How does portfolio insurance (PI) work? PI in its simplest form provides investors with upside participation (always less than 100%) in risky assets (for example, equities), at the same time guaranteeing a minimum return (always less than the risk-free rate) over a prespecified time horizon. In other words, the investor gives up some upside in exchange for a guaranteed return. Obviously, PI involves optionalities as the investor gets the best of equities and cash (minus the cost of this option). High volatility must increase the value of an investor’s right to ex-post achieve the best return, while low interest rates will increase the amount of assets that need to be held in riskless securities in order to satisfy the guaranteed return.
Is PI unattractive if either rates are low or volatility is high? High volatility increases option costs and hence reduces participation in the risky asset. However high volatility also raises the odds to experience large positive and negative return relative to a low volatility environment, which must exactly offset the volatility costs. It is true that low rates force insurance providers to set aside large amounts of capital to ensure that guarantees are met, but it is equally true that low rates decrease the opportunity costs of not investing into competing investments. Despite this, suppliers of PI reduce protection levels, change payoffs (for example, participation in average index rather than end of period index), take on tail risks (writing out of the money puts, digitals) or change the underlying (participation in price index, instead of performance index) in a flawed attempt to increase the attractiveness of insurance.

How is PI generated? PI can either be generated by buying or replicating the associated options. Direct replication by investors is often preferable as they do not expose the investor to intellectual risk (option profits and losses are viewed on a stand-alone basis rather than in a portfolio context, and hence associated with reckless speculation) nor do they generate excess profits for investment banks.

Can we measure insurance costs by the return differential between insured and uninsured assets? PI deliberately generates positively skewed distributions by transforming a low probability of large losses into a high probability of small returns. By their very construction insured portfolios will offer attractive average returns (generated by a few very large returns), with poor median returns (returns where 50% of all returns fall below). However the median is a much better measure of location, if returns are not symetrically distributed.

Rebalancing rules are key
PI is often implemented by replicating option like payoffs via dynamic hedging. The most common approach to this is the so called CPPI strategy (constant proportion PI). It is driven by a simple allocation rule providing investment (entry strategy) and disinvestment (exit strategy) decision.

How does CPPI work? The amount of investment into equities is calculated as the product of multiplier and cushion. The cushion is defined as the difference between the current value of the CPPI portfolio (for example 100) and the present value of the targeted floor (for example the present value of 100 in three years’ time amounts to 90 for a 3.6% discount rate). We can interpret the cushion as risk capital. The multiplier translates changes in risk capital into allocation changes. For example: a multiplier of three will lead to a 30% equity allocation in the above example while the remaining 70% is invested into the riskless asset (zero bonds with three years’ maturity). If the cushion increases from 10 to 12 we arrive at a 36% equity allocation.

How can we interpret the multiplier? Suppose we ignore interest rates for a moment and assume equities fall (instantaneously) by 33%. Hence our risky assets decreased from 30 to 20, but we are still left with 70 in the riskless asset. Total portfolio value declined to 90, which equals the targeted floor. It is not a coincidence that one over the multiplier equals 33%. CPPI provides investors with a stochastic guarantee. If markets drop instantaneously, or over any time period where we can not readjust the portfolio allocation (typically overnight but it could also happen over the course of a week, if exchanges are closed as on 11 September 2001) by more than the inverse of the multiplier, a CPPI strategy will fail to protect the floor. Note that a gradual loss of 33% or even far more accompanied by readjustments does not pose a problem.
What are the costs of a CPPI? As CPPI is effectively a trend-following strategy it will buy (sell) equities after they have risen (fallen). Buying high and selling low will lose money. However these costs are close to the costs of an equivalent option (in which case the option provider runs a similar strategy). Note that it is ex-post realised volatility that determines the volatility costs, rather than ex ante expected volatility. If volatility is larger than expected a CPPI strategy will result in lower returns than expected.

What effect has the multiplier? Large multipliers will result in high turnover, as well as volatility costs. A useful measure of the inherent costs of a CPPI strategy is to ask what fraction of the provided risk capital (cushion) would be lost if equities return the same as riskless bonds. This relationship is plotted in Figure 1 (one-year CPPI with 25% volatility on Euro Stoxx 50).
Large multipliers result in substantial volatility costs (up to 90% of cushion). Also note that volatility costs for large multipliers triple if volatility rises from 15% to 35%.

How can we derive the multiplier? Multipliers are principally derived from historical data (worst-case historical return) or from volatility estimates (worst-case return at prespecified confidence from volatility forecasting model or extreme value theory). If forecasts signal rising volatility, equity allocations might be reduced even if the portfolio value has risen. It can be shown that CPPI strategies are a special form of value at risk-based strategies.

What are the risks of CPPI? The obvious risk is gap risk – ie, the risk that equities fall by more than the inverse of the multiplier. However there are two more subtle, but equally important risks. The first risk is interest rate risk. Usually if equity markets decline in a crisis scenario, interest rates will be pushed down by save haven arguments. However, this lowers the rate at which the disinvested equity (equity sold in case of market crash) can be reinvested. The second often overlooked risk is the sensitivity to realized volatility.
Suppose we evaluate a CPPI with one year time horizon, floor of 100. Interest rates are assumed to be 3.5%. Figure 2 shows the expected participation (z-axis) of a CPPI strategy as a function of multiplier (y-axis) and underlying equity index performance (x-axis). While large multipliers deliver the highest performance for modest volatility levels, they result in disastrous performance for high volatility levels (delivering the floor even for high equity performance).

How can we improve CPPI? By definition it is difficult to improve on a forecast-free, transparent and rule-based strategy (and hence passive) without altering the concept. Deutsche Asset Management aims to limit turnover without compromising performance or taking on larger risks. Traditionally rebalancing rules aim to adjust portfolio weights back to target weights (set by CPPI formula), as soon as they fall outside a prespecified corridor. However marginal rebalancing back to the upper and lower barrier of the no trading corridor considerably saves trading costs as described in Figure 3.
For a 10% corridor (if target allocation is 5%, we trade if weights are above 5.5% or below 4.5%) our rebalancing rule saves about 50% in turnover for a multiplier of 4. Marginal rebalancing saves turnover at the boundaries (unless we are triggered very often) relative to central rebalancing, while at the same time maintaining the chance to move back inside the corridor as a result of reverting relative weights. We might suspect that risks have also risen as marginal rebalancing will force us to stay longer at the boundaries and hence face increased gap risk.
Figure 4 shows the contrary (based on large number of Monte Carlo simulations). For modest multipliers there is no increased gap risk at all. Investors will only experience slightly higher risks if multipliers become large (above eight). This methodology proves to be extremely valuable when underlyings are illiquid or if futures are not available for transactional Note that the exact corridor is a function of volatility and transaction costs and needs to be calibrated for different risky assets.
Who should buy PI?
Countless books on option pricing offer numerical guidance to evaluate even the most complicated exotic options but remain surprisingly silent on the question, who should buy PI. While everybody benefits from the pooling of independent insurance risks, engaging in PI is a zero-sum gain. For every buyer of PI there must be a seller. We need to find criteria for what type of investor would buy or should sell PI. Let us assume investors maximise expected utility from end of period wealth and exhibit decreasing marginal utility from wealth. Under these innocent assumptions, the finance literature makes three statements about PI.
Investors whose risk aversion falls faster (with increasing wealth) than the average risk aversion (risk aversion of the market) will demand convex strategies (PI). Those strategies translate rising wealth levels into larger equity allocations. Equally falling wealth levels will be accompanied by reduced equity allocations. Note that we need to compare individual risk aversion to average market risk aversion as we need to state the decision to buy PI within market equilibrium.
Investors with above-average expectations (higher return expectations than the market) will also invest in PI. Intuitively this makes sense as it is well known that PI will only provide investors with meaningful returns, if realised returns on the underlying stock market are considerably higher than average returns. In other words: investors with average risk aversion need to be optimistic about the market. This is not surprising as optimistic investors tend to invest more aggressively and hence need to protect their downside.
All path-dependent strategies should be avoided by above investors as every path-dependent strategy is dominated by a corresponding (same expected return) path independent strategy. Path dependency creates additional volatility of end of period wealth without increasing expected return.
Equipped with the above analysis we can investigate popular PI strategies. Traditional PI like protected put and CPPI are ideally suited for investors with decreasing relative risk aversion as their rebalancing rules translate rising wealth levels into increased equity weightings. Pension funds are a natural group of investors whose risk aversion increases faster with falling wealth than the average risk aversion in the market. The smaller a pension fund surplus becomes the larger the risk aversion. Insurance products that lock in intermediate wealth levels (ratcheting) or guarantee past highs destroy value for long-term investors as intermediate wealth obviously has no value for investors concerned about terminal wealth. In fact, investors throw away considerable amounts of money. Discount certificates (effectively covered call writing) translate rising wealth levels into falling equity allocations, while falling wealth levels lead to rising equity allocations. In order to find them appealing investors need to have an increased willingness to take on extra risk, if their wealth level is falling. It is unclear why retail investors and private clients (target clients for this kind of product) that are described as conservative would want to invest in strategies that generate large losses when they need money most (as their marginal utility from consumption will be high if wealth is down). Note that this also applies to writing deep out of the money puts to partially finance PI.
At this point it seems noteworthy to make investors aware of some additional features of PI often overlooked in practice.
PI typically applies to the very liquid asset classes (preferably with liquid future markets) as it makes hedging easier for the insurance provider. For long-term investors this means that they are giving up diversification and lose the liquidity premium. The same is true for insurance programmes on illiquid asset classes (like hedge funds). In this case the liquidity premium (and more in often quite intransparent transactions) is paid to the liquidity provider (insurance seller).
Investors often apply PI to a subset of their equity holdings. This generates overhedging. An insurance programme on foreign equities will hedge out risks that are not necessary to hedge as the correlation between foreign and domestic equities is less than one.
Single-period returns of insured portfolios exhibit the deliberately engineered significant (positive) skewness that makes the concept attractive to investors identified above. However, PI is often periodically reset (rolled over). Unfortunately the central limit theorem tells us that the product of independent and identically distributed returns will become approximately lognormal (after about 30 draws). Even though single-period return distributions exhibit significant skew, repeated investing into reset strategies will lose this property in the long run.
Repeatedly investing into annual PI on liquid asset classes on a subset of assets will lead to an underdiversified and overhedged portfolio that also loses its skewness in the long term.
Pension funds have been made painfully aware of the risks associated with their often large equity exposure. As a consequence traditional PI (protective put, CPPI) proved to be very popular. But is this the correct answer? Recall that PI provides the investor with the maximum of equities or cash (minus premium). Falling equity markets will result in a synthetic cash position. However, cash is one of the most risky assets in an asset liability context as it carries zero duration. Assuming a 30-year duration on liabilities a pension fund invested into 50% equities will see a 15% erosion of its surplus from equities alone if rates drop by 100 basis points. What pension funds really need is an option (insurance programme) that provides the best of equities and long bonds.
Liability-driven investors will find that traditional PI still expose them to large risks as holding cash proves to be risky for investors with long-term liabilities.

Forecast free – or not quite?
Equity protection strategies are often advertised as being forecast free. We have already seen above that this is not the case. Investors exhibiting average (market) risk aversion will only buy PI when they are more bullish than the market consensus. However, quantitative analysts are (very often to their own frustration) asked to come up with an ‘optimal’ protection product. Without knowledge of investors’ utility there is no optimal product. The practical answer is that the art in selling insurance programmes (or options) is to set up structures in a way taking risks that are not perceived to be risky. One obvious example is to repeatedly sell deep out-of-the-money puts to finance puts bought at higher strikes. While most of the times this strategy will cheapen PI there is one scenario where all gains are given back (when equity returns are sufficiently negative). All this strategy does is to bet against an infrequent event that might not have realised in the past and hence looks riskless to many investors.
What has been said above can also be turned around. Every concept has some scenarios where it performs better that other concepts. Investors need to make sure these scenarios are also theirs. Suppose you consider investing into either CPPI strategy or protective put. Hedging costs are paid upfront in a protective put strategy and depend on ex-ante expected volatility. For a CPPI strategy, however, costs depend on ex-post realised volatility. Investors expecting realised volatility to be lower than ex-ante expected volatility will prefer CPPI-type (replicating) strategies. Alternatively some investors might want to minimise the regret to invest into the wrong asset class. These investors want to buy an option (or hedging programme) to achieve the maximum of equities and bonds (minus premium). If the relative performance between both asset classes is large enough it can cover the costs of providing the option and still outperform a simple balanced portfolio. If the relative performance is small investors will have underperformed relative to a balanced fund. Relative to CPPI, this type of strategy does not protect your wealth as investors will loose out if both assets classes fall by 20% (loss amounts to 20% plus additional loss of premium). Choosing the optimal PI programme requires a consultative approach from your asset manager, preferably integrated into an ALM approach.
Bernd Scherer is head of Advanced Applications Group Europe, Deutsche Asset Management in Frankfurt

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