3.2 Portfolio Insurance

A major purpose of options is hedging, i.e. the protection of investments against market risk caused by random price movements. An example for active hedging with options is the portfolio insurance. That is to strike deals in order to change at a certain point of time the risk structure of a portfolio such that at a future point of time The portfolio insurance creates a risk structure of the portfolio which prevents extreme losses. For illustration purposes we consider at first a simple example.

Example 3.4  
An investor has a capital of 10500 EUR at his disposal to buy stocks whose current price is 100 EUR. Furthermore, put options on the same stock with a delivery price of $ K=100$ and a time to maturity of one year are quoted at a market price of 5 EUR per contract. We consider two investment alternatives.
Portfolio A:
Buying 105 stocks.
Portfolio B:
Buying 100 stocks for 10000 EUR and buying 100 put options for 500 EUR.
The price of the put options can be interpreted as the premium to insure the stocks against falling below a level of 10000 EUR. Denoting the stock price in one year by $ S_T$ the value of the non-insured portfolio is $ 105 \cdot S_T.$ This portfolio bears the full market risk that the stock price drops significantly below 100 EUR. The insured portfolio, however, is at least as worth as 10000 EUR since if $ S_T < 100$ the holder exercises the put options and sells the 100 stocks for 100 EUR each.

Should the stock price increase above 100 EUR the investor does not exercise the put which thus expires worthless. By buying the put some of the capital of portfolio $ B$ is sacrificed to insure against high losses. But, while the probabilities of high profits slightly decrease, the probabilities of high losses decrease to zero. Investing in portfolio $ B$ the investor looses at most 500 EUR which he paid for the put. Table 2.6 shows the impact of the stock price $ S_T$ in one year on both the insured and the non-insured portfolio values and returns.

Table 2.6: The effect of a portfolio insurance on portfolio value and return.
  Non-insured Insured Insured
  portfolio portfolio portfolio in %
Stock price $ S_{T}$ Value Return Value Return of the non-
$ [$ EUR$ ]$ $ [$ EUR$ ]$ % p.a. $ [$ EUR$ ]$ % p.a. insured portfolio
50 5250 -50 10000 -4.8 190
60 6300 -40 10000 -4.8 159
70 7350 -30 10000 -4.8 136
80 8400 -20 10000 -4.8 119
90 9450 -10 10000 -4.8 106
100 10500 0 10000 -4.8 95
110 11550 +10 11000 +4.8 95
120 12600 +20 12000 +14.3 95
130 13650 +30 13000 +23.8 95
140 14700 +40 14000 +33.3 95


The numerous conceivable strategies to insure portfolios can be classified by the frequency with which the positions in the portfolio have to rebalanced. Two approaches can be distinguished: The static strategy sketched in the previous example can be modified. Instead of hedging by means of put options the investor can chose between the following two strategies:
Strategy 1:
The investor buys an equal number of stocks and puts.
Strategy 2:
The investor buys bonds with a face value equal to the floor he is aiming at and for the remaining money he buys calls on the stock.
All strategies commonly practiced rely on modifications of the above basic strategies. While following the first strategy it is the put which guarantees that the invested capital does not drop below the floor, applying the second strategy it is the bond which insures the investor against falling prices. The stocks respectively the calls make up for the profits in case of rising prices. The equivalence of both strategies follows from the put-call parity, see Theorem 2.3.

Before deciding about what kind of portfolio insurance will be used some points have to be clarified:

1.
Which financial instruments are provided by the market, and what are their characteristics (coupons, volatilities, correlation with the market etc.)?
2.
Which ideas does the investor have about
-
the composition of the portfolio (which financial instruments),
-
the amount of capital to invest,
-
the investment horizon,
-
the floor (lower bound of the portfolio value) or rather the minimum return he is aiming at the end of the investment. Given the floor $ F$ and the capital invested $ V$ the possibly negative minimum return of a one year investment is given by $ \rho = \frac{F-V}{V}$.
The strategies 1 and 2 described above we illustrate in another example.

Example 3.5  
We proceed from the assumption that the investor has decided to invest in stock. Depending on the type of return of the object we distinguish two cases (for negative returns, as storage costs of real values for example, the approach can be applied analogously):
i) continuous dividend yield $ d$
ii) ex ante known discrete yields with a time 0 discounted total value of $ D_0.$
The data of the example is shown in Table 2.7. The volatility can be interpreted as a measure of variability of the stock price. The notion of volatility is an essential part of option pricing and will be treated extensively later. Placing our considerations at the beginning $ t=0$ of the investment the time to maturity is $ \tau=T - t = T.$ For both strategies the goal is to determine the number $ n$ of stocks and/or (European) options and their delivery price $ K.$


Table 2.7: Data of Example 2.5
Data of Example 2.5:  
Current point of time $ t$ 0
Available capital $ V$ 100000 EUR
Target floor $ F$ 95000 EUR
Investment horizon $ T$ 2 years
Current stock price $ S_0$ 100 EUR
Continuously compounded annual interest rate $ r$ 0.10
Annual stock volatility $ \sigma$ 0.30
Dividends during time to maturity  
Case i): continuous dividends $ d$ 0.02
Fall ii): dividends with present value $ D_0$ 5 EUR


Case i)
The stock pays a continuous dividend at rate $ d$ = 2% p.a. which he reinvests immediately. At maturity $ T$ the position in the stock grew from $ n$ stocks to $ ne^{d\tau}$ with $ \tau = T-0 =T.$ Thus, for strategy 1 he has to buy in $ t=0$ the same number of put options. Since the amount he wants to invest in $ t=0$ is $ V$ it must hold

$\displaystyle n\cdot S_0 + ne^{d\tau}\cdot P_{K,T}(S_0,\tau) = V.$ (3.21)

The investor chooses the put options delivery price $ K$ such that his capital after two years does not drop below the floor $ F$ he is aiming at. That is, exercising the puts in time $ T$ (if $ S_T
\leq K$) must give the floor $ F$ which gives the second condition

$\displaystyle ne^{d\tau}\cdot K = F \iff n=\frac{F}{K}e^{-d\tau}.$ (3.22)

Substituting equation (2.22) into equation (2.21) gives

$\displaystyle e^{-d\tau}S_0 + P_{K,T}(S_0,\tau) - \frac{V}{F}\cdot K = 0.$ (3.23)

Thanks to the Black-Scholes pricing formula for European options that will be derived later in Section 6.2 the put price is expressed as a function of the parameter $ K.$ The delivery price which he is looking for can be computed by solving equation (2.23) numerically, for example by means of the Newton-Raphson method. In this case, $ K$ is equal to $ 99.56$. To be sure that the capital value does not drop below the floor $ F=95\;000$ EUR he buys
$ n=\frac{F}{K}e^{-d\tau}=916.6$ stocks and $ n\cdot
e^{d\tau}=954$ puts with delivery price $ K=99.56.$ The price of the put option given by the Black-Scholes formula is $ 8.72$ EUR/put. 4455 SFEexerput.xpl

Following the corresponding strategy 2 he invests $ Fe^{-r\tau}=
77\;779.42$ EUR in bonds at time 0 which gives compounded to time $ T$ exactly the floor
$ F=95\;000$ EUR. For the remaining capital of $ V-Fe^{-r\tau} = 22\;220.58$ EUR he buys $ 954$ calls with delivery price $ K=99.56$ which have a price of $ 23.29$ EUR/call according to the Black-Scholes formula. From the put-call parity follows the equivalence of both strategies, i.e. both portfolios consisting of stocks and puts respectively zero bonds and calls have at each time $ t$ the same value:

$\displaystyle n\cdot S_t + ne^{d\tau}P_{K,T}(S_t,\tau) = nKe^{-b\tau} + ne^{d\tau}C_{K,T}(S_0,\tau)$ (3.24)

where $ \tau = T- t, b=r-d$ and $ nKe^{-b\tau} = Fe^{-r\tau}$ due to equation (2.22). Table 2.8 shows the risk decreasing effect of the insurance.


Table: The effect of a portfolio insurance in case i) on portfolio value and return. 4465 SFEoptman.xpl
  Non-insured Insured Insured
  portfolio portfolio portfolio in %
Stock price $ S_{T}$ Value Return Value Return of the non-
$ [$ EUR$ ]$ $ [$ EUR$ ]$ % p.a. $ [$ EUR$ ]$ % p.a. insured portfolio
70 72857 -27 95000 -5 130
80 83265 -17 95000 -5 114
90 93673 -6 95000 -5 101
100 104081 +4 95400 -5 92
110 114489 +15 104940 +5 92
120 124897 +25 114480 +14 92
130 135305 +35 124020 +24 92
140 145714 +46 133560 +34 92


Fig.: The effect of a portfolio insurance: While the straight line represents the value of the insured portfolio as a function of the stock price, the dotted line represents the value of the non-insured portfolio as a function of the stock price. 4469 SFEoptman.xpl
\includegraphics[width=1\defpicwidth]{optman.ps}

Case ii)
Until maturity the stock pays dividends with a time 0 discounted total value $ D_0=5$ EUR which are after distribution immediately invested in bonds. At time $ T$ the dividend yield has a compounded value of $ D_T = D_0e^{r\tau}=6.107$ EUR where $ \tau=T$ denotes the time to maturity. Reasoning as in case i) and taking the dividend $ D_T$ into account he buys $ n$ stocks respectively $ n$ puts, and obtains the following equations

$\displaystyle n\cdot S_0 + n P_{K,T}(S_0-D_0,\tau) = V$ (3.25)

and

$\displaystyle nK + n D_T = F .$ (3.26)

The substraction of the cash dividend $ D_0$ from the stock price $ S_0$ in the option price formula cannot be justified until we introduced the binomial model in Chapter 7. Briefly, in a perfect market the stock price decreases instantaneously by the amount of the distributed dividend. Otherwise, an arbitrage opportunity arises. Substituting equation (2.26) into equation (2.25) gives:

$\displaystyle S_0 + P_{K,T}(S_0-D_0,\tau) - \frac{V}{F}\cdot (K + D_T) = 0$ (3.27)

Solving the equations analogously as in case i) the number $ n$ of stocks and puts and the delivery price $ K$ for strategy 1 are obtained:

$\displaystyle K=96.42$   und $\displaystyle \quad n=\frac{F}{K+D_T}=926.55$

For strategy 2 he buys $ 926.55$ calls at a price of $ 23.99$ EUR/call with a delivery price $ K=96.42.$ He invests $ 95\;000
e^{-r\tau} = 77\;779.42$ in bonds. For case ii) the effect of the portfolio insurance for both strategies is shown in Table 2.9 taking into account the time $ T$ compounded total dividend.


Table 2.9: The effect of a portfolio insurance in case ii) on portfolio value and return.
  Non-insured Insured Insured
  portfolio portfolio portfolio in %
Stock price $ S_{T}$ Value Return Value Return of the non-
$ [$ EUR$ ]$ $ [$ EUR$ ]$ % p.a. $ [$ EUR$ ]$ % p.a. insured portfolio
70 76107 -24 94996 -5 125
80 86107 -14 94996 -5 110
90 96107 -4 94996 -5 99
96.42 102527 +3 94996 -5 93
100 106107 +6 98313 -2 93
110 116107 +16 107579 +8 93
120 126107 +26 116844 +17 93
130 136107 +36 126110 +26 93
140 146107 +46 135375 +35 93


The example shows how a portfolio insurance can be carried out by means of options in principle. In practice, the following problems frequently occur:

-
The number $ n$ of stocks and options is not an integer. In a perfect financial market financial instruments are perfectly divisible, in reality, however, this is not the case. The error resulting from rounding up or down to closest integer can be neglected only in large portfolios.
-
Puts and calls traded on the market do not cover the whole range of delivery prices. Thus, options with the computed delivery price are possibly not available. Furthermore, options typically expire in less than one year which makes static strategies only limited applicable when long-term investments are involved.
-
Finally, the market provides first of all American options which are more expensive than European options which are sufficient to insure the portfolio. The additional exercise opportunities offered by American options, are only of interest if the investor possibly has to close the portfolio early.
The fact that options are not traded at all delivery prices suggests to produce them by the delta hedge process described in Chapter 6. But since a dynamic strategy is involved transaction costs have to be taken into account and give rise to other problems. Finally, we point out that when insuring large portfolios it is convenient to hedge by means of index options, i.e. puts and calls on the DAX for example, not only from a cost saving point of view but also those options replace options on a single underlying which are not traded on the market. To compute the exact effect of an index option hedge the correlation of the portfolio with the index is needed. The latter correlation is obtained from the correlations of each individual stock contained in the portfolio with the index. Besides detailed model assumptions as the Capital Asset Pricing Model (CAPM see Section 10.4.1) which among others concern the stock returns are required.