Dynamic portfolio strategies
Portfolio Insurance refers to portfolio strategies that take into account the constraints, put onto portfolio wealth dynamics.
SmartFolio combines two types of portfolio insurance strategies into one:
Let denote the portion of initial wealth that investor wishes to secure.
- A portfolio strategy that consists in preventing discounted portfolio wealth from loosing a prespecified portion of its initial value.
- A portfolio strategy that guarantees preservation of a prespecified portion of accumulated profits.
Let denote the secured portion of accumulated income.
Note. If , then the corresponding portfolio insurance strategy is equivalent to securing portion of the maximum-to-date value of discounted wealth. In other words, an investor does not allow the maximum drawdown of his discounted wealth ever to exceed the given constant .
Construction of Portfolio Insurance Strategy
Imagine that an investor wishes to apply -portfolio insurance to some underlying portfolio strategy . Let denote discounted wealth of a final strategy at time . For simplicity lets assume that trades occur at discrete times .
Portfolio insurance rules are presented below:
- At the initial wealth is divided in two parts and . The former denotes initial value of secured wealth, while the latter denotes initial value of risk wealth.
The sum of secured wealth and risk wealth will be further referred to as aggregate wealth.
- Strategy is applied to risk wealth, while secured wealth is kept in a riskless asset.
- Let denote maximum-to-date value of aggregate wealth. Every time the aggregate wealth reaches its new historical maximum, the amount of extra profits is transferred to secured wealth and put into a riskless asset.
It is obvious, that formulated portfolio insurance rules satisfy wealth constraints, stated above. The reason why these very rules were chosen to determine portfolio insurance is explained by the following fact (similar problem is discussed in [Cvitanic, Karatzas; 1995]):
Let denote Merton portfolio for an investor with CRRA utility function. Then the stated above portfolio insurance rules, applied to , define a portfolio strategy that is optimal for the same investor in presence of the corresponding portfolio wealth constraints.