Constant proportion portfolio insurance (CPPI) allows an investor to limit downside risk while retaining some upside potential by maintaining an exposure to risky assets equal to a constant multiple m>1 of the 'cushion', the difference between the current portfolio value and the guaranteed amount. In diffusion models with continuous trading, this strategy has no downside risk, whereas in real markets this risk is non-negligible and grows with the multiplier value. We study the behavior of CPPI strategies in models where the price of the underlying portfolio may experience downward jumps. This allows to quantify the 'gap risk' of the portfolio while maintaining the analytical tractability of the continuous--time framework. We establish a direct relation between the value of the multiplier m and the risk of the insured portfolio, which allows to choose the multiplier based on the risk tolerance of the investor, and provide a Fourier transform method for computing the distribution of losses and various risk measures (VaR, expected loss, probability of loss) over a given time horizon. The results are applied to a jump-diffusion model with parameters estimated from market data.