A Comparative Investigation of Portfolio Insurance Strategies in
The Netherlands
ThesisMaster of Science in Business Administration Specialisation: Finance
University of Groningen Faculty of Economics and Business
Abstract
In 2010 a lot of Dutch pension funds had trouble in maintaining an asset/liability ratio of more than 105%. In this paper we look at the assets and how we can prevent the assets from falling in value while still holding the possibility to gain from rising asset prices. We therefore compose 3-‐month maturity portfolio using a risky and a riskless asset and test several strategies on these portfolios. We compare two portfolio insurance
techniques, namely CPPI and OBPI, with a buy-‐and-‐hold strategy using the AEX Total Return Index as a risky asset and the 3-‐month LIBOR interest for the risk-‐free asset. We find that that the OBPI strategy is the most effective strategy followed by the CPPI strategy and finally by the Buy-‐and-‐Hold strategy.
JEL classification G11, G23, G32 Keywords
Portfolio insurance, CPPI, OBPI, buy-‐and-‐hold
The Effectiveness of Portfolio Insurance Strategies
A Comparative Investigation of Portfolio Insurance Strategies in the Netherlands
Jord Jansen *
I. Introduction
Since the beginning of trade the world has seen booms and crashes resulting in high volatility for prices in both company stocks and commodities. Take for example the tulip mania. According to Thompson (2006) the peak price of this bubble was in February 1637. Prices were 20 times higher than in November 1636, only 3 months earlier. However, some time later, in May 1637, the prices of these tulips crashed and people who bought contracts for this flower went bankrupt. In recent times we also see bubbles, crashes and crises. Think, for example, of the Asian Crisis in 1997 or the
Internet Bubble of 2001 and of course the financial crises that started in 2007. For many companies this volatility in the market provides problems. For them it would be very helpful if there would be some kind of insurance against these price fluctuations. In this paper we will investigate what kind of insurance strategy provides the best protection against losses below a specified floor. The goal is to provide a better insight in the possibilities and efficiency of portfolio insurance (PI) strategies.
The idea of portfolio insurance was born on September 11, 1976. Hayne Leland was wondering what kind of product would be appealing for the financial industry. It was on this day that his brother mentioned that after the stock market decline a lot of pension funds had withdrawn large quantities of their assets from the stock market and were missing out on the bull market of 1975. According to Leland & Rubinstein (1976, p. 1),
Leland his brother said:” If only insurance were available, those funds could be attracted back to the market”. He knew a little about the Black & Scholes formula (Black & Scholes (1973)) for valuing options and other derivatives. He discovered that the Black &
Scholes formula could be extended to create the options synthetically. This brought him on the idea to write an official framework for portfolio insurance together with Mark Rubinstein. They produced a paper with this framework in 1976 called ‘The Evolution of Portfolio Insurance’. From than on option-‐based portfolio insurance (OBPI) with
synthetic put options became a very useful tool for insuring a certain value of ones portfolio.
However, investors would, in a painful manner, find out that this way of portfolio insurance has its limitations. The problem with OBPI is that it depends on a theoretical framework. The problem with this is, for example, that it assumes that some of the input variables are constant while in the real world they fluctuate. This is one of the reasons why OBPI did not work in the crash of 1987 in which the volatility in the market rose while OBPI models assumed it was constant. Therefore, positions were formed on wrong information and losses were made. Another important issue that is related to the former problem was pointed out by Jacobs (1999). He claims that the positive feedback system inherent to these strategies amplifies the market price movements. For example, in the crash of 1987 stock prices were falling, portfolio insurance strategies needed to sell stocks increasing the pressure on stock prices to fall even further. Because of the handicap that OBPI has, people searched for alternatives. One of these alternatives is the constant proportion portfolio insurance strategy (CPPI).
CPPI was first introduced by Perold (1986) in his paper ‘Constant Portfolio Insurance’. CPPI does not depend on the Black & Scholes framework, as the OBPI strategy. Hence, it does not have to cope with the assumptions of this model like a constant volatility and constant interest rate. However, this strategy also has its disadvantages. The biggest disadvantage is that it does not rely on a theoretical framework. Moreover, there are two factors in the model which have to be determined without any idea of what the
theoretical value of these variables should be.
So, above we see two dynamic portfolio insurance strategies. However, is there one strategy that outperforms the other or could it be that there is a “do nothing” strategy, as Perold and Sharpe (1988) call it, which outperforms these dynamic portfolio insurance strategies? One of the best-‐known “do nothing” strategies is the buy-‐and-‐hold strategy. The idea of this strategy is to allocate assets between risky assets (stock) and safe assets (bills) and than hold the position until maturity. The proportion allocated to safe assets provides a minimum return while the proportion allocated to stocks provides an upside potential return.
beneficiaries of portfolio insurance, we will focus and approach our research from the perspective of these funds. One important note that we make is that this research only focuses on the asset side of the asset/liability ratio that pension funds have to maintain. Hence, it is still possible for a pension fund to fail this ratio when its assets are
protected. This could, for example, happen when the interest rate decreases, which increases the liabilities.
Now that we know what portfolio insurance is, who could benefit from it, and given there are multiple strategies to provide this protection, we could ask ourselves the following: What is the best and most efficient way to protect ones assets from losses while still having the possibility to gain from increasing asset prices? This will be the main research question of this paper.
To give a complete answer to this question we need to find an answer to several different aspects of portfolio insurance. We will try to answer the following research questions in order to obtain a valid conclusion:
-‐ Do portfolio insurance techniques provide superior floor protection against losses compared to a simple buy-‐and-‐hold strategy?
-‐ Which portfolio insurance technique is the best in protecting the floor value of a portfolio?
-‐ Which technique gives the most return over the investment period?
-‐ How do the distributions of the different strategies compare to each other? -‐ What technique is the most cost-‐efficient?
We will answer these questions for two portfolio insurance strategies, namely OBPI and CPPI. To look whether these strategies actually improve insurance against losses we will compare these strategies with a buy-‐and-‐hold strategy by simulating portfolio insurance strategies with a length of 3 months using the AEX Total Return index and the 3-‐month LIBOR middle rate as the interest rate. The reason for choosing three months as the insurance period is because pension funds have to report their coverage ratio to ‘De Nederlandsche Bank’ every three months. We compare the results between the several strategies over 111 3-‐month periods to answer the above questions.
This paper adds to the existing literature by adding information on the performance of these strategies in the Netherlands.
In section one we will discuss the literature with respect to the dynamic portfolio
insurance strategies and their performance. Next, we will discuss the methodology used for our analysis followed by a description of the data. After that we will discuss the results and draw the conclusions.
II. Literature Review
To understand the idea and reasoning of this paper we need to explain the basic principles of portfolio insurance and how it works. Rubinstein (1985) explains that portfolio insurance protects the insurance taker against a devaluation of his portfolio value below a certain minimum value at time T while still giving him the opportunity to benefit from the upward potential on his investment. Of course, such protection is not for free and thus a fee has to be paid for this kind of pay-‐off structure.
Now that we know that portfolio insurance is about protecting a portfolio from sharp declines in value, it is a good idea to look at some of the properties of the pay-‐off
structure of these strategies. According to Rubinstein (1985) there are three important properties for the return pattern of a perfectly insured portfolio. First, the probability of experiencing losses below a certain floor during the insurance period is equal to zero. Second, the expected return of any profitable position is a predictable percentage of the expected return that would have been earned if all funds were invested in the risky assets underlying the portfolio. The third property depends on three assumptions. (i) There is a restriction, which obliges the portfolio to be invested only in one asset class and in cash loans. (ii) The expected rate of return on the risky asset is higher than the expected return on cash. (iii) The portfolio insurance is fairly priced. If these three assumptions hold then the third property states that, among all strategies that possess property one and property two, the insured portfolio will have the highest expected return amongst insured portfolios. These three properties are always present in a perfectly insured portfolio.
minimum value of their portfolio but can accept reasonable risk on the extra value of the portfolio. An example of the second investor type would be funds with managers who think they can produce above average returns through stock selection. So the paper of Leland (1980) supports our idea of pension funds being a consumer of portfolio insurance.
Next, we turn to the different kinds of portfolio insurance strategies beginning with the OBPI strategy. There are several different options to insure your portfolio within the OBPI strategy. It is for example possible to use exchange-‐traded European put options to insure the portfolio against losses. According to Rubinstein (1985) the advantage of using exchange-‐traded European put options is that it is 100 percent reliable in preventing losses. Furthermore, this strategy only depends on the price at maturity of the index, if dividends are reinvested, so that the portfolio is path independent.
The reason why path independence is a beneficial property is explained by Rubinstein (1985) and Bookstaber & Langsam (2000). The bottom line is that path dependency leads to extra uncertainty on the outcome of the portfolio value. The investor interested in portfolio insurance is not rewarded for bearing this extra risk and thus does not want to have it. Therefore, path independence is seen as a beneficial property when insuring a portfolio. As Bookstaber and Langsam (2000, p.3) state: “A strategy that is not path independent gives an uncertain payoff, and therefore violates the very premise of portfolio insurance: giving a known payoff.”
exercise early while European options have to be held to maturity. Because of this flexibility the cost of an American put option is higher than that of its European
counterpart. Since an investor, interested in insuring its portfolio on a certain date in the future, is not interested in early exercise he is not willing to pay for the extra costs an American option entails.
The use of exchange-‐traded European put options does not only bring advantages with it. MacKenzie (2004) states that there are several disadvantages with using exchange-‐ traded options for portfolio insurance. He says that although there have been organized option exchanges in the US from as early as 1973, most of the options available had short-‐term maturities so that problems arose when the expiration date of the insurance strategy was longer than the time to maturity of the option. Options have to be rolled over increasing the transaction costs of the strategy. Furthermore, there were
limitations in the amount of positions that could be accumulated. Another problem was that, especially in the 1970s and beginning of the 1980s, it was hard to find and collect options to insure a well-‐diversified portfolio. In these decades options were only traded on individual stocks and not on stock indices. The reason for this was that the Securities and Exchange Commission (SEC) was suspicious of derivatives like, for example, options because they feared that they would be used for destabilising speculation. Therefore, the SEC rejected proposals to introduce index options on the exchange. Rubinstein (1985) explains another important issue concerning the time to maturity of options. He explains that, if an option has to be rolled over because the insurance period is longer than the maturity of the put option, this has an impact on the path independency of the strategy. Reason for this is that the price of the option will differ with different market
return to fall in comparison with a put option that does not have to be rolled over. So, as the amount of rollovers becomes more frequent, path dependency increases.
One solution to these problems could be the use of the over-‐the-‐counter market.
Another financial institution could write a put option on the whole portfolio that needs to be insured. The benefits of this market are (i) the possibilities to match the maturity of the option to the maturity of the insurance period, (ii) the fact that only one option on the whole portfolio can be obtained so that the risk and return of the option exactly matches the risk and return of the portfolio, and (iii) path independence is restored. However, Hull (2009) and Bertrand & Prigent (2002) provide us with some
disadvantages for this method. First, the writer of the option may default so the
purchaser is subject to some credit risk. Second, over-‐the-‐counter options are not liquid due to the fact that they are adjusted to match one certain portfolio. Third, the prices of over-‐the-‐counter products are less competitive. So, although the over-‐the-‐counter market provides a solution to some problems mentioned above, it also has its disadvantages.
put and equalize it to the period to be insured and therefore there is no need to rollover options. Second, it is easier to insure a portfolio of assets with synthetic put options than with exchange-‐traded put options. This holds especially for the 70s and 80s since at that time options were only traded on stocks and not on indices. With synthetic puts you can choose the risky asset and make use of stock indices. Finally, the strike price of a
synthetic put can be chosen by the investor, therefore one does not have the problem of finding options with the right exercise prices.
Huu Do and Faff (2004) state that one of the problems with synthetic puts is that they are path dependent. Reason for this is that they are subject to replication error. Because of this error, funds are allocated to the wrong assets and the payoff becomes dependent on the future path of this asset. Another problem indicated by Huu Do and Faff (2004) is the necessity of frequent trading when a synthetic put strategy is used. Therefore, transaction costs will be relatively high. According to them, one solution to this problem could be the use of futures. Instead of trading in the asset itself one could short futures written on the stock. For example, if the risky asset falls in value one should increase the short position in futures and vice versa. Sutcliffe (2006) states some other benefits of trading in futures like greater liquidity, lower transaction costs and faster execution. However, there are several disadvantages with the use of futures. There exists the risk of a potential mismatch between the asset price and the futures price. Also, futures
contracts are standardized, hence, it is much more difficult to replicate a portfolio perfectly. Furthermore, if the maturity of the insurance strategy is longer than the maturity of the future, than the future has to be rolled over, so that path dependency increases.
So we see that there are different OBPI strategies for protecting the value of the portfolio which all have their own advantages and disadvantages. But there are also some other disadvantages that concern all of these OBPI strategies. Rubinstein (1985) gives several examples of disadvantages due to market imperfections or uncertainty for these OBPI strategies. The problems that he mentions are uncertain interest rates, uncertain volatility, security price jumps, and transaction costs.
Rubinstein explains that if there are uncertainties about the interest rate, it will be impossible to replicate long-‐term European options exactly since OBPI relies on
constant interest rates. He also states that uncertain volatility is a major impediment on dynamic insurance strategies since the predicted volatility is not only important for the pricing of options but also for all determinants. Hence, misestimation could have a major impact on the outcomes of an insurance strategy. Rendleman & O’Brien (1990) wrote a paper in which they investigate the effect of volatility misestimation on the outcome of a portfolio insurance strategy. They found that volatility misestimation has a significant impact on the ending payoffs. However, there results are based on the assumption that the volatility is constant over the whole insurance period, stock prices evolve according to a certain probability distribution, and a weekly revision of the portfolio. Moreover, the results of their research pertain only limited to pure option-‐replication insurance strategies.
misalignment in the allocation of resources and the risk exists that the portfolio value will deviate from its insured value. Another way in which this can happen is when the portfolio is not updated continuously. The ideal situation with OBPI strategies would be to update ones portfolio continuously. However, in real life this is impossible and
portfolios are updated, for example, daily. Thus, between every update there is a jump in prices so that positions cannot move to its new level in a smooth manner, which can lead to a portfolio value that is lower than the assigned floor.
Lastly Rubinstein explains that transaction costs are an extra cost that is made by changing the positions in the risky and riskless assets. These costs are not incorporated in the model for determining the correct positions. Therefore, positions are allocated in a slightly inefficient manner. According to Arnott & Clarke (1987) transaction costs can have two effects. First, it can lead to the possibility of missing the floor of your insurance strategy. However, these shortfalls are relatively small and should not be too alarming. Second, the transaction costs have an impact on the mean and median returns of the portfolio. The average return on an insured portfolio drops significantly against the rate of return without transaction costs.
There are some special cases of CPPI in which the strategy mimics other strategies. For example, Bertrand and Prigent (2002) showed that although CPPI does not depend on input variables (interest rates, variance) like the synthetic put strategy does, it can replicate a synthetic put strategy. They state that the synthetic put strategy can be considered as a generalized version of the CPPI approach. In this case the multiplier is a function of the value of the risky assets and hence it is variable itself. Furthermore, Black & Perold (1992) explain that a stop-‐loss strategy is another special case of the CPPI strategy. A stop-‐loss strategy orders to sell the asset if its value falls below a specified floor value. A stop-‐loss strategy can be presented as a CPPI strategy when the multiplier goes to infinity so that the complete portfolio is completely invested in the risky asset if the portfolio value is above the floor value. The assets completely switch to the riskless asset when the portfolio value is below the floor value. The last case is explained by Constantinou and Khuman (2009). They state that if the multiplier is set to one, the CPPI strategy is equal to a buy-‐and-‐hold strategy. If the multiplier is equal to one then it is impossible to fall below the floor since only the cushion is invested. So during the investment period there is no need to adjust the amount of money allocated to the riskless asset.
market pricing mechanism1. Furthermore, there is no need to agree on a definitive
expiration date for the insured portfolio, the CPPI strategy can be used as long as necessary. Another important property is pointed out by Merchant (2004). He states that with rising interest rates the likelihood of higher returns increases. The reason for this is that with higher interest rates, there is less cash needed to make sure that the minimum floor is achieved. Therefore more cash can be allocated to the risky asset, which increases the chance of a higher return.
Earlier we stated that CPPI is very popular because of its simplism, however, this simplism also comes with a downside. Bookstaber & Langsam (2000) explain that the CPPI strategy lacks a theoretical foundation. It is a heuristic that does not contain a foundation for analytical study like, for example, the Black-‐Scholes formula. So, values for the multiple and floor are chosen without any theoretical framework that can explain why these values are used. Furthermore, the CPPI technique has problems with price jumps. As is the case with OBPI techniques, CPPI strategies could break through the floor because the positions cannot change gradually with prices. Cont & Tankov (2009) refer to this risk as ‘Gap Risk’. They have investigated this problem and they introduce a jump parameter to adjust for jumps in prices so that the multiplier is corrected based on the investor’s risk aversion. Another problem has to do with the transaction costs. For example, Black & Perold (1992) have theoretically shown that in a CPPI strategy it is possible that, if the index ratio follows a geometric Brownian motion, the transaction costs can destroy the cushion above the floor as the trading frequency increases.
Another very important problem with the CPPI strategy is called the ‘cash lock’ problem (Brandl (2009)). This problem occurs when the value of the portfolio invested in risky
assets declines by as much as the size of the cushion. When this happens, the portfolio will be fully invested in the riskless assets and it is impossible to benefit from a price increase of the risky assets. This also explains the fact that the CPPI strategy is a path dependent strategy. For example, if a portfolio gets cash locked before maturity of the insurance period, it influences the future outcome of the portfolio value.
Now that we know that there exist inefficiencies in portfolio insurance strategies, it could be the case that the strategies have different levels of efficiency. There have been several papers that compare the CPPI and OBPI strategy. For example, Huu Do (2002) looked at the performance of these two strategies using the (old) Australian All
Ordinaries Index (AAOI) from 1992 up to 2000. He divided the sample period in 33 non-‐ overlapping three-‐month insurance periods. Furthermore, he uses transaction costs of 0.5%. Huu Do finds that CPPI dominates in floor protection with daily rebalancing. However, the synthetic put strategy works better when there is a value-‐based rebalancing trigger like, for example, a market movement (AAOI) of more than 2%. Furthermore, he finds that there is no justification to use dynamic portfolio insurance practises when CPPI and OBPI are restricted to trading in only one bill and ones index.
Black and Rouhani (1989) compared the pay-‐offs of both the CPPI and OBPI strategies. Their conclusion was that the CPPI strategy works better in a volatile market, and that the OBPI strategy works better in relatively calm markets.
is split in consecutive 2-‐month insurance periods. Furthermore, transaction costs are 0.8%. They find that both techniques provide a superior performance to the buy-‐and-‐ hold technique. However, the performance of CPPI really stands out against the OBPI technique.
Another paper by Bertrand and Prigent (2002) compares the OBPI strategy and the CPPI technique on several aspects. They compare the two strategies on their payoffs,
expectations, variance, skewness and kurtosis of the returns, possible property of
stochastic dominance, and some of the quantiles of the returns. They conclude that there is no dominance for the standard criteria of portfolio choices. Furthermore, they found that, conditionally to some events of the dynamics of the asset price, it is possible to prefer one strategy to the other. For example, in case of a large drop in asset prices the value of the portfolio under the CPPI strategy is always larger than under the OBPI strategy. Next to that they prove that the synthetic put method can be seen as a generalized CPPI method.
percentage of the initial insured investment rises, the probability that the CPPI portfolio value is higher than the value of its counterpart increases.
One can see that there already is quite some literature about this subject. However, conclusions in these papers are all different and thus there is no definitive answer. Therefore we try to add some information on this subject by performing this study. In the next section we will discuss the methodology that we use in this paper.
Table 1: Empirical Results: OBPI versus CPPI
Author (Year) Subject Method Conclusions
Black & Rouhani (1989)
Constant Proportion Portfolio Insurance and the Synthetic Put Option: A Comparison
Monte Carlo simulation and an analysis of the impact of actual/implied volatalities
(i) CPPI perfoms better in a bear market or a slightly bullish market (ii) OBPI works better in a
moderately bullish market Huu Do (2002) Relative Performance of
Dynamic portfolio Insurance Strategies: Australian Evidence
Simulations of the CPPI and OBPI strategy using the AAOI index using data from 1992 till 2000
(i) CPPI dominates in floor protection using daily rebalancing.
(ii) OBPI dominates in floor protection with trigger based rebalancing
(iii) PI is not supported if it is restricted to trading in one risky and one riskless asset
Bertrand & Prigent (2002)
Portfolio Insurance Strategies : OBPI versus CPPI
Comparison of both strategies in pay-offs, property of
stochastic dominance, expectations, variance, skewness and kurtosis of the returns. And the “Greeks” are studied.
(i) There is no dominance for the standard criteria of portfolio choices (ii) Conditional to some dynamics of asset prices one strategy can be preferred over the other.
Author (Year) Subject Method Conclusions
Bertrand & Prigent (2003)
Portfolio Insurance Strategies: A Comparison of Standard Methods when the Volatility of the Stock is Stochastic
Both methods are analyzed in a Black-Scholes model and in a stochastic volatility world modelled by a Ornstein-Uhlenbeck process
(i) Stochastic volatility slightly increases the expected return for OBPI but decreases it for CPPI (ii) CPPI is more affected by stochastic volatility
(iii) if the percentage of the initial insured investment rises, the probability that the CPPI portfolio value is higher than the OBPI value increases.
Er & Erdogan Aktan (2009)
Performance Of Portfolio Insurance Strategies: Evidence From Turkey
Simulations of the OBPI and CPPI strategy using data from the Turkish ISE-30 from 1997 till 2008
(i) Both OBPI and CPPI perform better than a buy-and-hold strategy (ii) CPPI performs better than OBPI
III. Methodology
In the introduction we stated the following research question: “What is the best and most efficient way to protect ones assets from losses while still having the possibility to gain from bull markets?” To answer this question we have to state two hypotheses. The first hypothesis is needed to answer the question whether portfolio insurance
techniques provide better protection than other techniques like the buy-‐and-‐hold strategy. The second hypothesis tests whether one of the two portfolio insurance techniques performs better than the other. Below we find the two hypotheses:
Hypothesis 1
H0 The performance of OBPI and CPPI as an insurance technique, is equal to the performance of a Buy-and-Hold strategy.
H1 The performance of OBPI and CPPI as an insurance technique is not equal to the performance of a Buy-and-Hold strategy.
Hypothesis 2
H0 The performance of OBPI as a portfolio insurance technique is equal to that of CPPI.
H1 The performance of OBPI as a portfolio insurance technique is not equal to that of CPPI.
A.1. OBPI
The OBPI strategy is based on the pricing formula of Black and Scholes (1973), which was extended by Merton (1973). To get a good understanding of how this strategy works, we will first explain the assumptions and principles of the OBPI strategy using exchange-‐traded put options.
The model for the theoretical value of a put option depends on several assumptions, some of which are already mentioned and explained in section II. The assumptions are:
1. The underlying asset (stock) price follows a geometric Brownian motion in which the expected return of the stock price and stock price volatility remains constant over time
2. There are no transactions costs or taxes.
3. There are no dividends on the stock during the life of the option. 4. There are no riskless arbitrage opportunities.
5. Security trading is continuous.
6. The world in which the option exists is assumed to be risk-‐neutral. 7. Investors borrow and lend at the same risk-‐free rate of interest. 8. The short-‐term risk-‐free interest rate is constant over time. 9. Short selling of securities with the use of proceeds is permitted. 10. The option uses the European exercise terms.
Using these assumptions one can obtain the value of a European put option by using the following formula: (1)
where St is the stock price at time t, K is the strike price of the option, r is the risk-‐free
interest rate, σ2 is the variance of the underlying asset, T is the time to maturity, and N(x) is the cumulative normal distribution function. This formula does not account for
dividends while in the real world there are dividends. Therefore we make use of the AEX Total Return Index, in which the dividends are reinvested in the index. Hence, we do not have to take into account a dividend yield in our calculations.
Now that we know how an exchange-‐traded put option is valued we will turn to the creation of a synthetic put option. To create a synthetic put we have to adjust equation (1). We add the factor St to both sides of the equation to obtain:
€
pt+ St = Ke
−rT
N(−d2) + [1 − N(−d1)]St (2)
On the left-‐hand-‐side we see that portfolio insurance can be obtained by investing St in
the asset and by buying a put on this asset that covers the value invested in the asset. On the right-‐hand-‐side we see that the same portfolio insurance can be obtained by
To find the values that have to be invested in stocks and bonds we assume that the total value of the portfolio at the start is equal to W0. If we would invest all of it in stocks than
€
nS0 = W0 (3)
Where n is the amount of stocks, S0 is the price of stocks at the start of the insurance
period.
The floor value, which is the minimum value of the portfolio at maturity, is equal to aW0.
Where a is the floor protected as a percentage of the initial portfolio value W0.
If we want to protect the portfolio we could buy a put for every stock held at the start of the insurance period. However, costs are involved in buying a put, these costs have to be paid out of the portfolio value. So the portfolio value is composed as:
€
b(P0+ S0) = W0 (4)
where b is the amount of bonds and puts that need to be bought for the portfolio to be protected. Because P0 and S0 have a positive value, b should be smaller than n.
We want to protect the portfolio from falling below the floor value aW0 at maturity.
Hence, at maturity:
€
b(PT + ST) ≥ aW0 (5)
This can also be written as:
€
bmax(ST,K) ≥ aW0 (6)
If we substitute (3) for W0 and rearrange the equation we obtain for K:
€
K = an
Now, if we assume that at the start of the insurance period St is equal to the portfolio
value W0 than n=1 so that equation (7) can be written as:
(8) Because b < n, b has to be smaller than 1. K is the floor value that we will use to calculate the put value.
Now that we explained how we can make a portfolio insurance strategy with exchange traded options using equation (4), we will explain how it is done using a risky and riskless asset. To make a synthetic portfolio insurance strategy we have to invest an amount equal to the delta of equation (4) in the risky asset and the other part in the riskless asset. The delta can be calculated as follows:
€
δW0
δSt
= b[N(d1) −1+1] = bN(d1)
(9) This is equal to the amount that we need to invest in the risky asset.
To calculate this amount we need to estimate a value for b. This can also be done using equation (2) and (4). Equation (4) can also be written as:
€
b[Ke−rT
N(−d2) − StN(−d1) + St] = W0
(10)
If we use equation (8) to replace K we see that the only variable that is not known is b. So if we solve formula (10) for b and put it into equation (9) we obtain the proportion to invest in risky assets.
As stated earlier we also need to invest in riskless assets to obtain a secured portfolio. The proportion invested in riskless assets is equal to:
(11) Where qr is the proportion of riskless assets.
So if an amount equal to the delta of the portfolio is invested into the risky asset, and the rest of the portfolio is put in the riskless asset, we should obtain an insured portfolio. One important thing to note is that the portfolio is only insured for a very short time since the delta of a portfolio changes with St. Hence, proportions in the risky and riskless
asset should be updated continuously to provide perfect portfolio insurance.
A.2. CPPI
The CPPI strategy also allocates the wealth of the portfolio between risky assets and non-‐risky assets. However, as stated before, the CPPI strategy does not rely on a theoretical framework for deciding how much of the portfolio should be designated to the risky asset. The amount of the portfolio designated to the risky asset is calculated as:
(12) Where is the exposure to the risky asset, Wt is wealth at time t, Ft is the floor value
If m > 1 than the CPPI strategy is leveraged which means that a part of the floor is used to invest in the risky asset. If the risky asset falls in value, a part of the risky assets is sold and invested in the riskless asset. An implication of leveraging is that the amount invested in the risk-‐free asset is not always equal to the present value of the floor during the investment period. Hence, the value invested in the risk-‐free asset is equal to
(13) Earlier in this paper we explained that the CPPI strategy is not based on a theoretical framework and thus there is no way to determine the value of Ft and m. However, to
compare this strategy with the OBPI strategy, we need to specify Ft and m. Therefore, we
adopt the approach of Kat (1994): the floor value in the CPPI strategy will be equal to the strike price of the OBPI strategy and will thus be set equal to aW0e-rT at the beginning
of the insurance period. Furthermore, m is chosen so that the initial allocation to the risky and non-‐risky asset is the same as in the OBPI strategy. Bertrand and Prigent (2002) provide us with an explanation for this approach.
As stated earlier, the OBPI strategy can be seen as a generalized CPPI approach
(Bertrand & Prigent, (2002)) with a variable multiplier. Both types of strategies obtain portfolio insurance with dynamic management of the proportions in risky and riskless assets. Bertrand and Prigent (2001) have shown how the multiplier for an OBPI strategy can be established. We know from Hull (2009) that put-‐call parity is
(14) We than adapt it so that the total value of the asset and option are equal to the initial portfolio value W0:
b( pt+ St) = b(ct+ Ke
−rT) = W
0 (15)
This can also be written as: € b( pt + St) = b[StN(d1) − Ke −rTN(d 2)] Cushion +bKe −rT Floor (16) In equation (16) we see that the first part on the right hand side of the equation is equal to the cushion of the CPPI strategy and the second part is equal to the floor. In the CPPI strategy the cushion is equal to the exposure to risky assets divided by the multiplier. With this information we can calculate the multiplier for the OBPI strategy as:
€ mt = Et stock (Wt − Ft) = N(d1)Wt StN(d1) − Ke −rT N(d2) (17)
This explains the approach from Kat (1994) in which he sets the proportions in the risky asset equal at the start of the insurance period for both portfolio insurance techniques using the multiplier. So at the beginning of each insurance period, the OBPI strategy is equal to the CPPI strategy.
A.3. Buy-and-Hold
The last strategy that we will discuss is the Buy-‐and-‐Hold strategy. This is by far the simplest of the three strategies that we discuss in this paper. The idea of the strategy is to allocate the portfolio between a risky and a riskless asset at the beginning of the period and wait till the end of the “insurance” period to see what the final value is. Let it be clear that this strategy is not a portfolio insurance strategy. It is possible for the portfolio value to fall below the set floor.
and CPPI strategy. Hence, the total portfolio value, using a buy-‐and-‐hold strategy, can be described by:
and (18) Where Wt is the portfolio value at time t, α is the proportion invested in the risk-‐free
asset, and β is the proportion invested in the risky asset.
A.4. Assumptions
Before we explain how we compare the several different strategies we have to make some assumptions that are important for giving some boundaries to this research. First, it is not possible to sell short. The reason for this is that during the investigated period it was not always possible to sell short due to trading restrictions. For example, in 2008 minister Bos prohibited naked short selling in the Netherlands2. Hence, to give a good
comparison over time, we restrict this research by taking away the possibility of short selling.
A.5. Comparison
There are several different ways in which we can test which strategy performs best in insuring a portfolio value and how dependent these outcomes are on misestimating variables. The first tests that we will perform test how many times the portfolio achieves the minimum required floor and what the variance is in floor protection. We only test these values at maturity. The percentage of times the floor is achieved is calculated as:
€
nachieved
ntotal
(19) Where nachieved is the amount of portfolios in which the floor is achieved and ntotal is the
total amount of portfolios.
We will also look at the variance of the value achieved at maturity. The standard deviation is calculated as:
€ (Wi,T − W ) 2 i=1 n
∑
ntotal −1 (20) Where Wi,T is the realized value of portfolio i at maturity,€
W is the average realized
value of the portfolios, and ntotal is the amount of portfolios created. Furthermore, we
will look at the average return that every strategy provides. The return is annualized and can be calculated as:
€
Ri = (Wi,T
W0 )
4
−1 (21) Where Ri,T is the annualized return on portfolio i, Wi,T is the realized value of portfolio i
at maturity and W0 is the initial value at the start of the portfolio insurance period.
The standard deviation of these returns will be calculated as: € (Ri,T − R)2 i=1 n
∑
ntotal −1 (22) Another factor that we will look at are the transaction costs created by every strategy. Transaction costs will be calculated as 1% of the value that is transferred from the risky asset to the riskless asset and vice versa. The costs of 1% are arbitrarily chosen since these costs are not incorporated into the model. Hence, they do not affect the results and the relative difference in transaction costs remains the same between the strategies.Next, we will test the distribution of the portfolio insurance outcomes. Measures that we use for that are skewness and kurtosis.
To test the robustness of the outcomes we will test several values for the different variables. First, we will test the impact of misestimating the variance. To do this we will add or deduct 10% in variance of the historical variance. We will then compare these results with the results found using the historical variance. Furthermore, we will change the floor value that has to be achieved at maturity. We will test the floor protection with a quarterly required floor value of -‐1.25% and 0% of the initial portfolio value.
