assets is equal to the present value of the cash outflows from the liabilities, then: Duration matching Another immunization method is duration matching. Here, a portfolio manager creates a bond portfolio with a duration equal to the duration of the liabilities. To make the match actually profitable under changing interest rates, the assets and liabilities are arranged so that the total convexity of the assets exceed the convexity of the liabilities. In other words, one can match the first derivatives (with respect to interest rate) of the price functions of the assets and liabilities and make sure that the second derivative of the asset price function is set to be greater than or equal to the second derivative of the liability price function. Rebalancing Immunization requires that the average durations of assets and liabilities be kept equal at all times. This makes it necessary to rebalance the portfolio investments regularly, because the years remaining in the planning period grow shorter with each passing year. Coupon income, reinvestment income, proceeds from maturities and sales proceeds must be reinvested in securities that will keep the portfolio's duration equal to the remaining years in the planning period. Immunization in practice An immunization strategy is designed so that as interest rates change, interest-rate risk and reinvestment risk will offset each other. However, as Dr. Frank Fabozzi points out, the Macauley duration metric and immunization theory are based on the assumption that any shifts in the yield curve during the planning period will be parallel, i.e. equal at each point in the term structure of interest rates. But when a non-parallel shift in the yield curve occurs, there is a risk that the portfolio will not be immunized even if its duration matches the liability duration. Immunization risk can be quantified so that a portfolio that minimizes this risk can be constructed. A principal component analysis of changes along the U.S. Government Treasury yield curve reveals that more than 90% of yield curve shifts are parallel shifts, followed by a smaller percentage of slope shifts and a small percentage of curvature shifts. Using that knowledge, an immunized portfolio can be created by creating long positions with durations at the long and short end of the curve, and a matching short position with a duration in the middle of the curve. These positions protect against parallel shifts and slope changes, in exchange for exposure to curvature