Portfolio Insurance and VaRoP. A Comparison

Contents

Introduction

Portfolio Insurance

Stop-loss strategy

Synthetic put strategy

Constant-Proportion-Portfolio-Insurance

VAR and VaRoP

Comparison

Stop Loss and VAR / VaRoP

CPPI and VAR / VaRoP

Summary and outlook

Appendix: Assumptions and discussion

Discussion of assumptions

Bibliography

Introduction

Investments in money and capital markets involve different loss potentials that market participants should be able to manage. Below is an overview and comparison of selected strategies to manage the risks.

Portfolio insurance (PI) strategies were developed in the 1980s. They are used to hedge portfolios or individual investments against price losses. The volume of assets hedged with these strategies is significant. Different forms of individual strategies have developed over the years.

Risk quantification and Value at Risk (VAR) strategies emerged around the same time. Risks of individual investments or portfolios were measured and different strategies were developed to take them into account in Value at Risk optimised portfolios (VaRoP). VaRoP is a strategy that calculates an optimal portfolio taking into account a given or permissible maximum VAR.

Both strategies are intended to protect portfolios from losses in value. Their similarities and differences are presented and summarised in this paper. Their applicability in practice is also examined.

Portfolio Insurance

Portfolio insurance is a form of programme trading, like block trading and index arbitrage. In block trading, market participants can offer extensive positions in individual assets or entire portfolios to a block trader without having to look for special counterparties. In index arbitrage, on the other hand, market participants exploit deviations of an index future or an index option from the respective theoretical values. With suitable transactions, they can then achieve almost risk-free excess returns on the spot or futures market.

The literature provides a variety of definitions for portfolio insurance.¹ Summarising these, the aim is to protect a portfolio against losses in value. This protection can cover the whole or parts of the portfolio. It regularly consists of interest-bearing financial securities, stocks and futures market instruments. To protect the portfolio, market participants reduce the amount of stocks if the portfolio value approaches the minimum value set ex ante in the event of price losses. At the same time, they increase the amount of interest-bearing financial securities. Conversely, market participants proceed when the portfolio's price increases. Portfolio insurance strategies follow predefined trading rules; there is no demand for forecasting future developments. They provide protection against systematic capital market risks and can be used for any form of capital investment with a risk premium.² The main strategies of portfolio insurance are the stop-loss strategy, the synthetic put strategy and the constant proportion strategy.

Stop-loss strategy

In the stop-loss strategy (SLS), market participants specify ex ante a minimum price below which the value of the portfolio or the stocks should not fall until the end of the investment period.³ If the value reaches the stop-loss price, the portfolio or the stocks are sold and the funds received are invested in risk-free interest-bearing securities. In a dynamic stop-loss strategy, the portfolio or the stocks are bought back if the corresponding values rise above the stop-loss price again. If the stop-loss price rises as the level of the risk-free interest rate during the investment period, then market participants realise a comparable return at the end of the investment period. They also maintain their participation in increases in the value of the portfolio or individual stocks. In practice, it has been shown that this strategy leads to the desired result.⁴

Synthetic put strategy

In the synthetic put strategy (SPI), market participants use long and short positions in stocks and interest-bearing financial securities. In this way, they can generate a theoretical value trend and the resulting cash flows of a put option. They use money market securities or medium-term investments, as well as etocks or entire portfolios for this purpose. The positions to be taken are derived from the option valuation theory according to the binomial model or the formula of Black and Scholes⁵. Formula 1 is as follows:

Formula 1: Put-call parity for option valuation:

Abbildung in dieser Leseprobe nicht enthalten

Market participants can duplicate the value of an option. To do this, they take a long position in t in the amount of Ct, a short position in the amount of -St and a discounted long position in the amount of K*(1+rf)-(T-t) in order to have built up the theoretical price of a put option with a long position in Pt. These positions must be continuously adjusted to changing market conditions and timing during the hedging period. At the end of the investment period, market participants hold a position in a portfolio that is synthetically hedged with a put option. Synthetic put strategies are easy to illustrate in theory. They should also be easy to apply in practice, especially for institutional market participants. Nevertheless, they have not been used to any significant extent in recent years. The reasons for this are not apparent; any obstacles to increased use presumably need to be removed.

Constant-Proportion-Portfolio-Insurance

In constant proportion portfolio insurance (CPPI) for stocks or other forms of investment, market participants specify a minimum value for the portfolio. This minimum value is the floor and is smaller than the portfolio value in t = 0. The floor increases by a certain percentage during the hedging period.⁶

The difference between the portfolio value and the floor is the cushion. This value is variable and results from the dynamic adjustment of the portfolio from formula 2:

Formula 2: Exposure of Constant Proportion Portfolio Insurance

Abbildung in dieser Leseprobe nicht enthalten

The exposure Et is the proportion of the portfolio in risky securities.⁷

If the value of the portfolio increases and Et > Et-1, then the shares of the risky positions are expanded according to formula 2. Here, stocks are bought and interest-bearing securities, ideally zero-coupon bonds, are sold.

In the event of price losses of the portfolio and/or increases in the value of the floor, if Et < Et-1, the risky positions are reduced accordingly. Stocks are sold and interest-bearing securities, at best zero-coupon bonds, are bought.

At the end of the investment period, the value of the portfolio is assumed to be at least equal to the floor. The⁸ probability of losses of the portfolio is thus reduced to zero. The probability of increases in the value of the hedged portfolio is maintained.⁹

The multiplier has a special theoretical meaning, as formula 2 shows. The larger the multiplier, the more extensive the purchases and sales, the more the composition of the portfolio changes.¹⁰

A suitable multiplier is essential if the portfolio value in T is to be above that of the floor. It seems obvious that with inappropriately large multipliers, the losses of exposure as a result of price jumps can be so great that they can no longer be compensated for by increases in the value of the interest-bearing securities.¹¹,¹²

VAR and VaRoP

The Value at Risk (VAR) measure shows the maximum loss that market participants can theoretically realise for a defined time horizon with a given confidence interval and standard deviation of their risk position.¹³ Risk positions can be stocks, interest-bearing securities, derivatives or entire portfolios thereof.

Market participants first determine the size and standard deviation of their risk position. They determine the degree of certainty for the confidence interval, e.g. 95% or 99% certainty.¹⁴ They also assume a certain probability distribution, usually the normal distribution. The time horizon must also be determined, usually from one day to one year. The market participants can then calculate the VAR.¹⁵

Essential to the calculation of VAR is the estimation of volatility. It can be determined in different ways, e.g. from historical data, via simulations or calculations of the implied volatility of options.¹⁶

The VAR is calculated using the formula 3.

Formula 3: Calculation VAR

Abbildung in dieser Leseprobe nicht enthalten

The procedure for calculating the daily VAR for a theoretical equity portfolio is illustrated with an example.¹⁷ The individual values required are:

Abbildung in dieser Leseprobe nicht enthalten

If the VAR is to be calculated for several combined positions, then the¹⁸ matrix calculation is suitable.

Volatilities and standard deviations change continuously. To assess the risk, market participants should also continuously calculate the respective VAR of the portfolio.¹⁹

VAR alone is not active management. It must be embedded in an extended management strategy²⁰, in this case the Value at Risk optimised Portfolio (VaRoP) strategy.

In VaRoP strategies, market participants first define their risk positions across all forms of investment. To do this, they calculate the VAR of the total position using formula 3.

They then determine their optimal portfolio, derived from portfolio theory²¹ and via Efficient Market Theory (EMT) and the capital market line.²² Given a risk-free interest rate, the market participants can then calculate the efficient capital market curve. This shows the maximum expected return for a portfolio composition at a given risk. Simplified, the procedure via VaRoP can be formally represented as follows:

Abbildung in dieser Leseprobe nicht enthalten

In t0, the market participants determine their optimal portfolio according to the EMT. At the same time, they calculate the VAR of this position. If the efficient portfolio is at or within an admissible risk range, no transactions are executed. However, if the portfolio is outside the permissible portfolio range, then the market participants adjust it through corresponding transactions. Since the determinants change regularly over time, continuous buying and selling transactions can be expected.

Comparison

Stop Loss and VAR / VaRoP

In the stop-loss strategy (SLS), the floor is subtracted from the assets Vt. This results in the position St to be invested in risky securities. St is greater than zero, or St is equal to zero. Formally represented as follows: Vt - Floor = St, with St > 0; St; 0

The risky asset St takes on this role in comparison with the VAR, without forecasts of volatility and without active management of the position St. There is only a steady adjustment of the size of St, according to the changes in Vt - Floor = St.

St is only similar to VaRoP with active management of the portfolio Vt. Here, a forecast of the volatility and an anticipatory adjustment of the portfolio is made, corresponding to the procedures of strategies with VaRoP. Formally, it can be represented as follows:

St = VAR without management

Abbildung in dieser Leseprobe nicht enthalten

Here St is to be compared with VaRoP. P should correspond to VAR, and St minus the discounted strike price of the option can also correspond to VAR. Also, the strike price of the option should take values of the floor. Formally, it is as follows:

Abbildung in dieser Leseprobe nicht enthalten

[...]

¹ See T. Ebertz and C. Schlenger, 5/1995. R. Hohmann, 1996, pp. 12-16.

² R. Uhlmann, 2008, p. 3 and 17.

³ See K. Quandt, 2-3.8.2002.

⁴ However, increased transaction costs due to more frequent adjustments of the portfolio have a negative effect. Problems also arise in the event of sudden, very pronounced price changes, such as a "flash crash". This also applies to other forms of portfolio insurance. See R. Benders and M. Maisch, 24.11.2009. R. Benders and M. Eberle, 10.5.2010. U. Rettberg, 10.5.2010. T. Riecke, 1.3.2007. O.V., 22.4.2015. B. Finke, 23.4.2015. K. Slodcyk and A. Dörner, 23.4.2015. C. Siedenbiedel, 24.4.2015.

⁵ On option pricing, see F. Black and M. Scholes, 1973, pp. 637-654. J. C. Cox and S. A. Ross, 1976, pp. 145-166. L. Jurgeit, 1989.

⁶ For example, around the interest rate for risk-free investments, here the interest rate for risk-free zero-coupon bonds with a comparable maturity. See also R. Uhlmann, 2008, pp. 31-32.

⁷ Risky securities here are accordingly stocks or also futures and options. See R. Hohmann, 1996, pp. 105-109 and the sources cited there.

⁸ If short sales are not permitted, Et can be represented as follows: Et = max [m*Qt ; 0 ] If borrowing is prohibited, formula 2 for determining Et is accordingly: Et = min[ m*(Vt - Gt) ; St ] If the floor is known ex ante, the portfolio value can be represented as follows: Vt = max [Gt ; Gt + Qt ]

⁹ This is the difference to a comprehensive hedge with futures, where the probability of participating in later price increases of the portfolio is partially or completely given up.

¹⁰ One way to determine the multiplier is for market participants to set the level of the multiplier ex ante. Then they determine the amount of the initial exposure or floor depending on the portfolio value. Then m = (Et / Gt). See R. Hohmann, 1996, p. 109. R. Uhlmann, 2008, pp. 38-41.

¹¹ See A. F. Perold, 1986, p. 7. S. Mantel, 2014, pp. 42-43.

¹² There is therefore a suggestion in the literature that the multiplier should not be greater than the reciprocal of the amount of the largest expected negative price jump. See R. Uhlmann, 2008, pp. 136-139, 143. Another interesting question here is whether the multiplier should be determined with the help of the maximum sustainable value at risk. This question cannot be answered here. See the other sections of this paper.

¹³ See B Jendruschewitz, 1997, pp. 6-7. P Jorion, 2002, pp. 22-25, 117. For different forms of risk, see ibid, 2002, pp. 15-21. M Choudhry, 2006, pp. 30-32. For definition and differentiation of risks, see ibid, pp. 3-7. For different quantitative measurements of risks, see ibid, pp. 9-11.

¹⁴ For the determination of the confidence interval, see B. Jendruschewitz, 1997, pp. 19, 32. M. Choudhry, 2006, pp. 23, 46-47.

¹⁵ The expected loss associated with an intercept below the normal distribution curve and the critical confidence interval, in relation to the defined time horizon and the calculated standard deviation, see B. Jendruschewitz, 1997, pp. 19-20, 26-29. M. Choudhry, 2006, pp. 24-26, 35-36, 51-52.

¹⁶ On this see B. Jendruschewitz, 1997, pp. 31-32, 35-37, 39-40, 50-61, 64-73.

¹⁷ See T.M. Guldiman, 1995. B Jandruschewitz, 1997, pp. 20, 30, 33, 36-39, 96-100. P. Jorion, 2002, pp. 108-113.

¹⁸ On matrix accounting, see L. Kruschwitz, 1995, p. 314. See also B. Jendruschewitz, 1997, pp. 31, 34, 45-47, 78-80. M.Choudhry, 2006, pp. 39-44.

¹⁹ For a summary and critique of the VAR, see B. Jendruschewitz, 1997, pp. 110-112.

²⁰ On divergent risk management via limits, see B. Jendruschewitz, 1997, pp. 20-22. On VAR for active risk management, see P. Jorion, 2002, pp. 383-387. On VAR for interest bearing financial instruments see M. Choudhry, 2006, pp. 62-86, for options see ibid, pp. 88-99, for Monte Carlo simulations see ibid, pp. 102-107.

²¹ See H.M. Markowitz, 1952, pp. 77-91. The same, 1970. L. Perridon, M. Steiner and A. Rathgeber, 2009, pp. 252-258. J. Berk and P. DeMarzo, 2017, pp. 401-409.

²² L. Perridon, M. Steiner and A. Rathgeber, 2009, pp. 263-267. J. Berk and P. DeMarzo, 2017, pp. 417-422.