Dynamic asset allocation for liability-driven
investors: Alternatives to conventional
portfolio insurance strategies
By: Leander MeijeringS3517241
l.f.meijering@student.rug.nl University of Groningen Faculty of Economics and Business
MSc. Finance Supervisor: Dr. A. Plantinga1
Date: 12-6-2019 Abstract
This paper introduces three alternatives to a Constant Proportion Portfolio Insurance (CPPI) strategy. These three strategies enable the investor to have an exposure to stocks when the portfolio value is below the target level. This paper finds that all three strategies are an improvement on the CPPI strategy based on a range of performance measures. The results show that of the three alternative strategies, the two strategies that uses a multiplier conditional on the coverage ratio dominate. The strategy that uses a dynamic multiplier based on prospect theory shows a slight dominance overall.
Key words: Dynamic asset allocation, prospect theory, liability-driven investment, Monte Carlo simulation.
JEL classifications: G11, G15, G17.
Word count: 12,676 (excluding abstract).
1 I would like to thank Dr. A. Plantinga for his helpful comments throughout the entire process of
2
1. Introduction
Liability-driven investing (LDI), is a framework used to manage current investment assets in order to meet future liabilities and is implemented by about 75 percent of defined-benefit pension plans. LDI is often implemented by institutional investors, such as pension funds and individual investors whom are concerned with meeting future liabilities, such as retirement and trust funds. In this paper the focus is on pension and retirement related LDI applicable to both institutional and individual investors. For a pension fund these liabilities are defined as the retirement benefits paid off to their clients. For individual investors these liabilities are defined as the future payments to provide a living during their pension. Since 2008 the funded status ratio of pension funds, the percentage of future liabilities they are able to cover, has decreased on average from 105.3 percent in 2008 to around 80 percent in 20152.
Nowadays people speak of a so called ‘pension crisis’ which presumably all started back in 2008 with the fall of Lehman Brothers3. Merton (2014) raised concerns about retirement
planning, which – according to him – started back in 2000 in the wake of the dot-com crash. Major firms in the US went bankrupt because they were unable to meet their obligations under defined-benefit plans4. The main issue is that pension funds cannot cover all future
retirement benefits to clients, which could be attributed to adverse market conditions – for example the financial crisis of 2008 and dot-com crash – or demographic developments such as ageing of the population5. This could induce individual investors to provide for their own
pension as they cannot be sure that all of their expected retirement benefits are being paid out.
In defining their asset allocation strategy, investors should account for future liabilities. Investors are restricted by their liabilities in making investment decisions. Portfolio insurance strategies are appropriate for investors who need to limit their downside risk but still desire upside potential (Perold and Black, 1992). This makes portfolio insurance strategies an ideal tool for liability-driven investors. The downside protection safeguards the coverage of their future liabilities but still allows for upside potential. Constant Proportion Portfolio Insurance (CPPI) and Time Invariant Portfolio Protection (TIPP) are two examples of portfolio insurance strategies.
2 See for instance: https://www.institutionalinvestor.com/article/b14z9ql0d332sd/matching-current-assets-to-future-liabilities and https://www.forbes.com/sites/johnmauldin/2019/05/20/the-coming-pension-crisis-is-so-big-that-its-a-problem-for-everyone/#3eac75e737fc.
3 See https://marketupdate.nl/columns/de-val-van-lehman-luidde-de-pensioencrisis-in/
4 In a defined-benefit plan the employer pays a benefit which is a predetermined formula of age,
tenure, and current and past wages. The employer bears the risk of meeting retirement benefits.
3 CPPI invests a constant multiple of the cushion in stocks up to the borrowing limit, where the cushion is the difference between wealth and a specified floor. The floor should be set equal to the present value of the future liabilities (Black & Jones, 1987). TIPP is a similar strategy, introduced by Estep and Kritzman (1988), however they adjust the floor continuously to a specified percentage of the highest value the portfolio reaches.
Implementing these strategies for investors with a liability structure can give rise to some complications. For instance, if the present value of the future liabilities is greater than the value of the portfolio, TIPP and CPPI strategies are hard to implement. As these strategies force the investor to immediately invest their full proceeds in bonds. Historical data shows that bonds on average generate a lower return than equity (Siegel, 1992). Being fully invested in bonds can prevent the investor from ever reaching the aspired level that is required to cope with the future liabilities. This is for example the situation in the Netherlands, where a significant number of pension funds struggle with covering their
expected liabilities. The present value of the pension benefit obligations exceeds the value of the retirement plan assets for the main pension funds in the Netherlands6. This is not only
relevant to pension funds and other institutional investor but also for individual investors. For individual investors who use a portfolio insurance strategy as a retirement strategy, it is important that the strategy generates a return that is sufficiently large to support the spending throughout retirement (Estrada and Kritzman,2018). When the present value of expected retirement spending (target level) exceeds the value of the portfolio, a standard portfolio insurance strategy such as a CPPI or TIPP strategy can cause the same problem; there is no longer certainty of reaching the floor (target level)7.
This – the situation where the target level exceeds the portfolio value – has major implications for the risk preference of an investor according to the prospect theory – introduced by Kahneman and Tversky (1979) – that assumes that investors value losses and gains differently. Expected utility was prevalent for multiple decades as the dominant model for decision making under uncertain circumstances (Tversky and Kahneman, 1992).
However, evidence suggests that the expected utility is violated in systematic ways (Starmer, 2000) and that expected utility gives inconsistent outcomes (Hershey and Schoemaker, 1985). Tversky and Kahneman (1979) introduced the prospect theory, ‘which explains the major violations of the expected utility theory in choices between risky prospects with a
6 See https://www.elsevierweekblad.nl/economie/opinie/2019/01/kabinet-kan-pensioencrisis-niet-oplossen-669257/
7 It is crucial to note that in a situation where the portfolio values does not cover the expected
4 small number of outcomes’. The prospect theory is generated by a value function that is concave for gains and convex for losses8. This means that investors are more concerned with
limiting their downside risk than their upside potential. Critical to this value function is the reference point from which gains and losses are measured9. I will illustrate this by giving an
example. Consider two funds within the framework of LDI, one of the funds is underfunded – the value of the portfolio cannot cover the target level – and the other fund is overfunded – the value of the portfolio exceeds the target level. According to prospect theory the fund that is underfunded will employ a value function that is convex and the fund that is
overfunded will employ a value function that is concave. In this paper the concern is with the fund that is underfunded.
Prospect theory assumes that investors are loss averse and therefore gains and losses should be valued differently. Benartzi and Thaler (1993) define loss aversion as the tendency for individuals to be more sensitive to reductions in their levels of well-being than to
increases of the same magnitude. This is nicely illustrated by ‘the marshmallow
experiment’10, where one child is given one marshmallow and this makes the child happy.
Similarly another child is given two marshmallows, however with a twist; right before the child can eat the marshmallows one marshmallow is taken away and the child is not so happy. The outcome for both children is the same: one marshmallow. However, they derive very different levels of utility from the experiment – the level of utility associated with giving up one marshmallow is lower than the level of utility associated with acquiring a
marshmallow. This anomaly – the asymmetry in value of one marshmallow for both children – is what Kahneman and Tversky (1984) call loss aversion. They find that a loss of €X is more aversive than a gain of €X is attractive. They elaborate on this by stating that most
respondents in a sample of undergraduates refused to stake $10 on the toss of a coin if they stood to win less than $30. This implies that the attractiveness of the possible gain is not nearly sufficient to compensate for the aversiveness of the possible loss. Over the last decades several other empirical studies have tried to find evidence for this phenomena. Abdellaoui, Bleichrodt and L’Haridon (2008) find that risk seeking and concave utility can concur under prospect theory and they offer support for the validity of prospect theory. Loss aversion is also a convenient concept to explain other phenomena. For example, Benartzi and Thaler (1995) find that the concept of loss aversion can explain the equity premium puzzle. It follows that investors are risk averse for gains and risk seeking for losses, they
8 A concave value function characterizes a risk averse investor and a convex value function
characterizes a risk seeking investor.
9 The reference point is usually the status quo. See for example Odean (1998) or Tversky and
Kahneman (1979).
5 dislike losses so much they are willing to take additional risk to ‘jump out of it’. This is
supported by the findings of Fiegenbaum (1989) who finds strong evidence that prospect theory can explain the tradeoff between risk and return. Organizations that are below their target level are found to be risk-takers while organizations that are above their target level are risk-averters. Organizations are the perfect equivalent for liability driven investors. This fuels the relevance and importance of this paper in the sense that investors – according to the prospect theory and supported by empirical evidence – become risk seeking when losses arise. Whereas the CPPI strategy, in a situation where losses cause the floor to exceed the portfolio value, forces the investor to act opposite and take even lesser risk. This makes the prospect theory a convenient and applicable concept for liability driven investors.
I propose the following three alternative strategies to overcome the limitations of the CPPI strategy in the situation mentioned above:
1. Floor Adjustable Strategy (FAS): the liabilities are redefined, at a minimum,
acceptable level, such that the present value of the liabilities is less than the value of the portfolio. After redefining the liabilities the investor can continue with a portfolio insurance strategy (CPPI).
2. Multiplier Switch Strategy (MSS): the investor takes additional risk to restore the value of the portfolio above the present value of the liabilities. A concave strategy can be used to achieve this. When the value of the portfolio is above the expected liabilities the investor can continue with a convex strategy (CPPI).
3. Dynamic Multiplier Strategy (DMS): this is a CPPI strategy with a dynamic multiplier, which accounts for the risk preferences of an investor under the prospect theory. The objective is to find the most suitable alternative strategy that solves the problem of using portfolio insurance strategies (CPPI), for a liability driven investor when the value of the expected liabilities exceed the value of the assets. The research question is defined as: “What is the most optimal dynamic asset allocation strategy for liability driven investors when the expected liabilities exceed the value of the retirement plan assets?”
All three strategies are tested using a Monte Carlo simulation as well as a historical
6 The results show that these alternative strategies do exactly what they are meant to do: overcome the limitations of the CPPI strategy. The results suggest that all three strategies outperform the CPPI strategy in a situation where the portfolio value is below the target level based on the performance measures. Furthermore I find that of the three strategies, The MSS and DMS outperform the FAS. The MSS and DMS show similar results with a slight edge towards the DMS, however this edge is considerable small.
Before continuing reading it is probably useful to get familiar with some notations that will be used throughout the paper. A lot of equivalent terms will be used interchangeably throughout the paper. Appendix A gives an overview of these terms and notations.
2. Portfolio insurance strategies
2.1 Constant Proportion Portfolio Insurance (CPPI)
Liabilty-driven investors in general would like to limit their downside risk to avoid cuts in their future retirement benefits (Veldhuizen, 2014). Portfolio insurance strategies, such as CPPI and TIPP, can provide in this demand.
CPPI was first introduced by Black and Jones (1987) and Perold (1986), as a less complex portfolio insurance strategy than option based insurance strategies. A CPPI strategy without leverage and short-sale restrictions calculates the exposure to stocks at time t, 𝑒𝑡, as:
𝑒𝑡 = 𝑚 × 𝐶𝑡 (1) where, 𝑚 is the multiplier greater than one and 𝐶𝑡 is the cushion at time t.
The exposure to stocks is determined by a constant multiple of the so called cushion, which is the difference between the value of the portfolio and the pre-specified floor, hence:
𝐶𝑡 = 𝑊𝑡− 𝐹 (2)
where, 𝑊𝑡 represents the current value of the portfolio or wealth at time t and 𝐹𝑇 is the
present value of the floor. A CPPI strategy is known as a convex strategy because of its convex payoff diagram. This strategy is equivalent to buying stocks as they rise and sell stocks as they fall. According to Perold and Sharpe (1995) this strategy will do relatively well in continuously rising or falling markets and hence will benefit from momentum in markets. On the other hand however this strategy will perform relatively poor in a flat but fluctuating market (high volatility) and hence will be hurt by mean reversals.
CPPI strategies are very similar to buy and hold strategies, since a buy and hold strategy is a CPPI strategy with a multiplier of one and a floor equal to the value invested in bonds
7 needed when the current exposure differs from 𝑒𝑡. Perold and Black (1992) distinguish two
types of rebalancing: discrete and continuous rebalancing. With discrete rebalancing the difference between exposure and m times the cushion is measured by the index ratio, differential moves in the stocks vs. bonds, are magnified in the cushion. Perold and Black (1992) show that any move in the index ratio will put the portfolio out of balance. Under continuous rebalancing the index ratio follows geometric Brownian motion, which includes a volatility cost factor.
In practice the CPPI strategy is usually implemented such that the exposure to stocks can vary between 0% and 100% of the total assets invested (Dichtl & Drobetz, 2010). Therefore, short sale restrictions are added to the model:
𝑒𝑡= max[𝑚𝑖𝑛(𝑚 × 𝐶𝑡, 𝑊𝑡), 0] (3)
Figure 2.1: The change in the exposure to stocks with respect to the portfolio value for a CPPI strategy.
The exposure diagram above (figure 2.1) perfectly illustrates the problem defined in the introduction. The diagram shows that when the value of the portfolio decreases below the threshold level of €100.000 (equivalent to a coverage ratio of one) the exposure is given by a flat line at 0%. This shows that when the value of the portfolio falls below the floor, the investor is forced to invest 100% in bonds. Once the portfolio value has surpassed the threshold level of €100.000 the exposure is given by a concave curve. The flat line at the end of the curve is caused by the short-sale restrictions imposed in equation (3) above.
Remember the example of the two funds in the introduction. The exposure to stocks of the fund that is underfunded is represented by the flat line. This shows that the exposure to stocks for the underfunded fund is indeed zero. The concave curve shows the exposure to
0% 20% 40% 60% 80% 100% 120% 20 40 60 80 100 120 140 160 180 200 220 Exp o su re , e (in %)
Portfolio value, 𝑊 (in thousands €)
8 stocks for the overfunded fund. This illustrates that the main problem – central in this paper – only applies for underfunded investors.
2.2 Time Invariant Portfolio Protection (TIPP)
TIPP was first introduced by Estep and Kritzman in 1988, as an improvement to CPPI. CPPI ensures the investor of a certain floor, which the portfolio value cannot go below. However, if stock market increases and so does the exposure to stocks, the cushion increases and so does the potential fall in value of the portfolio. In other words, a CPPI strategy doesn’t give protection of interim capital gains (Dichtl & Drobitz, 2010). A TIPP strategy provides a protection of interim capital gains, by adding a fluctuating floor to the model. The floor increases with the value of the portfolio, hence the fall of this portfolio decreases. After specifying the initial floor and fixed multiplier the following steps are required (Estep & Kritzman, 1988):
- Calculation of the current portfolio value (stocks plus bonds).
- Estimate the proposed floor by multiplying the portfolio value with the floor percentage.
- If proposed floor is above the previous floor, than the proposed floor becomes the new floor, otherwise the old floor remains.
- Calculate new exposure by using equation (3).
A major downside of this strategy is discussed by Choie and Seff (1989). Once the floor is adjusted upwards it never comes down again. This opens the possibility that the value of the portfolio reaches or falls below the current floor, and hence the investor ends up more and more in the risk-free asset (if the market declines with more than 1
𝑚 ).
3. Alternative strategies
To overcome the limitations of a CPPI strategy, I introduce three alternatives strategies. The main objective of these strategies is to enable an exposure to stocks when the portfolio value is below the target level. The strategies should be more equipped to deal with
investors risk preferences, under the prospect theory, in situations of underfunding and get better results.
3.1 Strategy 1 – Floor Adjustable Strategy (FAS)
9 the initial target level. The exposure to stocks, 𝑒𝑡, is given by equation (4) if the condition
𝜋𝑡 ≥ 1 is satisfied. If 𝜋𝑡< 1 than 𝐹 is adjusted and the exposure 𝑒𝑡 is given by equation (5).
The adjustment consists of a δ decrease in the target level 𝐹, where δ is set such that the expected liabilities are a portion of 90% of the value of the assets. The floor 𝐹𝑡 – which now
changes over time – is made conditional on the coverage ratio 𝜋𝑡. The coverage ratio is
defined as the value of the portfolio over the target level (𝑊𝐹𝑡). This implies that the strategy can take the following two forms:
𝑒𝑡= max[𝑚𝑖𝑛(𝑚 × 𝐶𝑡, 𝑊𝑡), 0] 𝑖𝑓 𝜋𝑡≥ 1 (4)
𝑒𝑡 = max[𝑚𝑖𝑛(𝑚 × 𝐶𝑡∗, 𝑊𝑡), 0] 𝑖𝑓 𝜋𝑡 < 1 (5)
where 𝐹 is the initial target level and a constant and the adjusted cushion at time t, 𝐶𝑡∗, is
presented as:
𝐶𝑡∗ = 𝑊𝑡− 𝐹𝑡 (6)
where 𝐹𝑡 is an decreasing function of 𝐹 and is defined as:
𝐹𝑡 = 𝐹 × δ (7)
where δ is defined as: δ =(𝑊𝑡× 𝜏)
𝐹 (8) where τ is a constant and set at 90%11. This strategy allows for exposure to stocks when 𝜋
𝑡 <
1 by resetting the target level at a temporary lower level. The exposure diagram below in figure 3.1 illustrates how this strategy operates in practice. We see a constant level of exposure until a value of €100.000 (equivalent to a coverage ratio of one) which is caused by equation (1), where the floor is held at a constant level of 90% of the portfolio value. After the value of €100.000 is transcend, equation (4) is in place and the we see a classic concave shape that typifies a CPPI strategy (see figure 2.1), until an exposure of 100% is reached. The flat line in figure 3.1 – which represents the exposure to stocks for an underfunded investor – shows that this strategy allows for a constant exposure to stocks of 20% in a situation of underfunding. The exposure to stocks for an overfunded investor is similar to the concave curve in the regular CPPI strategy.
11 This is an arbitrary chosen number, however there is some reasoning behind it. The relative low
10
Figure 3.1: The change in the exposure to stocks with respect to the portfolio value for the FAS.
3.2 Strategy 2 – Multiplier Switch Strategy (MSS)
The procedure for this strategy is analogous to the procedure of the CPPI strategy. Therefore equation (3) is applicable to this strategy. To be able to have an exposure to stocks when 𝜋 < 1, I introduce a negative multiplier. Therefore a non-positive constraint to the multiplier 𝑚 is imposed, such that 𝑚 < 0. The exposure to stocks can be presented as:
𝑒𝑡 = max[𝑚𝑖𝑛(𝑚∗× 𝐶𝑡, 𝑊𝑇), 0] (9)
where, 𝑚∗ is the negative of the multiplier in equation (3). And the cushion at time t is presented as:
𝐶𝑡 = 𝑊𝑡− 𝐹 (10)
where, 𝐹 represents the initial target level. The major downside of this strategy is that when 𝑊𝑇 ≥ 𝐹 , the exposure to stocks is zero and the investor is forced to invest 100% in the risk-free asset. At this point we are back at the original problem that fueled this paper. To overcome this problem I introduce a slight modification of this strategy. The multiplier 𝑚 is made conditional on the coverage ratio. The coverage ratio is defined as the value of the portfolio over the target level (𝑊
𝐹). The strategy can now take the following two forms:
𝑒𝑡= max[𝑚𝑖𝑛(𝑚 × 𝐶𝑡, 𝑊𝑡), 0] 𝑖𝑓 𝜋𝑡≥ 1 (11)
𝑒𝑡 = max[𝑚𝑖𝑛(𝑚∗× 𝐶𝑡, 𝑊𝑡), 0] 𝑖𝑓 𝜋𝑡 < 1 (12)
When the value of the assets is back at the level that it satisfies 𝜋𝑡 ≥ 1, the original CPPI
strategy is used again. This overcomes the limitations of equation (9). The exposure diagram
0% 20% 40% 60% 80% 100% 120% 20 40 60 80 100 120 140 160 180 200 220 Exp o su re , E t(in %)
Portfolio value, Wt (in thousands €)
11 below shows a unique shape. Due to the short-sale constraints the exposure to stocks is limited to 100% and therefore we see a straight line at both ends of the curve. As the portfolio value increases and moves toward €100.000, corresponding with a coverage ratio of one, the exposure to stocks decreases sharply after €60.000. This is easily explained by looking at the cushion. The cushion decreases when moving towards a portfolio value of €100.000, and this causes the exposure to shrink. After the threshold level of €100.000 is transcend ( 𝜋𝑡 ≥ 1) the curve transforms into a concave curve and equivalent to a regular CPPI strategy, until a value of €200.000 is reached and we observe a flat line again that caps the exposure to 100%.
Figure 3.2: The change in the exposure to stocks with respect to the portfolio value for the FAS.
However, this strategy does give rise to some issues. Once the CPPI strategy is active, the same issues of CPPI discussed in the previous chapter, apply here. The main issue is that once the target level is transcend the need for exposure to stocks presumably decreases for a liability driven investor12, while the exposure to stocks increases. A TIPP strategy can solve
this problem of high exposure to stocks but can give rise to other issues (Choie and Seff, 1989). In addition, a TIPP strategy increases the floor as wealth increases, however in this particular case the floor is linked to the expected liabilities or target level of an liability-driven investor and it might not be appropriate to increase this target level as the expected liabilities are assumed to be constant.
12 This is based on the prospect theory (Tversky and Kahneman, 1979) and it is assumed that this
theory applies and holds for liability-driven investors.
0% 20% 40% 60% 80% 100% 120% 20 40 60 80 100 120 140 160 180 200 220 Exp o su re , e (in %)
Portfolio value, W (in thousands €)
12
3.3 Strategy 3 – Dynamic Multiplier Strategy (DMS)
This strategy uses a dynamic multiplier similar to the one developed in the paper of Yao and Li (2016). The dynamic multiplier in this strategy follows from the prospect theory and is based on the risk preference of the investor. As explained in the introduction, prospect theory assumes different risk preferences for gains and losses: risk aversion for gains and risk seeking for losses. I assume that for gains (𝜋𝑡>1) the level of risk aversion of a liability driver
investor exhibit increasing relative risk aversion (IRRA), where the level of risk aversion increases as the wealth (value of the assets) increases and moves towards or transcends the target level. This means that an investors need for risk decreases once the target level has been reached. For losses (𝜋𝑡<1) the concept of loss aversion applies and I assume that the
investor becomes risk seeking. This means that an investors need for risk increases once the target level has been breached downwards.
This results in a multiplier that is a linear function of the coverage ratio:
𝑚𝑡= max [𝛼 + 𝛽𝜋𝑡, 0] 𝑖𝑓 𝜋𝑡 ≥ 1 (13)
𝑚𝑡= 𝛼 + 𝛽𝜋𝑡 𝑖𝑓 𝜋𝑡 < 1 (14) Where α and β are both constant and different from zero. The parameters α and β are set such that m=-2 when 𝜋𝑡=1 in (8) and m=2 when 𝜋𝑡=1 in (9). Figure 1 shows the change in the
dynamic multiplier with respect to the coverage ratio.
Figure 3.3: The change of the multiplier with respect to the coverage ratio.
Figure 3.3 shows that equation (13) has the shape of a hockey stick. This is really convenient as it prevents the multiplier 𝑚𝑡 from becoming negative when the cushion 𝐶𝑡 is positive and
give a negative exposure to stocks. Figure 3.3 also shows that when 𝜋𝑡 ≥ 1 the multiplier
decreases as 𝜋𝑡 increases, which accounts for the risk aversion of an investor. From figure
13 3.3 it can be deduced that when 𝜋𝑡< 1 the multiplier increases as 𝜋𝑡 decreases, which
accounts for the loss aversion of an investor. It follows that equation (13) accounts for the risk averse behavior of an investor when confronted with gains, so equation (13) is the equivalent of the concave value function in the prospect theory of Tversky and Kahneman (1979). Equation (14) accounts for the risk seeking behavior of an investor when confronted with losses, so equation (14) can be seen as the equivalent of the convex value function in the prospect theory of Tversky and Kahneman (1979). The jump is located at 𝜋𝑡=1, because the cushion 𝐶𝑡 is zero in that case and the exposure to stocks is zero independent of the
multiplier. The exposure to stocks is now defined as:
𝑒𝑡 = max[min (𝑚𝑡× 𝐶𝑡, 𝑊𝑡),0] (15)
The exposure diagram below shows a similar shape as the exposure diagram of MSS. This is not a surprise since both strategies work with a negative and a positive multiplier conditional on the coverage ratio. Due to the short-sale constraints the exposure is limited to 100%, that is why we see a flat curve at first. As wealth increases towards the critical point where it equals the floor (𝜋𝑡=1) we see a sharp decline in the exposure towards zero when 𝑊𝑡 = 𝐹
(𝐶𝑡 = 0 and 𝜋𝑡=1). When 𝜋𝑡>1 we see the curve change into a parabola. This seems
counterintuitive since the multiplier is decreasing, so we would expect the exposure to decrease. However, the decreasing exposure due to the decreasing multiplier is neutralized by an increasing cushion. Therefore, we see an increasing exposure until €140.000, after that we see the exposure decrease towards zero. The exposure remains at zero after the
€200.000 level has been transcend due to the short-sale constraints. Therefore we also see a flat line at the end of the curve.
Figure 3.4: the change in the exposure to stocks with respect to the portfolio value. 0% 20% 40% 60% 80% 100% 120% 20 40 60 80 100 120 140 160 180 200 Exp o su re , e (in %)
Portfolio value, W (in thousands €)
14
3.4 Hypothesis
The FAS, 2 and 3 are all expected to be improvements on the regular CPPI strategy. The exposure diagrams all indicate that, in the situation where the portfolio value is exceeded by the floor, the three alternative strategies take more risk and are therefore expected to perform better. In the situation where portfolio value exceeds the floor, the FAS and 2 show similar exposure diagrams and only the DMS slightly deviates. Given the properties and exposure diagrams of the different strategies it is expected that the three alternative
strategies all outperform the CPPI strategy in the situation where the coverage ratio is below one. The longer the time horizon the higher the probability that the coverage ratio will recover above one. This implies that the shorter the time horizon the higher the outperformance of the alternative strategies is expected to be.
H1: All strategies outperform a regular CPPI strategy, which is taken as a benchmark, in the starting situation where the expected liabilities exceeds the portfolio value by a significant amount and an investment horizon of 10 years based on the portfolio performance
measures.
To check the robustness of these results and detect possible differences in the results, I also test this hypothesis with a 20 year investment horizon.
MSS and 3 show similar exposure diagrams when the coverage ratio is below one, in
contrast with the FAS that shows a different exposure diagram. They deviate in the fact that the FAS adjusts the floor and MSS and 3 adjusts the multiplier. This has the implication that in the case of a coverage ratio that is slightly below one, the cushion in the MSS and 3 is very small and so is the exposure to stocks whereas the cushion for the FAS is artificially larger and thus creates a larger exposure to stocks. Within the range of a coverage ratio of 0.9 and 1 the exposure to stocks is greater for the FAS than for the MSS and DMS. Therefore, in the starting situation where 𝜋𝑡=1, the FAS is expected to outperform the MSS and DMS.
H2: The FAS outperforms the MSS and DMS in the starting situation where the portfolio value equals the expected liabilities and an investment horizon of 10 years based on the portfolio performance measures.
Again to check the robustness of the results I also test this hypothesis with a 20 year investment horizon.
15 10% of the value of the assets13. This implicates that in the starting situation where 𝜋
𝑡=0.5 ,
The MSS and DMS are expected to generate a larger exposure to stocks than the FAS and therefore outperform the FAS.
H3: The MSS and DMS outperform the FAS in the starting situation where the expected liabilities exceed the portfolio value by a significant amount and an investment horizon of 10 years based on the portfolio performance measures.
In the long run, when the coverage ratio is expected to transcend the threshold level of one, it would be reasonable to assume that these differences converge and that the two
strategies will show more similar outcomes. However, since the MSS and DMS are expected to transcend the coverage of one quicker, we would still need to see an outperformance by the MSS and DMS over the FAS. Therefore this hypothesis is also tested with a 20 year investment horizon.
The DMS is a modified and presumably improved version of the MSS and has a different exposure diagram. It shows that between a value of €60.000 and the threshold level of €100.000 the slope for the MSS is slightly steeper. This means that the DMS allows for a larger exposure to stocks than the MSS. When the portfolio value transcends the threshold level, the exposure to stocks shows a significant difference. The DMS uses a dynamic
multiplier that accounts for increasing risk aversion of a liability driven investor. As the value of the assets increases and the positive difference with the expected liabilities increases, the need for exposure to stocks decreases. The dynamic multiplier that is a function of the coverage ratio ensures a decreasing exposure to stocks as the positive value of the cushion increases. Considering that the prospect theory holds and applies for liability driven
investors14, this strategy should outperform the MSS, since it accounts for the risk
preferences of an investor under the prospect theory.
H4: The DMS outperforms the MSS in the starting situation where the expected liabilities exceed the portfolio value by a significant amount and an investment horizon of 10 years based on the portfolio performance measures.
The downside of this strategy compared to the other two is that the upside potential is limited, once the threshold level has been surpassed the exposure to stocks decreases until the exposure is zero. The probability that this threshold level will be transcend increases with the investment horizon. Therefore, I also test this hypothesis with a 20 year investment horizon.
13 Note that the cushion is only limited in the case where the coverage ratio is below one and
equation (2) applies.
16
4. Research method and data
4.1 Data collectionI collect the data necessary for the simulations from one database. I use the online database of Robert Shiller, which contains monthly prices of the S&P 500 index over the period 1871-2018, to calculate monthly returns that represents the stocks in the three models. All dividends are assumed to be immediately reinvested. The monthly returns 𝑟𝑡 for the S&P
500 index at time t, are given by the following formula: 𝑟𝑡 =(𝑃𝑡+1− 𝑃𝑡+ 𝐷𝑡)
𝑃𝑡 (16) where, 𝑃𝑡+1 relates to the quoted spot price of the S&P 500 in one month, 𝑃𝑡 is the actual
quoted spot price of the S&P 500 at time 𝑡 and 𝐷𝑡 represents the dividends paid out. These
returns are converted into continuous compounded returns, 𝑟𝑡∗, using the following
formula:
𝑟𝑡∗= ln(1 + 𝑟𝑡) (17) The constant mean return model is used to calculate the average monthly return over the period 1871-2018, which serves as an input in the Monte Carlo situation (Mackinlay, 1997). The average continuous compounded return for the S&P 500 index, 𝑅̅, is given by:
𝑅̅ =𝑖 1 𝑁∑ 𝑟𝑡 𝑁 𝑡=0 + 𝜁𝑡 (18)
where, 𝑁 represents the investment horizon, 𝜁𝑡 is the time period 𝑡 disturbance with
𝐸(𝜁𝑖𝑡) = 0 and 𝑣𝑎𝑟(𝜁𝑖𝑡) = 𝜎𝜁2𝑖.
The online database of Robert Shiller is also used to collect data on the 10 year US treasury yield, which represents the bonds in the three strategies. I use the same 1871-2018 period to calculate the average yield on the 10 year US treasury bonds. The average yield, 𝑌̅, is given by the following formula:
𝑌̅ = 1 𝑁∑ 𝑦𝑡
𝑁
𝑡=0
+ 𝜁𝑡 (19)
where, 𝑁 represents the investment horizon, 𝜁𝑖𝑡 is the time period 𝑡 disturbance for security
𝑖 with 𝐸(𝜁𝑖𝑡) = 0 and 𝑣𝑎𝑟(𝜁𝑖𝑡) = 𝜎𝜁𝑖
2. This average yield (19) is on an annual basis, the
monthly average yield, 𝑌̅∗, is given by:
17
4.2 Simulation
I use a similar setup as Cesari and Cremonini (2002) to employ a Monte Carlo simulation where I test the performance of each alternative on a hypothetical portfolio. I generate 120 (10 years) and 240 (20 years) monthly returns for stocks and bonds from two normal
distributions, one that characterizes stocks and one that characterizes bonds. The S&P 500 index is used to represent a well-diversified portfolio of stocks. The 10 year US treasury yields are used to represent bonds. The analysis is based on two different starting situations. Scenario 1 represents an initial situation where the value of the portfolio is €100,000
portfolio and the target level is constant at a level of €100,000 and a time horizon of 10 and 20 years. In addition I assume an initial withdrawal rate of 4% on an annual basis (Estrada and Kritzman, 2018). This is equivalent to a coverage ratio of one. Scenario 2 represents an initial situation where the value of the portfolio is €50,000 and the expected liabilities (floor) are constant at a level of €100,000 and a time horizon of 10 and 20 years15. This is equivalent
to a coverage ratio of 0.5. In addition I follow Estrada and Kritzman (2018) and assume an initial withdrawal rate of 4% on an annual basis. The multiplier for each strategy, except the DMS, is set at two16.
In total there are four strategies each being tested in four situations, which results in a total of 16 portfolios, as can be seen in table 1. All portfolios are rebalanced on a monthly basis and the simulation process is repeated for a total of 10,000 runs.
To see the empirical implication – how would the alternative strategies actually have performed in the past – of these strategies I also employ a historical simulation to test the performance of each proposed strategy on the same hypothetical portfolio as for the Monte Carlo simulation. I use monthly returns of S&P 500 index and 10 year US treasury yields over a 10 and 20 year horizon. Again an initial withdrawal rate of 4% on an annual basis is used and the portfolio is rebalanced on a monthly basis. For the historical simulation the yields need to be transformed into monthly bond returns. Using the yield, 𝑌̅, from equation (11) the monthly bond returns can be calculated. However, some strict assumptions are needed:
1. At the start of every investment period (10 or 20 years) the 10 year treasury bond is assumed to sell at par.
2. After every investment period of 10 or 20 years the bond is rolled over into a new bond (selling at par) with a maturity of 10 or 20 years.
15 Note that this doesn’t apply for the FAS, where the floor is not constant.
16The multiplier for the DMS depends on the coverage ratio and is dynamic by nature. In the status
18 The bond returns are split up into two components, a ‘price return’ and a ‘coupon return’. The price return, 𝑟𝑡𝑝, is calculated as the change in the bond price and is defined as:
𝑟𝑡𝑝 =𝐵𝑡− 𝐵𝑡−1
𝐵𝑡−1 (21) where, 𝐵𝑡 represents the market price of the bond at time t and is defined as:
𝐵𝑡 =𝐶𝐹𝑡 𝑌 (1 − 1 (1 + 𝑌)𝑛) + 𝐹𝑉 (1 + 𝑌)𝑛 (22)
where, 𝐶𝐹𝑡 is the cashflow from the coupon payment at time t and FV is the face value of
the bond and set at €100,-. The calculation of the monthly coupon return, 𝑟𝑡𝑐, is simply
dividing the annual coupon return by twelve months. The annual coupon return is set equal to the yield, Y, on the first month of every investment period. The coupon is constant over the remainder of the investment period. Finally the price return, 𝑟𝑡𝑝, and coupon return, 𝑟𝑡𝑐, are added together and converted into continuously compounded returns:
𝑟𝑡𝐵= ln[1 + (𝑟𝑡𝑝+ 𝑟𝑡𝑐)] (23) In total there are fourteen consecutive 10 year periods and seven consecutive 20 year periods from 1871 to 2011 and a total of 1,400 monthly returns. Further summary statistics are displayed below in table 1.
The Monte Carlo simulation has several advantages over the historical simulation. All circumstances under the Monte Carlo simulation are equal for each strategy. A Monte Carlo simulation is more convenient for employing statistical tests and make inferences about the statistical significance of the results. The number of simulated investment periods
19 Table 1
Summary statistics
The table below shows some summary statistics on the underlying data and both simulations. Panel A shows the statistics on the S&P 500 and US 10y treasury yield. Observations represents the number of monthly returns observed in the period 1871-2019. Panel B shows the statistics of both
simulations under scenario 1 and a 10 year horizon. The observations represents the number of investment periods of 10 year. Panel C shows the statistics on both simulations under scenario 2 and a 20 year horizon. The observations represents the number of investment periods of 20 year.
Panel A: Stock & bond data
SP500 US treasury yield Mean 0,72% 0,37% Standard deviation 4.03% 2.29% Sharpe ratio 0.18 0.16 Observations 1776 1776 Time period 1871-2019 1871-2019
Panel B: Simulation scenario 1
10 year Historical Monte Carlo
Observations 14 10,000
Withdrawal rate % 4% 4%
Initial portfolio value € 100,000 € 100,000 Initial target level € 100,000 € 100,000
Panel C: Simulation scenario 2
20 year Historical Monte Carlo
Observations 7 10,000
Withdrawal rate % 4% 4%
Initial portfolio value € 50,000 € 50,000 Initial target level € 100,000 € 100,000
4.3 Portfolio performance measures
There are many different tools to measure the performance of a portfolio and in addition asset allocation strategies. The Sharpe ratio, introduced by Sharpe (1966), is a
well-established and broadly used performance ratio in asset allocation models. The Sharpe ratio describes the expected excess return per unit of risk. The Sharpe ratio is derived from the capital asset pricing model (CAPM), where the slope of the capital market line (CML)
represents the Sharpe ratio. The Sharpe ratio of a portfolio usually takes the following form: 𝑆𝑟𝑝 =
(𝐸(𝑟𝑝) − 𝑟𝑓)
20 where, 𝐸(𝑟𝑝) represents the expected return on a given portfolio, 𝑟𝑓 is the return on the
risk-free asset17 and 𝜎
𝑝 is the volatility of a given portfolio, measured by the standard
deviation. The Sharpe ratio will be used as a tool to measure the performance of the three strategies.
Within the framework of LDI, the funded status is prevalent amongst pension funds. The funded status compares the assets to the liabilities of a pension plan18. The funded ratio is a
measurement tool of the funded status and Rust (2006) defines it as “the ratio of the plan’s assets over its liabilities”. This funded ratio is closely related to the coverage ratio mentioned in Estrada and Kritzman (2018), this ratio can be used for both institutional investors and individual investors. In this paper the coverage ratio, 𝜋𝑡 , is defined as the fraction of the
target level that is covered by the value of the portfolio: 𝜋𝑡 =𝑊𝑡
𝐹 (25) where, 𝑊𝑡 represents the value of the retirement plan assets at time t which is equated to
𝑊𝑡 in equations (2) and (3), 𝐹 is the present value of the expected liabilities which is
equated to 𝐹 in equations (2) and (3). As discussed in the introduction, portfolio insurance strategies such as CPPI and TIPP give rise to complications when 𝐹 > 𝑊𝑡. To relate this
problem to the coverage ratio, 𝐹 > 𝑊𝑡 is equivalent to 𝜋𝑡 < 1. Later on in the paper, the
coverage ratio will be presented in percentages to highlight small differences between the strategies19. This enables the use of the coverage ratio as a tool to measure the performance
of the three strategies.
Estrada and Kritzman (2018) argue that the coverage ratio by itself is not enough to measure the performance of alternative investment strategies. They state in their paper: “investors are much more likely to be displeased with outcomes that fall short of full supporting retirement spending than pleased with outcomes that generates surpluses.” This statement is an implication of the prospect theory and assumes that investors are loss averse and the coverage ratio should account for this asymmetry in preferences. Therefore, Estrada and Kritzman (2018) link the coverage ratio to a kinked utility function.
𝑈(𝜋) = 𝜋 1−𝛾− 1 1 − 𝛾 𝑖𝑓 𝜋 ≥ 1 (26) 𝑈(𝜋) =1 1−𝛾− 1 1 − 𝛾 − 𝜆(1 − 𝜋) 𝑖𝑓 𝜋 < 1 (27)
21 where, U represents utility, 𝜋 is the coverage ratio, 𝛾 the coefficient of risk aversion and λ is a linear penalty coefficient when π <1. The kink, also known as the ‘reference point’, is located at the status quo – a coverage ratio of one – which gives the utility function a convenient property (see figure 4.1). The penalty now only takes place if the coverage ratio falls below one. When the coverage ratio is above one, the investor is assumed to exhibit increasing relative risk aversion, so the investor derives increasing satisfaction but at a diminishing rate. When the coverage ratio is below one the loss aversion is illustrated by a steep down-slope. A kinked utility function describes the preference of an investor under the prospect theory (Tversky and Kahneman, 1979). It fits ideally in the framework of liability-driven investment where a downside breach of the target level outweighs the upside breach of the target level. This is now a convenient performance measure that takes into account the differences in preferences of an investor under the prospect theory.
Figure 4.1: The kinked utility function of the coverage ratio, π.
In addition to this coverage ratio we introduce the failure rate to measure the performance of the three strategies. The failure rate, Ω, is defined as the probability that the coverage ratio fails to remain at or above 1 at the end of the investment horizon:
Ω =1
𝑇 ∑(𝜋 < 1)
𝑇
1
(28) where, T is the number of simulation runs. An important shortcoming of this failure rate is that it only measures the failure rate at the end of the investment horizon and thus does not account for intermediate failure rates during the investment horizon. To account for the
-0,10 -0,05 0,00 0,05 0,10 0,15 90% 95% 100% 105% 110% 115% 120% 125% 130% 135% 140% 145% 150% U tility Coverage ratio, π
22 failure rate over the entire investment horizon, an average failure rate, Ω̅, is introduced which is given by:
Ω̅ = 1
𝑁 ∑(𝜋 < 1)
𝑁
𝑡=0
(29) where, 𝑁 denotes the investment horizon.
Table 2
Outcomes simulation
Table 2 shows summary statistics, including the prevalent performance measures, of the outcomes of the Monte Carlo simulation for every strategy, including the CPPI strategy. Panel A shows the outcomes in scenario 1 – the starting situation where the target level is equal to the portfolio value – for a 10 year and 20 year horizon. Panel B shows the outcomes in scenario 2 – the starting situation where the target level exceeds the portfolio value by a significant amount – for a 10 year and 20 year horizon. These numbers below are all averages or medians of 10,000 simulations of 10 and 20 year investment periods.
Panel A: Scenario 1
10 year CPPI FAS MSS DMS
Median terminal value € 165,142 € 184,171 € 162,956 € 161,980
Average return E[r] 0.49% 0.58% 0.48% 0.41%
Standard deviation σ 2.95% 3.06% 2.51% 1.97%
Failure rate Ω 2.15% 0.13% 1.35% 1.85%
Average failure rate Ω̅ 10.45% 6.02% 9.81% 9.68%
Average coverage π 135.91% 143.85% 135.90% 131.22%
U(π) 0.07 0.08 0.07 0.07
Sharpe ratio 𝑆𝑟𝑝 0.17 0.19 0.19 0.21
Average exposure20 42.01% 49.33% 43.44% 21.25%
Turnover 4.36% 5.27% 5.02% 3.62%
20 year CPPI FAS MSS DMS
Median terminal value € 383,421 € 354,101 € 383,359 € 297,803
Average return E[r] 0.64% 0.60% 0.64% 0.55%
Standard deviation σ 3.05% 3.54% 3.05% 2.74%
Failure rate Ω 0.00% 0.04% 0.00% 0.08%
Average failure rate Ω̅ 3.48% 5.56% 3.34% 4.58%
Average coverage π 232.31% 216% 231.77% 204.75% U(π) 0.09 0.09 0.09 0.10 Sharpe ratio 𝑆𝑟𝑝 0.21 0.17 0.21 0.20 Average exposure 70.21% 67.23% 70.92% 13.37% Turnover 2.95% 3.65% 3.12% 2.92%
20 This is the average exposure to stocks as a fraction of the portfolio value of 10,000 simulated
23 Table 2 – Continued
Panel B: Scenario 2
10 year CPPI FAS MSS DMS
Median terminal value € 72,037 € 89,733 € 93,784 € 95,483
Average return E[r] 0.33% 0.50% 0.56% 0.58%
Standard deviation σ 2.11% 2.44% 3.21% 3.28%
Failure rate Ω 91.95% 70.88% 58.70% 55.53%
Average failure rate Ω̅ 98.30% 94.88% 86.88% 85.21%
Average coverage π 62.89% 69.28% 74.24% 74.95%
U(π) -0.74 -0.62 -0.55 -0.54
Sharpe ratio 𝑆𝑟𝑝 0.16 0.20 0.18 0.18
Average exposure 0.48% 20.08% 66.38% 68.49%
Turnover 0.11% 1.92% 5.29% 4.92%
20 year CPPI FAS MSS DMS
Median terminal value € 145,985 € 131,604 € 183,387 € 181,437
Average return E[r] 0.51% 0.46% 0.62% 0.58%
Standard deviation σ 2.59% 2.59% 3.55% 3.62%
Failure rate Ω 12.86% 16.58% 6.02% 5.97%
Average failure rate Ω̅ 65.99% 70.20% 50.74% 49.93%
Average coverage π 95.60% 90.07% 115.47% 111.77% U(π) -0.48 -0.39 -0.26 -0.26 Sharpe ratio 𝑆𝑟𝑝 0.20 0.18 0.18 0.16 Average exposure 16.82% 27.36% 63.05% 46.67% Turnover 1.60% 2.85% 4.76% 4.00% 4.4 Statistical tests
24 𝑡 = (∑ 𝐷)/𝑁 √∑ 𝐷2−((∑ 𝐷) 2) 𝑁 (𝑁 − 1)(𝑁) (30)
where, ∑ 𝐷 is the sum of the differences between the paired samples, ∑ 𝐷2 is the sum of squared differences between the paired samples, (∑ 𝐷)2 is the sum of the difference squared and N is the sample size.
5. Results
Table 3 shows the differences between the CPPI strategy and the three proposed strategies in the situation of underfunding. The results suggest that all strategies outperform a CPPI strategy. Some of the differences seem to be rather small, but are nonetheless significant. This is a perfect illustration of the statistical power of a Monte Carlo simulation, that is driven by the possibility to extend the sample size to an almost unlimited number of sample runs21. The outperformance is explained by the nature and properties of the different
strategies22. We see that for all the three strategies at least in 70% of the 10.000 simulated
investment periods the coverage ratio is higher than for the CPPI strategy. This is consistent with the differences between the means of the coverage ratios. The utility-based coverage shows similar results: all three strategies have a higher value than the CPPI strategy. Again at least in 70% of the 10.000 simulated investment periods this utility-based coverage ratio is higher for the three strategies in comparison to the CPPI strategy. The average failure rate emerges in the sense that this performance measure shows the greatest dominance of the three strategies. In 90% of the simulated investment periods the average failure rate is lower for all three strategies in comparison with the CPPI strategy. Lastly, the differences in Sharpe ratios are small but still significant distinguishable. The main findings from panels A, B and C from table 3 suggest that the alternative strategies do exactly what they are meant to do and solve the problem of a CPPI strategy in the situation where the floor exceeds the value of the portfolio. Overall the first three panels of table 3 show a clear outperformance of every strategy over the CPPI strategy.
Does this strong dominance also hold if the investment period is extended, increasing the probability to ‘jump out’ of a coverage ratio below one? Panels D, E, and F shows some very different results. The outcomes for the FAS relative to the CPPI in panel A have turned around. Extending the investment period with 10 years clearly has a negative effect on the
21 Note that the sample in this paper contains 10.000 sample runs (10.000 investment periods of 10
and 20 years), which explains the high statistical significance of the differences in performance measures.
25 FAS. This seems counterintuitive since the exposure to stocks follow the same pattern when the coverage ratio is above one. When the coverage ratio is below one, the FAS generates a constant exposure of 20% to stocks whereas the CPPI strategy generates a constant
exposure of 0% to stocks. It would be expected that the FAS outperforms the CPPI strategy, as is the case for a 10 year investment horizon. The MSS and DMS still outperform the CPPI strategy over a period of 20 year, however with slightly less dominance than over a 10 year period. In addition, over a 20 year period the CPPI strategy seems to generate higher Sharpe ratios than the alternative strategies. This is caused by the fact that the CPPI strategy has a higher exposure to bonds. This might seem odd, however bonds outperform stocks for sufficiently long holding periods (Hodges, Taylor and Yoder, 1997). The long holding periods could be the explanation for why we see this outperformance of CPPI Sharpe ratio only at the 20 year period. To summarize, the results for the 20 year investment period are in line – although the effect is smaller – with the 10 year period and expectations except for the FAS.
Table 3
Hypothesis 1
Table 3 shows the outcomes of the difference between a CPPI strategy and the other three
strategies. SAF %, MSS % , DMS % and CPPI% in each panel shows the fraction of 10.000 simulation runs the performance measure outperforms the comparing strategy. SAF, MSS, DMS and CPPI corresponds to the means of the performance measures as in table 1. Panel A test the proposition that the FAS outperforms a CPPI strategy in scenario 2 and a 10 year horizon. Panel B test the proposition that the MSS outperforms a CPPI strategy in scenario 2 and a 10 year horizon. Panel C test the proposition that the DMS outperforms a CPPI strategy in scenario 2 and a 10 year horizon. Panel D, E and F correspond with panel A, B and C respectively with a 20 year horizon. ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively.
Panel A: FAS vs. CPPI
26 Table 3 – Continued Panel C: DMS vs. CPPI 10 year π U(π) Ω̅ Sr(p) DMS % 74.7% 74.8% 92.4% 59.3% CPPI % 25.3% 25.2% 7.6% 40.7% DMS 74.95% -0.54 85.21% 0.18 CPPI 62.89% -0.74 98.30% 0.16 Difference 12.06%*** 0.20 -13.09% 0.02***
Panel D: FAS vs. CPPI
20 year π U(π) Ω̅ Sr(p) FAS % 43.6% 52.3% 44.4% 42.1% CPPI % 56.4% 47.7% 55.6% 57.9% FAS 90.07% -0.39 70.20% 0.18 CPPI 95.60% -0.48 65.99% 0.20 Difference -5.53%*** 0.09*** 4.21%*** -0.02*** Panel E: MSS vs. CPPI 20 year π U(π) Ω̅ Sr(p) MSS % 68.2% 62.3% 71.6% 44.2% CPPI % 31.8% 37.7% 28.5% 55.8% MSS 115.47% -0.26 50.74% 0.18 CPPI 95.60% -0.48 65.99% 0.20 Difference 19.86%*** 0.22*** -15.25%*** -0.01*** Panel F: DMS vs. CPPI 20 year π U(π) Ω̅ Sr(p) DMS % 68.0% 62.0% 72.6% 38.7% CPPI % 32.0% 38.0% 27.4% 61.3% DMS 111.77% -0.26 49.93% 0.16 CPPI 95.60% -0.48 65.99% 0.20 Difference 16.17%*** 0.22*** -16.06%*** -0.04***
27 where the portfolio value equals the target level. The results from panel A and B show a relative small but significant difference between the FAS and MSS as well as between the FAS and DMS. All performance measures are in favor of the FAS, except for the Sharpe ratio. The Sharpe ratio of the FAS and MSS are more or less equal and any small difference is not significant. The DMS has a higher Sharpe ratio than the FAS. In this scenario the relative small cushion in the MSS and DMS in contrast with the artificially larger cushion in the FAS, creates a larger exposure to stocks in the FAS23. This larger exposure to stocks enables the
FAS to outperform the MSS and DMS.
Again the investment period is extended to 20 years to see if these results still hold. Panels C and D show the results of a 20 year investment period. From the perspective of the FAS the results are similar with the 20 year investment period in table 3. The FAS seems to perform poorly for longer investment horizons. The results have turned around and the MSS now outperforms the FAS contrary to the results of a 10 year horizon. The results in panel D are a bit more ambiguous, where the FAS generates a higher coverage ratio, however the utility- based coverage ratio and the failure rate are in favor of the DMS. This may seem rather odd, however remember that the coverage ratio does not account for the asymmetry in
preferences of an investor (Estrada and Kritzman, 2018). It is possible that the coverage ratio is higher and the utility-based coverage ratio lower. This would imply that the DMS is more equipped to avoid coverage ratios below one and thus is a better fit for a liability driven investor under the prospect theory. The results that stands out most is that the FAS
performs rather poorly for a 20 year investment period relative to the 10 year period. This is peculiar, since especially the FAS and MSS are identical when the coverage ratio is above one24. So for the 20 year period – when the probability of transcending the coverage ratio of
one increases – the results are expected to be similar as for the 10 year period. The historical simulation outcomes are not useful in this context. The failure rate is practically zero, which implies that all strategies - except for the DMS - employ the same strategy. Therefore the results are very similar for these strategies.
23 This refers to scenario 1 introduced in chapter 4: Research method and data. In this scenario the
portfolio value is equal to the floor, which is equivalent to a coverage ratio of one.
28 Table 4
Hypothesis 2
Table 4 shows the outcomes of the difference between the FAS and MSS and FAS and DMS. The properties of table 4 are identical to table 3. Panel A test the proposition that the FAS outperforms the MSS in scenario 1 and a 10 year horizon. Panel B test the proposition that the FAS outperforms the DMS 3 in scenario 1 and a 10 year horizon. Panel C and D correspond with panel A and B, respectively only with a 20 year horizon instead of a 10 year horizon. ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively.
Panel A: FAS vs. MSS 10 year π U(π) Ω̅ Sr(p) FAS % 59.98% 59.96% 61.79% 49.19% MSS % 40.02% 40.04% 38.21% 50.81% FAS 143.85% 0.08 6.02% 0.19 MSS 135.90% 0.07 9.81% 0.19 Difference 7.95%*** 0.01*** -3.79%*** 0.00 Panel B: FAS vs. DMS 10 year π U(π) Ω̅ Sr(p) FAS % 63.83% 59.44% 61.59% 38.59% DMS % 36.17% 40.56% 38.41% 61.41% FAS 143.85% 0.08 6.02% 0.19 DMS 131.22% 0.07 9.68% 0.21 Difference 12.63%*** 0.01*** -3.66%*** -0.02*** Panel C: FAS vs. MSS 20 year π U(π) Ω̅ Sr(p) FAS % 42.74% 42.15% 49.06% 31.36% MSS % 57.26% 57.85% 50.94% 68.64% FAS 216.22% 0.09 4.42% 0.17 MSS 231.77% 0.09 3.34% 0.21 Difference -15.55%*** 0.00 1.07%*** -0.04*** Panel D: FAS vs. DMS 20 year π U(π) Ω̅ Sr(p) FAS % 53.87% 41.53% 51.18% 32.57% DMS % 46.13% 58.47% 48.82% 67.43% FAS 216.22% 0.09 5.56% 0.17 DMS 204.75% 0.10 4.58% 0.20 Difference 11.47%*** 0.01*** 0.98%*** -0.03***
29 stocks than the DMS and MSS this is probably caused by the finding of Hodges et al. (1997), that bonds tend to have higher Sharpe ratios than stocks. These findings are supported by the outcomes of the historical simulation, where we can clearly see that the strategies with a relatively high exposure to bonds have higher Sharpe ratios. The limited cushion in the FAS prevents this strategy from having a relatively large exposure to stocks when the coverage ratio is well below one.
Panels C and D show the results for a 20 year period and the results are consistent with previous results: the FAS performs poorly for longer investment periods. The dominance by the MSS and DMS is even more outstanding. In the situation where the target level exceeds the portfolio value by a significant amount, the MSS and DMS outperform the FAS. The outcomes from the historical simulation are in line with these findings. The results from the historical simulation show a clear outperformance by the MSS and DMS except for the Sharpe ratio, which is again higher for the strategy with a higher exposure to bonds. We know from hypothesis one that all three strategies outperform the CPPI strategy in this situation. To answer the research question: “What is the most optimal dynamic asset
allocation strategy for liability driven investors when the expected liabilities exceed the value of the portfolios?” one last test is needed.
Table 5 Hypothesis 3
Table 5 shows the outcomes of the difference between the FAS and MSS and FAS and DMS in scenario 2. The properties of table 5 are identical to table 3. Panel A test the proposition that the MSS outperforms the FAS in scenario 2 and a 10 year horizon. Panel B test the proposition that the DMS outperforms the FAS 3 in scenario 2 and a 10 year horizon. Panel C and D correspond with panel A and B, respectively only with a 20 year horizon instead of a 10 year horizon. ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively.
30 Table 5 – Continued Panel C: MSS vs. FAS 20 year π U(π) Ω̅ Sr(p) MSS % 74.14% 74.23% 75.34% 51.55% FAS % 25.86% 25.77% 24.66% 48.45% MSS 115.47% -0.26 50.74% 0.18 FAS 90.07% -0.39 70.20% 0.18 Difference 25.40%*** 0.13*** -19.46%*** 0.00 Panel D: DMS vs. FAS 20 year π U(π) Ω̅ Sr(p) DMS % 74.07% 73.74% 76.20% 45.87% FAS % 25.93% 26.26% 23.80% 54.13% DMS 111.77% -0.26 49.93% 0.16 FAS 90.07% -0.39 70.20% 0.18 Difference 21.70%*** 0.13*** -20.27%*** -0.02***
Table 6 shows the difference between the MSS and DMS in a situation of underfunding, to determine which strategy is the most optimal in the context of this paper. The first thing to notice from table 6 is the resemblance between the outcomes of the two strategies. Despite this resemblance, the differences are all significant except for the Sharpe ratio in panel A and the utility-based coverage ratio in panel B. Overall, panel A shows that the DMS outperforms the MSS by a minuscule margin. However, since the differences are this small, it would be misleading and bold to say that the DMS is the clear better of the two. Because of the small differences it is really convenient that we have a 20 year period as a robustness check.
Table 6 Hypothesis 4
Table 6 shows the outcomes of the difference between the DMS and MSS. The properties of table 8 are identical to table 3. Panel A test the proposition that the DMS outperforms the MSS in scenario 2 and a 10 year horizon. Panel B corresponds with panel A only with a 20 year horizon instead of a 10 year horizon. ***, **, and * denote significance at the 1%, 5%, and 10% levels, respectively.
31 Table 5 – Continued Panel B: DMS vs. MSS 20 year π U(π) Ω ̅ Sr(p) DMS % 49,05% 51,01% 51,82% 45,60% MSS % 50,95% 48,99% 48,18% 54,40% DMS 111,40% -0,26 50,86% 0,16 MSS 115,47% -0,26 50,74% 0,18 Difference -4,06%*** 0,00 0,13%*** -0,02***
Panel B shows the results for the 20 year period. Unfortunately the results are similar to panel A in the sense that the differences are very small. The MSS now generates a larger coverage ratio and the utility-based coverage ratio shows no differences. The DMS does still generate a slightly lower failure rate but also a lower Sharpe ratio. It is not an astonishment that the results are similar since both strategies show relative similar exposure diagrams. However, since the DMS accounts for the risk preferences of the investor, a more
considerable difference in the utility-based coverage ratio would be in line with expectations.
The results of the historical simulation suggests that the DMS has a slight edge on The MSS for both investment periods. However, as explained in chapter four, the number of
investment periods are too small to draw any statistical inferences. Nonetheless, it does give an indication how the strategies would have performed in the past. The results show that on average the DMS would have been the most optimal strategy to use for a liability driven investor over the period 1871-2011.
6. Conclusion
32 upside potential. This has the implication that investors are risk averse when confronted with gains and become risk seeking when confronted with losses. The CPPI strategy forces the investor to act in the opposite way, taking virtually zero risk when confronted with losses, decreasing the probability to ‘jump out’ of these losses. Therefore, I propose three alternative strategies as a solution to this shortcoming of the CPPI strategy:
1. Floor Adjustable Strategy (FAS): the liabilities are redefined, at a minimum acceptable level, such that the present value of the liabilities is less than the value of the
portfolio. After redefining the liabilities the investor can continue with a CPPI strategy.
2. Multiplier Switch Strategy (MSS): the investor takes additional risk to restore the value of the portfolio above the present value of the liabilities. A convex strategy can be used to achieve this by using a negative multiplier. When the value of the
portfolio is above the expected liabilities the investor can continue with a concave strategy with a positive multiplier (CPPI).
3. Dynamic Multiplier Strategy (DMS): the investor uses a CPPI strategy with a dynamic multiplier, which accounts for the risk preferences of the investor.
The main findings suggest that all three strategies are an improvement on the CPPI strategy. When investors are in situations where their portfolio value is below their target level, all proposed strategies outperform a CPPI strategy. Moreover the DMS and MSS, in particular, seem to work very well and are – according to the results in this paper – a significant improvement on the CPPI strategy. The FAS underperforms relative to the DMS and MSS, however it is still ought to be an improvement on the CPPI strategy. The DMS and MSS show very similar results – which is not surprising given the similar properties of both strategies – with a slight edge toward the DMS. This leads to the conclusion that the DMS is the most optimal asset allocation strategy for liability-driven investors25. These relative
groundbreaking findings could have extensive implications for liability-driven investors. Many pension funds are struggling with their coverage ratios and find themselves in a situation of underfunding26. This paper shows that more conventional asset allocation
strategies, such as a CPPI strategy, performs poorly in a situation of underfunding. The three strategies – especially the DMS and MSS – proposed in this paper offer an improvement to a CPPI strategy in the situation of underfunding. Since the so called ‘pension crisis’ could
25 Note that the differences between the MSS and DMS are considerable small in such a way that
many investors would be indifferent between the two strategies. However, the differences are significant, which makes strategy 3 – from a rational point of view – the most optimal.
