Constant proportion portfolio insurance

• Bond floor

The bond floor is the value below which the value of the CPPI portfolio should never fall in order to be able to ensure the payment of all future due cash flows (including notional guarantee at maturity).

• Multiplier

Unlike a regular bond + call strategy which only allocates the remaining dollar amount on top of the bond value (say the bond to pay 100 is worth 80, the remaining cash value is 20), the CPPI leverages the cash amount. The multiplier is usually 4 or 5, meaning you do not invest 80 in the bond and 20 in the equity, rather m*(100-bond) in the equity and the remainder in the zero coupon bond.

• Gap

A measure of the proportion of the equity part compared to the cushion, or (CPPI-bond floor)/equity. Theoretically, this should equal 1/multiplier and the investor uses periodic rebalancing of the portfolio to attempt to maintain this.

• Rules

If the gap remains between an upper and a lower trigger band (resp. releverage and deleverage triggers), the strategy does not trade. It effectively reduces transaction costs, but the drawback is that whenever a trade event to reallocate the weights to the theoretical values happen, the prices have either shifted quite a bit high or low, resulting in the CPPI effectively buying (due to leverage) high and selling low.

• Risks

As dynamic trading strategies assume that capital markets trade in a continuous fashion, gap risk is the main concern of CPPI writer, since a sudden drop in the risky underlying trading instrument(s) could reduce the overall CPPI net asset value below the value of the bond floor needed to guarantee the capital at maturity. In the models initially introduced by Black and Jones[2] Black & Rouhani,[3] this risk does not materialize: to measure it one needs to take into account sudden moves (jumps) in prices.[5] Such sudden price moves may make it impossible to shift the position from the risky assets to the bond, leading the structure to a state where it is impossible to guarantee principal at maturity. With this feature being ensured by contract with the buyer, the writer has to put up money of his own to cover for the difference (the issuer has effectively written a put option on the structure NAV). Banks generally charge a small "protection" or "gap" fee to cover this risk, usually as a function of the notional leveraged exposure.

Fundamentally, the CPPI strategy can be understood as ${\displaystyle V_{risky}=m\cdot (V-F)}$ where ${\displaystyle V_{risky}}$ is the value of assets in the risky portfolio, ${\displaystyle V}$ is the starting value of assets in the total portfolio, ${\displaystyle F}$ is the asset level below which the total portfolio should not fall, and ${\displaystyle m\geq 1}$ is the multiplier. Because the percentage of the portfolio invested in the risky asset at any given time can vary, the dynamics of a CPPI strategy are complex. The average return and variance of a CPPI strategy across investment period ${\displaystyle [t,T]}$ are[6]

${\displaystyle \mu _{CPPI}={\frac {F/V}{e^{rt-rT}}}V+Ce^{Tr+Tm\mu -Tmr}}$

${\displaystyle \sigma _{CPPI}^{2}=C^{2}e^{2T(r+m\mu -mr)}\left(e^{Tm^{2}\sigma ^{2}}-1\right)}$

where ${\displaystyle \mu }$ and ${\displaystyle \sigma ^{2}}$ are the average return and variance of the risky asset, respectively, and ${\displaystyle C=\left(1-{\frac {F/V}{e^{rt-rT}}}e^{-rT}\right)V}$.

In some CPPI structured products, the multipliers are constant. Say for a 3 asset CPPI, we have a ratio of x:y:100%-x-y as the third asset is the safe and riskless equivalent asset like cash or bonds. At the end of each period, the exposure is rebalanced. Say we have a note of $1 million, and the initial allocations are 100k, 200k, and 700k. After period one, the market value changes to 120k:80k:600k. We now rebalance to increase exposure on the outperforming asset and reduce exposure to the worst-performing asset. Asset A is the best performer, so its rebalanced to be left at 120k, B is the worst performer, to its rebalanced to 60k, and C is the remaining, 800k-120k-60k=620k. We are now back to the original fixed weights of 120:60:620 or ratio-wise 2:1:remaining.

• Portfolio insurance