A buy and hold strategy has been called a linear investment strategy because portfolio returns are a linear function of stock returns. The share purchases and sales involved in constant-mix and Constant proportion portfolio insurance (CPPI) strategies introduce nonlinearities in the relationship.

For constant-mix strategies, the relationship between portfolio returns and stock returns is concave; that is, portfolio return increases at a decreasing rate with positive stock returns and decreases at an increasing rate with negative stock returns.

In contrast, a Constant proportion portfolio insurance (CPPI) Buy and Hold strategy is convex. Portfolio return increases at an increasing rate with positive stock returns, and it decreases at a decreasing rate with negative stock returns. Concave and convex strategies graph as mirror images of each other on either side of a buy-and-hold strategy. Convex strategies represent the purchase of portfolio insurance, concave strategies the sale of portfolio insurance. That is, convex strategies dynamically establish a floor value while concave strategies provide or sell the liquidity to convex strategies.

**Summary of Linear Investment Buy and Hold Strategy**

Exhibit 11-9 summarizes the prior discussion of Perold-Sharpe analysis.

It is important to recognize that we have focused the discussion ofperformance in Exhibit 11-9 and the text on return performance, not risk (except to mention the downside risk protection in the Constant proportion portfolio insurance (CPPI) and stock/bills buy-and-hold strategies).

Finally, the appropriateness of buy and hold strategy, constant-mix, and constant-proportion portfolio insurance strategies for an investor depends on the investor's risk tolerance, the types of risk with which she is concerned (e.g., floor values or downside risk), and asset-class return expectations, as Example 11-9 illustrates.

**EXAMPLE 11-9 Strategies for Different Investors**

For each of the following cases, suggest the appropriate strategy:

1. Jonathan Hansen, 25 years old, has a risk tolerance that increases by 20 percent for each 20 percent increase in wealth. He wants to remain invested in equities at all times.

2. Elaine Cash has a $1 million portfolio split between stocks and money market instruments in a ratio of 70/30. Her risk tolerance increases more than proportionately with changes in wealth, and she wants to speculate on a flat market or moderate bull market.

3. Jeanne Roger has a €2 million portfolio. She does not want portfolio value to drop below €1 million but also does not want to incur the drag on returns of holding a large part of her portfolio in cash equivalents.

*Solution to Problem 1:* Given his proportional risk tolerance (constant relative risk tolerance) and desire to remain invested in equities at all times, a constant-mix strategy is appropriate for Hansen.

*Solution to Problem 2:* Her risk tolerance is greater than that of a constant-mix investor, yet Cash's forecasts include the possibility of a flat market in which CPPI would do poorly. A buy-and-hold strategy is appropriate for Cash.

*Solution to Problem 3:* The concern for downside risk suggests either a buy-and-hold strategy with €1 million in cash equivalents as a floor or dynamically providing the floor with a CPPI strategy. The buy-and-hold strategy would incur the greater cash drag, so the Constant proportion portfolio insurance (CPPI) strategy is appropriate.

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