the reinsurance in the event of a negative fund performance. We assume no default and thus neglect the credit risk. Note that from the manager’s perspective nothing changes, as (2.2) = (2.4). The transaction involves only the investor and reinsurer.

As this paper focuses on the up-front premium paid by the investor, we hereafter focus on the payoff of the reinsurance in (2.5). Note that the payoff function in (2.5) is defined assuming no liquidation barrier, ie, it is computed using the terminal fund value at the end of the investment horizon $T$. In order to incorporate the aforementioned gap risk and thus reflect a fund’s level of liquidity, we need to make some alterations to the payoff function of the reinsurance stated in (2.5). We set a liquidation barrier at the lower end of the first tranche, ie, at $K=(1-c_{\mathrm{M}})X_{0}$. If the fund value breaches this barrier at any time $\tau\in(0,T)$, all operations stop and the remaining assets are fully liquidated. Reflecting a fund’s underlying level of liquidity, we introduce the parameter $\varTheta$. As a measure of liquidity, $\varTheta$ is defined as the time a fund needs to liquidate its assets. The payoff function is defined as

 $\displaystyle V_{\mathrm{R}}(\tau+\varTheta)=m_{\mathrm{R}}\mathrm{e}^{r(\tau+% \varTheta)}X_{0}-[K-X_{\tau+\varTheta}]^{+},\quad\tau\in(0,T),$ (2.6)

where $\tau=\inf\{t>0;X_{t}\leq K\}$. In the event that the fund value does not breach the liquidation barrier, we have

 $[K-X]^{+}\lx@stackrel{{\scriptstyle X>K}}{{=}}0,$

ie, the reinsurer receives and keeps the full premium. The benchmark value we use is $\varTheta=\frac{1}{252}$, ie, one day until the full liquidation of the fund. In practice, the liquidation process of a fund depends heavily on the composition of its financial assets, and the liquidation time is different for each asset. In the following sections, we derive the price of the up-front premium depending on the fund’s liquidity.

## 3 Analytic valuation using geometric Brownian motion

Let the hedge fund performance $X=\{X_{t}\}_{t\in[0,T]}$ follow a GBM. Under the risk-neutral measure $\mathbb{Q}$, we have

 $\displaystyle\mathrm{d}X_{t}=X_{t}(r\mathrm{d}t+\sigma\mathrm{d}\tilde{W}_{t})% \implies X_{t}=X_{0}\exp((r-\tfrac{1}{2}\sigma^{2})t+\sigma\tilde{W}_{t}),$ (3.1)

where $r$ is the risk-free rate, $\sigma>0$ is the fund’s volatility and $\tilde{W}_{t}$ is a $\mathbb{Q}$-Brownian motion. As above, let $\tau:=\inf\{0 be the stopping time at which the fund value hits the barrier $K=(1-c_{m})X_{0}$. At time $\tau$, the fund is shut down and all its assets are liquidated. Using (3.1), we have

 $\displaystyle\tau$ $\displaystyle:=\inf\{0 (3.2)

where

 $Y_{t}=\underbrace{(r-\tfrac{1}{2}\sigma^{2})}_{\mu}t+\sigma\tilde{W}_{t}\quad% \text{and}\quad a=\ln\bigg{(}\frac{K}{X_{0}}\bigg{)}.$

For fixed $a$, the first hitting time $\tau$ in (3.2) follows an inverse Gaussian distribution (see Kwok 2008), $\tau\sim\operatorname{IG}(\mu^{*},\lambda^{*})$, where

 $\mu^{*}=\frac{a}{\mu},\qquad\lambda^{*}=\frac{a^{2}}{\sigma^{2}}$

and the density function $f$ is

 $\displaystyle f(\tau,\mu^{*},\lambda^{*})$ $\displaystyle=\sqrt{\frac{\lambda^{*}}{2\pi\tau^{3}}}\exp\bigg{(}{-}\frac{% \lambda^{*}(\tau-\mu^{*})^{2}}{2(\mu^{*})^{2}\tau}\bigg{)}$ $\displaystyle\qquad\iff f(\tau,a,\mu)=\frac{|a|}{\sigma\sqrt{2\pi\tau^{3}}}% \exp\bigg{(}{-}\frac{(a-\mu\tau)^{2}}{2\sigma^{2}\tau}\bigg{)}.$ (3.3)

We derive the value of the up-front premium $m_{\mathrm{R}}$ as a function of a fund’s liquidity by using the fair value approach stated in Djerroud et al (2016), where the fair value for the premium is obtained by setting the present value of the expected payoff equal to the initial investment, ie, $V_{\mathrm{R}}(0)=0$. In our setting, we have

 $\displaystyle V_{\mathrm{R}}(0)=\mathbb{E}_{\mathbb{Q}}[\mathrm{e}^{-r(\tau+% \varTheta)}V_{\mathrm{R}}(\tau+\varTheta)]=0.$ (3.4)

The payoff made by the reinsurer is that of a put option issued at time $\tau$ with strike $K=(1-c_{\mathrm{M}})X_{0}$ and exercise time $\tau+\varTheta$. For simplicity and without loss of generality, we set the initial investment to $X_{0}=1$. From (2.6), the terminal value of the reinsurer at time $\tau+\varTheta$ is thus

 $\displaystyle V_{\mathrm{R}}(\tau+\varTheta)=m_{\mathrm{R}}\mathrm{e}^{r(\tau+% \varTheta)}X_{0}-[K-X_{\tau+\varTheta}]^{+}=m_{\mathrm{R}}\mathrm{e}^{r(\tau+% \varTheta)}-[K-X_{\tau+\varTheta}]^{+}.$ (3.5)

At time $\tau$, the time of issuance of the put option, the value of the reinsurer’s position is

 $\displaystyle V_{\mathrm{R}}(\tau)=\mathbb{E}_{\mathbb{Q}}[\mathrm{e}^{-r% \varTheta}V_{\mathrm{R}}(\tau+\varTheta)\mid\mathcal{F}_{\tau}]=m_{\mathrm{R}}% \mathrm{e}^{r\tau}-V_{1}(\varTheta).$ (3.6)

We have $K=X_{\tau}$ by definition, as the option is issued when the fund breaches the barrier. Thus, $V_{1}(\varTheta)$ follows the well-known Black–Scholes option formula:

 $\displaystyle V_{1}(\varTheta)$ $\displaystyle=\mathbb{E}_{\mathbb{Q}}[\mathrm{e}^{-r\varTheta}[K-X_{\tau+% \varTheta}]^{+}\mid\mathcal{F}_{\tau}]=K(\mathrm{e}^{-r\varTheta}N(-d_{2})-N(-% d_{1})),$ $\displaystyle\qquad\text{with~{}}d_{1}=\frac{1}{\sigma}(r+\tfrac{1}{2}\sigma^{% 2})\sqrt{\varTheta},\quad d_{2}=d_{1}-\sigma\sqrt{\varTheta},$ (3.7)

where $N(\cdot)$ is the standard normal cumulative distribution function. Note that $V_{1}(\varTheta)$ does not depend on $\tau$, but depends solely on $\varTheta$, ie, $V_{1}(\varTheta)$ is a function of liquidity. At time $t=0$, the initial value of the reinsurer is

 $\displaystyle V_{\mathrm{R}}(0)$ $\displaystyle=\mathbb{E}_{\mathbb{Q}}[\mathrm{e}^{-r\tau}V_{\mathrm{R}}(\tau)% \mid\mathcal{F}_{0}]$ $\displaystyle=m_{\mathrm{R}}-V_{1}(\varTheta)\mathbb{E}_{\mathbb{Q}}[\mathrm{e% }^{-r\tau}\mid\mathcal{F}_{0}]$ $\displaystyle=m_{\mathrm{R}}-V_{1}(\varTheta)V_{2}(T),$ (3.8)

where

 $\displaystyle V_{2}(T)=\mathbb{E}_{\mathbb{Q}}[\mathrm{e}^{-r\tau}\mid\mathcal% {F}_{0}]=\int_{0}^{T}\mathrm{e}^{-rt}f(t)\mathrm{d}t.$ (3.9)

The above integral can be evaluated analytically (see Kwok 2008):

 $\displaystyle V_{2}(T)=\bigg{(}\frac{K}{X_{0}}\bigg{)}^{\!\alpha_{+}}N\bigg{(}% \delta\frac{\ln(K/X_{0})+\beta T}{\sigma\sqrt{T}}\bigg{)}+\bigg{(}\frac{K}{X_{% 0}}\bigg{)}^{\!\alpha_{-}}N\bigg{(}\delta\frac{\ln(K/X_{0})-\beta T}{\sigma% \sqrt{T}}\bigg{)},$ $\displaystyle~{}~{}\text{with~{}}\beta=\sqrt{(r-\frac{1}{2}\sigma^{2})^{2}+2r% \sigma^{2}},\quad\alpha_{\pm}=\frac{r-\tfrac{1}{2}\sigma^{2}\pm\beta}{\sigma^{% 2}},\quad\delta=\operatorname{sgn}\bigg{(}\ln\bigg{(}\frac{X_{0}}{K}\bigg{)}\!% \bigg{)}.$ (3.10)

We thus obtain

 $m_{\mathrm{R}}=V_{1}(\varTheta)V_{2}(T).$ (3.11)

The premium is composed of two parts: $V_{1}(\varTheta)$ represents the value of the issued option, and $V_{2}(T)$ incorporates the (discounted) probability of breaching the liquidation barrier throughout the entire investment horizon $T$.

Note that some practitioners might prefer results for a model that is fitted to historical data. Hence, we replace the risk-free drift with an empirical drift $b$, which implies that

 $\log\bigg{(}\frac{X_{t+\varDelta}}{X_{t}}\bigg{)}\sim N((b-\tfrac{1}{2}\sigma^% {2})\varDelta,\sigma^{2}\varDelta)=N(\mu_{\mathrm{emp}}\varDelta,\sigma_{% \mathrm{emp}}^{2}\varDelta)$

for a daily setting with $\varDelta=\frac{1}{252}$, as log returns are normally distributed. Best estimates for drift and volatility are $\hat{\sigma}=\sigma_{\mathrm{emp}}$ and $\hat{b}=\mu_{\mathrm{emp}}+\frac{1}{2}\sigma_{\mathrm{emp}}^{2}$, where

 $\mu_{\mathrm{emp}}=252\bigg{(}\frac{1}{N}\sum_{n=1}^{N}R_{n}\bigg{)}\quad\text% {and}\quad\sigma_{\mathrm{emp}}=\sqrt{\frac{252}{N-1}\sum_{n=1}^{N}(R_{n}-\mu_% {\mathrm{emp}})},$

with daily log return $R_{n}$. The empirical yearly returns $\mu_{\mathrm{emp}}$ and volatility $\sigma_{\mathrm{emp}}$ for selected hedge fund indexes (see Section 4 for details) can be found in Table 1.33 3 Estimating drift rates is notoriously difficult and nearly impossible to do in practice (see Merton (1980), the original paper on the problem). We follow a simple approach and use mean and standard deviation of log returns. We point out here that we do not price the contract using the risk-neutral measure $\mathbb{Q}$ in the following. The crucial assumptions needed to justify the replication arguments that underlie risk-neutral valuation might not hold in the hedge fund context; for example, as we specifically allow illiquid asset classes, the arbitrage-free dynamic hedging strategy underlying the classic Black–Scholes framework is rarely attainable. Therefore, we evaluate the present value of the expected payoff (represented by $V_{1}^{\mathbb{P}}(\varTheta)$ in (3.12)). Investors might be interested in this value, which is still available in closed form. For (3.7), we have

 $\displaystyle V_{1}^{\mathbb{P}}(\varTheta)$ $\displaystyle=\mathbb{E}_{\mathbb{P}}[\mathrm{e}^{-r\varTheta}[K-X_{\tau+% \varTheta}]^{+}\mid\mathcal{F}_{\tau}]=K(\mathrm{e}^{-r\varTheta}N(-d_{2})-% \mathrm{e}^{(b-r)\varTheta}N(-d_{1})),$ $\displaystyle\qquad\text{with~{}}d_{1}=\frac{(b+\frac{1}{2}\sigma^{2})\sqrt{% \varTheta}}{\sigma},\quad d_{2}=d_{1}-\sigma\sqrt{\varTheta},$ (3.12)

and for (3.10) we have

 $\displaystyle V_{2}^{\mathbb{P}}(T)$ $\displaystyle=\bigg{(}\frac{K}{X_{0}}\bigg{)}^{\!\alpha_{+}}N\bigg{(}\delta% \frac{\ln(K/X_{0})+\beta T}{\sigma\sqrt{T}}\bigg{)}+\bigg{(}\frac{K}{X_{0}}% \bigg{)}^{\!\alpha_{-}}N\bigg{(}\delta\frac{\ln(K/X_{0})-\beta T}{\sigma\sqrt{% T}}\bigg{)},$ (3.13)

with

 $\beta=\sqrt{(b-\tfrac{1}{2}\sigma^{2})^{2}+2r\sigma^{2}},\quad\alpha_{\pm}=% \frac{(b-\frac{1}{2}\sigma^{2})\pm\beta}{\sigma^{2}},\quad\delta=\operatorname% {sgn}\bigg{(}\ln\bigg{(}\frac{X_{0}}{K}\bigg{)}\!\bigg{)}.$

We thus obtain

 $\displaystyle m_{\mathrm{R}}^{\mathbb{P}}=V_{1}^{\mathbb{P}}(\varTheta)V_{2}^{% \mathbb{P}}(T).$ (3.14)

## 4 Valuation in a Markov-switching framework

We consider a Markov-switching model and evaluate the payoff of the reinsurance to compute the premium $m_{\mathrm{R}}$. Markov-switching or regime-switching models were first introduced by Hamilton (1989) and have since found regular application in the financial world. These models create returns similar to actual hedge fund returns and show typical hedge fund characteristics, ie, skewness, volatility clusters and fat tails. We assume a two-state Markov-switching model with state space $\mathbb{S}=\{1,2\}$ for the continuous-time Markov chain $\epsilon(t)$, where state 1 represents the “normal” market and state 2 the “stressed” market. $\epsilon(t)$ is generated by the intensity matrix

 $\displaystyle Q=\begin{bmatrix}-\lambda_{1}&\lambda_{1}\\ \lambda_{2}&-\lambda_{2}\end{bmatrix},$

where $\lambda_{1}$ and $\lambda_{2}$ are the transition rates for switching states. The fund value satisfies the following stochastic differential equation:

 $\displaystyle\mathrm{d}X_{t}=X_{t}(\mu_{\epsilon(t)}\mathrm{d}t+\sigma_{% \epsilon(t)}\mathrm{d}W_{t}),\quad X_{0}>0,$

where $\epsilon(t)\in\{1,2\}$, and $\mu_{\epsilon(t)}$ and $\sigma_{\epsilon(t)}$ represent the respective state’s return rate and volatility. In order to use the risk-neutral martingale pricing methodology, we need an equivalent martingale measure $\mathbb{Q}$. We follow the approach in Henriksen (2011) by using the Girsanov theorem and assume unchanged transition probabilities under the risk-neutral measure $\mathbb{Q}$ (see Bollen (1998) and Hardy (2001) for the same approach). The resulting price process for constant $r$ is given by

 $\displaystyle X_{t}=X_{0}\exp\bigg{\{}\int_{0}^{t}r-\tfrac{1}{2}\sigma_{% \epsilon(u)}^{2}\mathrm{d}u+\int_{0}^{t}\sigma_{\epsilon(u)}\mathrm{d}\tilde{W% }_{u}\bigg{\}},\quad X_{0}>0.$

For another approach, see Elliott et al (2005), where a regime-switching Esscher transform is applied to obtain an equivalent measure. Note that the Markov-switching modeled market is incomplete, as there is an additional stochastic dimension in the form of the regime-switching, and a unique $\mathbb{Q}$ measure cannot be determined (see, for example, Elliott and Swishchuk (2007) for a proof).

When performing simulations, we implement the following discretized version of the regime-switching Markov process on an equidistant grid:

 $\displaystyle X_{t_{i}}$ $\displaystyle=X_{0}\mathrm{e}^{R_{t_{i}}}\quad\forall i\in\{0,\dots,k\},~{}X_{% 0}>0,$ $\displaystyle R_{t_{i}}$ $\displaystyle=R_{t_{i-1}}+((r-\tfrac{1}{2}\sigma_{\tilde{\epsilon}(t_{i})}^{2}% )\varDelta+\sigma_{\tilde{\epsilon}(t_{i})}\sqrt{\varDelta}\eta_{t_{i}}),% \qquad R_{t_{0}}=0,$

where $0=t_{0}, $\varDelta\equiv\varDelta t_{j}:=t_{j+1}-t_{j}=T/k$, $\eta_{t_{i}}$ are independent and identically distributed standard normal random variables and $\tilde{\epsilon}(t)$ is a discretization of the continuous-state Markov chain $\epsilon(t)$ with transition matrix

 $\displaystyle P=\begin{bmatrix}(1-p)&p\\ q&(1-q)\end{bmatrix},$

where $p$ and $q$ are the transition probabilities for switching from one state to the other:

 $\displaystyle p$ $\displaystyle=\mathbb{P}(\tilde{\epsilon}(t_{i+1})=1\mid\tilde{\epsilon}(t_{i}% )=2),$ $\displaystyle q$ $\displaystyle=\mathbb{P}(\tilde{\epsilon}(t_{i+1})=2\mid\tilde{\epsilon}(t_{i}% )=1).$

The time-homogeneous Markov chain $\tilde{\epsilon}$ has a stationary distribution

 $\pi_{1}=\frac{q}{q+p}\quad\text{and}\quad\pi_{2}=\frac{p}{q+p}.$

We assume daily time steps and consider one year to be made up of 252 trading days, ie, $\varDelta=\frac{1}{252}$.

For the numerical approach, we simulate $N=100\,000$ paths of the discretized Markov-switching process.44 4 Simulations were conducted using MATLAB and R. For each sample path $X_{t}^{n}$, $n=1,\dots,N$, the reinsurer’s value at time $t=\tau+\varTheta$ is given by

 $\displaystyle V_{\mathrm{R}}^{n}(\tau^{n}+\varTheta)$ $\displaystyle=m_{\mathrm{R}}^{n}\mathrm{e}^{r(\tau^{n}+\varTheta)}X_{0}-[K-X^{% n}_{\tau^{n}+\varTheta}]^{+}.$ (4.1)

At time $t=0$, the reinsurer’s value is thus

 $\displaystyle V_{\mathrm{R}}^{n}(0)=m_{\mathrm{R}}^{n}X_{0}-\mathrm{e}^{-r(% \tau^{n}+\varTheta)}[K-X^{n}_{\tau^{n}+\varTheta}]^{+}.$ (4.2)

Similar to the analytical valuation, we use the fair value approach stated in Djerroud et al (2016) to derive the up-front premium $m_{\mathrm{R}}$ and set the present value of the expected payoff equal to zero such that

 $\frac{1}{N}\sum_{n=1}^{N}V_{\mathrm{R}}^{n}(0)=0,$ (4.3)

ie, using (4.2) and $X_{0}=1$, we find

 $\frac{1}{N}\sum_{n=1}^{N}(m_{\mathrm{R}}X_{0}-\mathrm{e}^{-r(\tau^{n}+% \varTheta)}[K-X_{\tau^{n}+\varTheta}^{n}]^{+})=0.$ (4.4)

We thus obtain

 $\displaystyle m_{\mathrm{R}}=\frac{1}{N}\sum_{n=1}^{N}\mathrm{e}^{-r(\tau^{n}+% \varTheta)}[K-X_{\tau^{n}+\varTheta}^{n}]^{+}.$ (4.5)

The following illustrates the algorithm employed.

1. (1)

Simulate the (discretized) Markov chain $\tilde{\epsilon}^{n}(t)$ for path $n$ with state space $\mathbb{S}=\{1,2\}$ for $t\in\{\frac{1}{252},\frac{2}{252},\dots,T\}$ in a daily setting with 252 trading days, where $T=1$, ie, one year.

2. (2)

Simulate the daily trajectory $X_{t}^{n}=X_{0}\mathrm{e}^{R_{t}^{n}}$ with

 $R_{t}^{n}=R_{t-1}^{n}+\tfrac{1}{252}(r-\tfrac{1}{2}\sigma_{i}^{2})+\sqrt{% \tfrac{1}{252}}\sigma_{i}\eta,$

where $R_{0}=0$, $\eta\sim N(0,1)$ and $i\in\{1,2\}$ denotes the state of the market.

3. (3)

If $X_{t}^{n}<0.9X_{0}$ for any $0, set the stopping time $\tau^{n}=\inf\{0 and use $X_{\tau^{n}+\varTheta}^{n}$ to evaluate the respective payoff and $\tau^{n}+\varTheta$ to discount the payoff.

4. (4)

Repeat steps (1)–(3) a sufficient number of times (here, $N=100\,000$) and apply (4.5) to find the expected up-front premium $m_{\mathrm{R}}$.

As it is preferable to decrease an estimate’s variance, we also apply the antithetic variates method in our simulation. This method exploits the symmetry of the randomly generated independent and identically distributed standard normal variables $\eta\sim N(0,1)$ in step (1) of our algorithm above. It holds for $\eta\sim N(0,1)$ that $\smash{\eta\lx@stackrel{{\scriptstyle d}}{{=}}-\eta}$, and thus we can replace $\eta$ by $-\eta$ in step (1) without changing the trajectory’s distribution. For the estimated trajectory

 $X^{n}_{\mathrm{antithetic}}:=\frac{X^{n}_{\eta}+X^{n}_{-\eta}}{2},$

it holds that $\operatorname{var}(X^{n}_{\mathrm{antithetic}})\leq\operatorname{var}(X^{n})$ (see Kroese et al (2013) for details).

Again, some practitioners might be interested in the actual discounted expected payoff of their investment. Analogously to Section 3 (using the GBM model), we replace the risk-free drift with empirical yearly returns $\mu_{1}$, $\mu_{2}$ and volatility $\sigma_{1}$, $\sigma_{2}$ for selected hedge fund indexes (see the details below) that can be found in Table 1. Note that the same arguments as in Section 3 on the difficulty of estimating empirical drifts and risk-neutral pricing hold in the Markov-switching model. We do not price the contract, but evaluate the present value of the expected payoff.

We fit the parameters of the Markov-switching model to real-world historical data. To obtain reasonable parameters and to cover most of the hedge fund universe, we analyze five main hedge fund strategies: equity hedge, equity market global, neutral, macro and merger arbitrage. Saunders et al (2013) use these strategies as proxies for all available strategies. Our primary source of data was the open source database for daily fund returns operated by the Chicago-based company Hedge Fund Research (HFR). The observation period starts on April 1, 2003, and ends on December 28, 2018. Parameter values for $\mu_{1}$, $\mu_{2}$, $p$ and $q$ are estimated using the well-known Baum–Welch algorithm (Baum et al 1970) in the R-routine “Baum–Welch” from the HiddenMarkov package. As initial values for this maximum likelihood estimation, we employ a heuristic proposed in Ernst et al (2009). Finally, we use Viterbi’s algorithm to create the most likely sequence of states (see Viterbi (1967) for details), compute the respective mean returns and average all five strategies’ returns and transition probabilities. Another approach to obtain the required parameters is estimation by moment matching (see Höcht et al (2009) for details).

Figure 1 illustrates the resulting crisis periods for the exemplary HFRXEH index using the heuristic and the Baum–Welch algorithm. Table 1 contains all analyzed HFRX indexes and their respective numerical parameter values and historical data for a single-regime (Black–Scholes) market.

## 5 Results

We set the initial parameters for both approaches to the following values:

• $T=1$ (one-year investment horizon);

• $X_{0}=1$ (initial investment);

• $K=(1-c_{\mathrm{M}})X_{0}=0.9$ (liquidation barrier);

• $r=0.01$ (risk-free rate).

In the following, the unit of the liquidity measure $\varTheta$ is one day, ie, $\varTheta=1$ represents a daily liquidation window and was implemented in the model as $\frac{1}{252}$.

### 5.1 Analytical results using the GBM approach

Figures 2 and 3 show the results for the analytical approach using a GBM framework. The premium $m_{\mathrm{R}}$ is generally an increasing function of the underlying fund’s volatility and liquidity. Figure 2 illustrates the value of premiums for fixed levels of liquidity (daily, weekly, biweekly and monthly, ie, $\varTheta\in\{1,5,10,20\}$) under the risk-neutral valuation (part (a)) and discounted expected payoff valuation (part (b)) using empirical values as a convex function of the volatility $\sigma$. Regardless of the liquidity window, it takes an annual volatility higher than approximately 5% to observe a noticeable impact on the premium $m_{\mathrm{R}}$. Assuming very liquid assets ($\varTheta=1$), we see premiums below 40 basis points (bps) even for extreme volatility ($\sigma=25$%). The premium using the risk-neutral valuation tends to be a bit higher than that computed from the discounted expected payoff valuation using empirical values (keeping all other parameters constant). Figure 3 illustrates the premium for fixed levels of volatility as a function of a fund’s liquidity and the concave relationship between the premium and the liquidity. Again, for a low level of volatility ($\sigma=5$%), there is only a marginal impact on the premium compared with the extreme levels of volatility in the risk-neutral valuation approach (Figure 3(a)). Figure 3(b) illustrates the valuation of the discounted expected payoff. Even for a one-month liquidation window ($\varTheta=20$) the premium is at a reasonable value of $m_{\mathrm{R}}\approx 0.7$bps.

### 5.2 Numerical results using the Markov-switching approach

Figures 46 show results for the numerical approach using a simulation with a Markov-switching model as the underlying framework. The stationary distribution

 $\pi=(\pi_{1},\pi_{2})=\bigg{(}\frac{q}{p+q},\frac{p}{p+q}\bigg{)}=(0.8317,0.16% 83)$

for $p=0.0175$ and $q=0.0865$ indicates a high probability of the market being governed by state 1, ie, the “good” market environment. Hence, Figures 4 and 5 illustrate the value of the premium depending on $\sigma_{1}$ for fixed levels of $\sigma_{2}$ and different levels of liquidity ($\varTheta$ in days). Similar to our results in Section 3, Figure 4 (initial state is “good”) illustrates the premium as a convex function of a fund’s volatility. For low levels of volatility in both states ($\sigma_{1/2}\leq 5$%), we find a marginal impact on the premium. However, assuming very liquid assets ($\varTheta=1$) even for extreme values in both states, ie, $\sigma_{1/2}=25$%, the premium is significantly lower ($m_{\mathrm{R}}\approx 74$bps) than in the single-regime model (for $\sigma=25$%, we find $m_{\mathrm{R}}\approx 170$bps). For lower levels of volatility in state 1 ($\sigma_{1}\leq 10$%), the volatility of state 2 has a much greater impact on the premium than a high volatility in state 1 ($\sigma_{1}\approx 25$%).

Figure 5 illustrates our results for a market with initial state “stressed”. As expected, the premiums are higher than when assuming a “good” initial market. In Figure 6, we see the premium using the discounted expected payoff valuation for empirical values depending on the initial state of the market. Assuming a “stressed” initial market, the premium is roughly twice as high as for a “good” initial market. Except for a daily liquidation window, even for a “stressed” market, the premium is below that using a single-regime model (see Figure 3). For a one-month liquidation window, premiums are approximately 0.3bps for a “good” market and approximately 0.5bps for a “stressed” market. For comparison, the premium for the same level of liquidity in the single-regime model is approximately 0.7bps. The results indicate that the selection of the applied model matters substantially for an investor. As the actual state of the market is not an observable variable, we also include a “weighted average”. The Baum–Welch algorithm indicates that the HFRX indexes in Table 1 spent an average of 84.27% in the “good” state during the period between April 1, 2003, and December 28, 2018. Using this number as a proxy for the probability that the market is in a “good” state, we compute a “weighted average premium”.

## 6 Backtesting

### 6.1 Setup and assumptions

This section elaborates on the backtesting of the suggested premium in the previous sections. It aims to evaluate the practical application of the concept and how it would have performed in recent years. As the concept of insuring hedge fund portfolio losses beyond the first-loss tranche is completely new and no work has been done so far (to the best of our knowledge), there are no predefined ways to evaluate the backtest or to benchmark the results of the proposed premium to other approaches. Hence, this section serves as an initial starting point in improving the practical application of the concepts mentioned in this paper. The general idea of backtesting in this context is the evaluation of the computed premium using real-world historical data. In order to obtain valid results, the look-ahead bias has to be avoided. This bias occurs when at certain times data or information that was not available at the actual point in time of the event is used for a simulation or evaluation of those events. The HFRX data set used for Table 1 is the main source of data used here. It contains daily returns from April 1, 2003, through December 28, 2018. Fund-dependent parameters, eg, returns, volatility and transition probabilities, are determined using the HFRX data set. The annualized risk-free rates $r$ are extracted from the Kenneth R. French website,5 which provides one-month Treasury bill returns starting in July 1926. The following list describes the backtesting approach implemented in this section.

1. (1)

Define a starting point, investment target (eg, HFRXGL) and crucial parameters (eg, liquidity measure $\varTheta$, fees, managerial deposit $c_{\mathrm{M}}$, initial investment $X_{0}$, etc).

2. (2)

Estimate all the parameters needed to compute the premium for a one-year insurance at a specific point in time that depends on the predefined valuation approach, ie, the GBM framework or Markov-switching framework.

3. (3)

Determine the premium and thus the initial values for the investor, the manager and the reinsurer at that specific point in time.

4. (4)

Let the hedge fund performance develop according to the historical data.

5. (5)
1. (a)

In the event that the fund value breaches the barrier at any time during the one-year investment horizon, all assets are liquidated and the payoff is evaluated using the final value, which depends on the underlying liquidity.

2. (b)

In the event of no breach, evaluate the one-year performance at the end of the period.

6. (6)

(Following step (5)(b) only) start a new investment period until the liquidation event occurs or a predefined stopping criterion, eg, no historical data is available anymore, is reached.

Note that we are using the risk-neutral approach. The following elaborates how the specific parameters at a certain point in time are obtained and how the fund performance is evaluated. First, we illustrate the implemented algorithm. Let the initial investments for the investor $X_{\mathrm{I}}(i)$, the manager $X_{\mathrm{M}}(i)$ and the reinsurer $X_{\mathrm{R}}(i)$ for investment period $i$, $i=1,2,\dots,T-1$, be given by

 $\displaystyle X_{\mathrm{I}}(i)$ $\displaystyle=\hat{V}_{\mathrm{I}}(i-1)=X_{i},$ $\displaystyle X_{\mathrm{M}}(i)$ $\displaystyle=\hat{V}_{\mathrm{M}}(i-1),$ $\displaystyle X_{\mathrm{R}}(i)$ $\displaystyle=\hat{V}_{\mathrm{R}}(i-1),$

where for $i=0$ the following values are set: $X_{\mathrm{I}}(0)=X_{0}$, $X_{\mathrm{M}}(0)=0$ and $X_{\mathrm{R}}(0)=0$. Note that in this paper $X_{0}=1$. At the beginning of the investment period $i$, the premium $\smash{m_{R_{i}}}$ is computed and the initial values for the investor $V_{\mathrm{I}}(i)$, the manager $V_{\mathrm{M}}(i)$ and the reinsurer $V_{\mathrm{R}}(i)$ for investment period $i$, $i=0,1,\dots,T-1$, are given by

 $\displaystyle V_{\mathrm{I}}(i)$ $\displaystyle=(1-m_{R_{i}})X_{\mathrm{I}}(i),$ $\displaystyle V_{\mathrm{M}}(i)$ $\displaystyle=X_{\mathrm{M}}(i),$ $\displaystyle V_{\mathrm{R}}(i)$ $\displaystyle=X_{\mathrm{R}}(i)+m_{R_{i}}X_{\mathrm{I}}(i).$

The final values for the investor $\hat{V}_{\mathrm{I}}(i)$, the manager $\hat{V}_{\mathrm{M}}(i)$ and the reinsurer $\hat{V}_{\mathrm{R}}(i)$ for investment period $i$, $i=0,1,\dots,T-1$ are given by

 $\displaystyle\hat{V}_{\mathrm{I}}(i)$ $\displaystyle=X_{i,i+1}-m_{R_{i}}\mathrm{e}^{r}X_{\mathrm{I}}(i)$ $\displaystyle\qquad-\alpha_{\mathrm{M}}[X_{i,i+1}-X_{\mathrm{I}}(i)]^{+}+[X_{% \mathrm{I}}(i)-X_{i,i+1}]^{+},$ (6.1) $\displaystyle\hat{V}_{\mathrm{M}}(i)$ $\displaystyle=\mathrm{e}^{r}V_{\mathrm{M}}(i)+\alpha_{\mathrm{M}}[X_{i,i+1}-X_% {\mathrm{I}}(i)]^{+}$ $\displaystyle\qquad-[X_{\mathrm{I}}(i)-X_{i,i+1}]^{+}+[(1-c_{\mathrm{M}})X_{% \mathrm{I}}(i)-X_{i,i+1}]^{+},$ (6.2) $\displaystyle\hat{V}_{\mathrm{R}}(i)$ $\displaystyle=\mathrm{e}^{r}V_{\mathrm{R}}(i)-[(1-c_{\mathrm{M}})X_{\mathrm{I}% }(i)-X_{i,i+1}]^{+},$ (6.3)

where $X_{i,i+1}$ denotes the one-year price development of the initial investment $X_{\mathrm{I}}(i)$ in period $i$. Hence, this is the final value of the initial investment $X_{\mathrm{I}}(i)$ for this period, which is used as initial investment in period $(i+1)$. Note that, for a one-period investment horizon and no flat management fee ($m_{\mathrm{M}}=0$), (6.1)–(6.3) yield (2.3)–(2.5). Now, if for any $\tau\in[0,1]$ it holds that $X_{i,i+\tau}\leq(1-c_{\mathrm{M}})X_{i}=(1-c_{m})X_{\mathrm{I}}(i)$, the hedge fund is liquidated depending on $\varTheta$, eg, in this paper $\varTheta\in\{\frac{1}{252},\frac{5}{252},\frac{10}{252},\frac{20}{252}\}$, and the final value is set to the following:

 $\displaystyle\hat{V}_{\mathrm{I}}(i)$ $\displaystyle=X_{i,i+\tau+\varTheta}-m_{R_{i}}\mathrm{e}^{r(\tau+\varTheta)}X_% {\mathrm{I}}(i)$ $\displaystyle\qquad-\alpha_{\mathrm{M}}[X_{i,i+\tau+\varTheta}-X_{\mathrm{I}}(% i)]^{+}+[X_{\mathrm{I}}(i)-X_{i,i+\tau+\varTheta}]^{+},$ $\displaystyle\hat{V}_{\mathrm{M}}(i)$ $\displaystyle=\mathrm{e}^{r(\tau+\varTheta)}V_{\mathrm{M}}(i)+\alpha_{\mathrm{% M}}[X_{i,i+\tau+\varTheta}-X_{\mathrm{I}}(i)]^{+}-[X_{\mathrm{I}}(i)-X_{i,i+% \tau+\varTheta}]^{+}$ $\displaystyle\qquad+[(1-c_{\mathrm{M}})X_{\mathrm{I}}(i)-X_{i,i+\tau+\varTheta% }]^{+},$ $\displaystyle\hat{V}_{\mathrm{R}}(i)$ $\displaystyle=\mathrm{e}^{r(\tau+\varTheta)}V_{\mathrm{R}}(i)-[(1-c_{\mathrm{M% }})X_{\mathrm{I}}(i)-X_{i,i+\tau+\varTheta}]^{+}.$

Note that we employ daily time steps in our backtest. This allows for a smoother performance graph.

As already mentioned, the approach is a simplification, and several assumptions and specifications are made.

Rolling windows.

Premiums are computed based on parameters obtained from a two-year rolling window. In the GBM approach, the estimated volatility is simply the sample volatility. For the Markov-switching approach, we use the Baum–Welch algorithm according to Section 4 to estimate volatility and transition probabilities.

No reinvestment.

This applies to the manager and reinsurer only. All premiums and performance fees are not reinvested in the hedge fund but transferred to a risk-free investment account. In other words, the up-front premiums are compounded using the risk-free rate at the time of occurrence. Premiums from following investment periods are added to the account and compounded until either the liquidation event is triggered or the ultimate investment horizon is reached. We are aware that in practice the premiums collected by an insurance company are subject to individual asset–liability management processes. In some cases, it is even plausible to reinvest parts of the premium or the entire premium in the fund and participate in upside performance. As this would overly complicate interparty relationships, we implement the simpler approach. The manager’s value follows similar assumptions. In practice, the manager commonly owns a stake of the fund and reinvests performance fees. In our approach, all performance fees are paid at the end of the investment period and are transferred to a risk-free investment account and compounded.

No consumption.

The investor reinvests all available funds in the following investment period, ie, the final value of investment period $t$ is the initial investment of period $t+1$. During the investment period, the investor cannot redeem their stake. In practice, this is enforced by lockup periods; however, these usually only last up to two years. The manager’s value is not subject to any consumption either. In practice, the manager takes out funds either according to a predefined schedule or at will (eg, to pay staff salaries, etc). In our approach, the manager does not own a personal stake in the initial fund, and compounded performance fees are even used to offset potential losses in the liquidation event. The same argument holds for the reinsurer and is linked to the no-reinvestment assumption above.

Investment horizon.

We implement consecutive independent one-year investment periods. All fees are paid annually: the premium is paid up front and the performance fee is paid at the end of the respective investment period. The performance fee is obtained according to a predefined schedule and in the course of the one-year period. The investment periods are independent from each other, ie, every year is implemented as if it were a stand-alone investment, and the results are accumulated.

Hurdle rates.

The liquidation barrier is reset at the beginning of each period and is only valid for that period. With the beginning of the following period, the liquidation barrier is reset depending on the new initial investment. In practice, many hedge funds have hurdle rates or high watermarks in place, which we ignore in this simplified approach.

If a portfolio is made up of multiple indexes, we implement a simple buy-and-hold strategy. A fictitious initial portfolio with equally weighted indexes is the basis for the calculation of the portfolio’s (weighted) returns and volatility, ie, during the investment periods, weights are adjusted depending on the individual performance of the underlying indexes when calculating returns and not kept constant. There is no dynamic rebalancing at any time.

Time schedule.

The starting date for any portfolio is April 1, 2005. This allows for two complete years of daily return data for the first rolling window. Each year, a new premium is computed on April 1, and the last trading day (and the evaluation of the portfolio) for this period is March 31. If no liquidation process is triggered, April 1, 2017, is the last start date for a one-year period. In total, this allows a maximum overall investment horizon of 13 years (April 1, 2005, through March 31, 2018).

### 6.2 Results

Note that in the following we refer to $\varTheta$ in daily units, ie, $\varTheta=1$ represents a one-day liquidation window, which is implemented in the simulation using $\varTheta=\frac{1}{252}$. Figures 712 show the results of the backtesting implemented using the GBM valuation according to Section 3.

Figures 7 and 8 illustrate the HFRX Equity Hedge Index (HFRXEH) for $\varTheta=1$ and $\varTheta=20$. In Figure 7, the reinsurer shows a profit for $\varTheta=1$, whereas they show a massive deficit for $\varTheta=20$. After breaching the liquidation barrier, the price of the underlying fund quickly increased but then largely decreased over the following period. In this figure, two features are visible. First, during any investment period (illustrated by the area between two ticks on the time axis), the investor’s value never drops below the respective period’s initial value. In other words, the initial value is the lower limit for the investor’s value in any period. This is illustrated by the flat parts in the top plot. Second, note the cap on the manager’s obligations, illustrated by the flat line at the end of the second plot. Here, the reinsurance is triggered.

Figures 9 and 10 illustrate the HFRX Macro/CTA Index (HFRXM) and represent examples where the reinsurer is able to cover losses beyond the first-loss tranche using the collected premiums. In Figure 9 ($\varTheta=1$), all losses are offset by the premium. In Figure 10 ($\varTheta=20$), the fund value actually rises above the level of the managerial deposit, ie, the reinsurance is not held liable; however, the collected premiums would have offset losses occurring at any time between the barrier breach and full liquidation. Also, note the difference in the multiplier due to the increase in $\varTheta$: the value of the reinsurer reaches a maximum of $V_{\mathrm{R}}\approx 18\times 10^{-4}=18$bps for $\varTheta=1$ and a maximum of $V_{\mathrm{R}}\approx 7.5\times 10^{-3}=75$bps for $\varTheta=20$.

Figure 11 shows the HFRX ED: Merger Arbitrage Index (HFRXMA), where the liquidation process is not triggered during the entire investment horizon. Here, the investor steadily builds up their value. Note the jump in the premium for the reinsurance when markets are very volatile (illustrated by rising volatility in the bottom graph) in April 2009 and 2010. In Figure 12, we implemented an equally weighted portfolio of all five HFRX indexes listed in Table 1. Due to the diversification, there is no liquidation breach even though some indexes show a massive negative performance during the financial crisis.

Figures 1316 show results for the Markov-switching approach according to Section 4. Figures 13 and 14 show roughly the same results for the HFRXEH Index as Figures 7 and 8 (GBM approach) except that the premium amount is higher for the Markov-switching approach. Figures 16 (Markov-switching) and 10 (GBM) yield similar results for the HFRXM Index. However, the maximum value of the reinsurer in the Markov-switching model is more than twice as high ($V_{\mathrm{R}}\approx 16\times 10^{-3}=160$bps) due to higher premiums.

To summarize the above, we obtain mixed results for both the GBM and the Markov-switching approaches. In some cases, losses can be absorbed by the premium paid. However, especially during the financial crisis, when sudden crashes caused funds to suffer heavy losses, the premium was not remotely high enough.

## 7 Conclusion

We suggest an up-front premium to a reinsurance party in exchange for full portfolio protection, which goes beyond the insurance of the first tranche of losses by a first-loss scheme. The additional downside protection extends the first tranche by considering a second tranche that will insure the investor against all losses, not just the first level. We propose both an analytical closed-form solution based on a derivative-pricing approach with a GBM as the underlying framework and a numerical solution using a simulation based on a Markov-switching model. In both approaches, we assume liquidity to be the key parameter, and thus the premium is derived as a function of the underlying fund’s liquidity. A simplified backtesting method delivers mixed results: for some hedge fund indexes, historical losses were too high to be covered by the calculated premium, especially during the financial crisis. However, in some cases the reinsurer was able to offset losses by the premiums collected.

There are several extensions and questions that could be the subject of future research. In this paper, we suggest a total of two tranches. This framework could be extended, creating a fund structure similar to an asset-backed security containing several tranches, each with a different risk and return profile. The parameters could also be altered to a point where both the reinsurer and the manager earn a fixed fee and a performance-dependent fee. Finally, in our approach the fund’s underlying liquidity is assumed to be homogeneous with daily liquidation steps. Combining our approach with certain liquidation time distributions, eg, an exponential or Weibull distribution, as suggested in Bordagand et al (2017), and the optimization of the liquidation process that comes along with the issue could be considered.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

## Acknowledgements

We wish to express our gratitude to Michael Ege for some very useful insights on the business aspects of this paper as well as for interesting comments on earlier drafts.

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