Constant Proportion Portfolio Insurance

The Constant Proportion Portfolio Insurance (CPPI) is a dynamic portfolio - hedging strategy .

history

The portfolio hedging strategies have their origins in the 1980s. They arose due to long-term price losses on the national and international securities markets (e.g. the sharp rise in oil prices on the stock exchanges in the early 1980s, which resulted in a sustained decline in prices ). The long-term upward movement of share prices on the stock exchanges in the 1990s pushed hedging strategies into the background. The idea of ​​portfolio hedging was initiated in particular by the work of Black and Jones (1987) and Black and Perold (1992).

Basic concept

The aim of portfolio hedging strategies is to limit the risk of loss in the event of falling prices on the securities markets and, at the same time, to enable participation in rising securities markets. The instruments of the CPPI strategy include risky financial investments (stocks) and risk-free fixed-income investments ( money market fund certificates ). Since this is a dynamic strategy , a permanent portfolio shift between risky and risk-free investments takes place during the entire observation period. The CPPI strategy assumes an ex-ante minimum portfolio value.

In order to present complex facts of the basic concept of the CPPI strategy as simply and understandably as possible, the following assumptions are made:

• Shares without dividends are traded on the market.
• There is no possibility of borrowing.
• No short sales are allowed.
• It is possible to buy any number of securities (any number of shares).
• There is a constant interest rate on the market throughout the investment period.

Initial situation:

Initial fortune	This amount is available to the investor at the time available ${\displaystyle t_{0}}$	€ 100,000
running time	The investment period is one trading year	250 days
Risk-free interest	The market has a constant interest rate throughout the investment period (nominal annual interest rate)	5%
Minimum portfolio value (floor)	The portfolio value may not fall below this value at the end of the investment period ( ) ${\displaystyle T}$	€ 100,000
multiplier	Reflects the investor's willingness to take risks	5

The values ​​of a portfolio secured with the CPPI strategy for the first three points in time are shown in the following table. The initial situation is zero.

Share price	40	60	90	100	110	120	140
Share portion	0	0	0.1182	0.2438	0.3516	0.4893	0.8052
Zero bond portion	1	1	0.8818	0.7562	0.6484	0.5107	0.1948
Portfolio value	92811.55	92792.99	97446.13	100,000	102339.22	105483.27	113449.61
Floor	95180.50	95161.47	95142.44	95123.42	95142.44	95161.47	95180.50
Cushion	0	0	2303.68	4876.58	7196.62	10321.80	18269.11
Exposure	0	0	11518.42	24382.91	35983.10	51609.01	91345.54
Reserve asset	92811.55	92792.99	85927.70	75617.09	66355.96	53872.26	22104.07
time	3	2	1	0	1	2	3

As can be seen from the initial situation presented above, the initial assets are € 100,000. As a result, the portfolio value is €. Floor in is the minimum portfolio value discounted from to and is calculated as follows: ${\displaystyle W(t_{0})=100\,000}$ ${\displaystyle t_{0}}$ ${\displaystyle T}$ ${\displaystyle t_{0}}$

${\displaystyle {\begin{array}{l}F(t)=F(T)\cdot \left({\frac {1}{1+{\frac {r}{T}}}}\right)^{T-t}\\F(t_{0})=100{.}000\cdot \left({\frac {1}{1+{\frac {0{,}05}{250}}}}\right)^{250}=95{.}123{,}42\\\end{array}}}$

The cushion or safety buffer is the difference between the current portfolio value ( ) and the floor ( ). This amount can be fully risked (the total loss can be accepted) and nevertheless the minimum portfolio value is guaranteed at point in time ( ) due to the proportion in the risk-free investment. ${\displaystyle W(t)}$ ${\displaystyle F(t)}$ ${\displaystyle T}$

${\displaystyle {\begin{array}{l}C(t)=\max(W(t)-F(t);\;0)\\C(t_{0})=\max(4{.}876{,}58;\;0)=4{.}876{,}58\\\end{array}}}$

In the above calculation the determination of the Cushion is shown at the time . Based on the assumption that short sales are not allowed, the cushion cannot go negative, i.e. H. it can be minimally zero. ${\displaystyle t_{0}}$

The exposure is the amount invested in risky assets. This share of shares is not only made up of the cushion, but is a constant multiple of it, as the probability of a total share loss in a day or overnight is very low.

${\displaystyle E(t)={\begin{cases}C(t)\cdot m&\mathrm {ohne\;Kreditbeschr{\ddot {a}}nkung} \\\mathrm {min} \left(m\cdot C(t);\;b\cdot W(t)\right)\qquad b\leq 1&\mathrm {mit\;Kreditbeschr{\ddot {a}}nkung} \end{cases}}}$

${\displaystyle E(t_{0})=\mathrm {min} \left(5\cdot 4876{,}50;\;1\cdot 100{.}000\right)=24{.}382{,}91}$(with m = 5, see table above)

The use of this strategy assumes that borrowing is prohibited. In the event of credit restrictions, the exposure is calculated as in the above formula and determined with specific numerical values ​​for the point in time . It stands for the maximum proportion of the assets that can be invested in a risky asset. This means that a maximum of 100% ( ) of the assets can be invested with risk. ${\displaystyle t_{0}}$ ${\displaystyle b}$ ${\displaystyle b=1}$

The remaining assets are called the reserve assets and are invested in risk-free securities. This amount is calculated as follows:

${\displaystyle {\begin{array}{l}R(t)=W(t)-E(t)\\R(t_{0})=100{.}000-24{.}382{,}91=75{.}617{,}09\\\end{array}}}$

This results in the following distribution of assets at the time : ${\displaystyle t_{0}}$

Equity weighting or risk-free fraction: ${\displaystyle {\frac {E(t_{0})}{W(t_{0})}}=0{,}2438}$ ${\displaystyle {\frac {R(t_{0})}{W(t_{0})}}=0{,}7562}$

In the case of a rising share price, in this example, the share return for the time ( ) is 9.53%. ${\displaystyle t_{1}}$

The stock return was calculated as follows:

${\displaystyle s(t)=\ln(S(t))-\ln(S(t-1))}$

for period 1:

${\displaystyle s(1)=\ln(110)-\ln(100)}$

The development of risk-free investment of at : ${\displaystyle t_{0}}$ ${\displaystyle t_{1}}$

${\displaystyle R(t_{0})\cdot \left({1+{\frac {r}{T-t_{0}}}}\right)}$

The portfolio value is thus in : ${\displaystyle t_{1}}$

${\displaystyle {\begin{array}{l}W(t_{1})=E(t_{0})\cdot (1+s(t_{1}))+R(t_{0})\cdot \left({1+{\frac {r}{T-t_{0}}}}\right)\\\\W(t_{1})=24{.}382{,}91\cdot (1{,}0953)+75{.}617{,}09\cdot (1{,}0002)\\W(t_{1})=102{.}339{,}06\\\end{array}}}$

The calculation of the values ​​for the other points in time is carried out analogously.

Simulation of the CPPI strategy in VBA

In the upper section of the graphic on the left, the development of the CPPI portfolio (white line), the share price (red line), the Cushion (light blue line) and the floor (yellow line) is shown. The lower section of the graphic shows the percentage portfolio breakdown of the risky (blue area) and risk-free investments (red area). As the graphic shows, a rising share price trend was simulated here. It can be clearly seen that the market portfolio or the pure equity portfolio has a higher value than the CPPI strategy for almost the entire investment period. This is due to the fact that part of the assets was invested in a risk-free investment, which, when share prices rise, generates a lower return than a risky investment. This distance can be controlled by choosing a suitable multiplier, provided that the value of the pure equity portfolio is above the floor. The composition of the portfolio in the lower section of the graph shows that the CPPI strategy is procyclical and trend-following. This means that when the share price rises, the share of stocks in the portfolio is increased and when the share price falls, it is reduced. If the share price rises so far that it exceeds the portfolio value, 100% of the assets are invested in a risky asset. This can be seen in the times or . ${\displaystyle m\cdot C(t)}$ ${\displaystyle W(t)}$ ${\displaystyle t=101}$ ${\displaystyle t=196}$

In contrast to other portfolio hedging strategies, CPPI permanently adjusts the portfolio to various influencing factors such as stock markets, interest rates and volatility. However, constant portfolio shifts can lead to high transaction costs. With rising markets, opportunity costs arise because the CPPI strategy achieves a lower return than market performance. The market movement is only reflected one-to-one in the portfolio's performance from an equity quota of 100%. If the share price falls so low during the investment period that the cushion falls to zero and rises again in the further course, you can no longer benefit from this price increase, which is also a disadvantage. It should also be noted that the CPPI strategy is flexible and easy to use. However, the use of this strategy is only recommended in falling markets, as the simulation has shown, especially if the price decline turns out to be long-lasting, such as during the first oil crisis in 1973/74 and the second oil crisis in the Years 1979/80.

literature

• Fischer Black , André F. Perold: Theory of constant proportion portfolio insurance . In: Journal of economic dynamics & control . tape 16 , no. 3/4 , 1992, ISSN  0165-1889 , pp. 403-426 .
• John C. Hull : Options, Futures, and Other Derivatives . 6th edition. Pearson Studium, Munich 2006, ISBN 3-8273-7142-2 (wi - economy).
• Thomas Zimmerer: Theoretical and practical aspects of constant proportion portfolio insurance. Capital preservation or absolute return concept? 2006, online (PDF; 536.90 kB) accessed on February 5, 2011, (previously: Constant Proportion Portfolio Insurance. Value protection or absolute return concept? In: Finanzbetrieb. 2, 2006, ISSN  1437-8981 , p. 97-106; and together with H. Meyer: Constant Proportion Portfolio Insurance. Optimization of the strategy parameters. In: Finanz Betrieb. 3, 2006, pp. 163-171).

Web links

• Waldemar Kemler, Peter Schaffner, Ling Cao:  ( Page no longer available , search in web archives ) Dynamic hedging of equity portfolios. Constant Proportion Portfolio Insurance (CPPI). PDF and Excel table.