Fisher and Roman Weil framed the issue as follows: to immunize a portfolio, "the average duration of the bond portfolio must be set equal to the remaining time in the planning horizon, and the market value of assets must be greater than or equal to the present value of the liabilities discounted at the internal rate of return of the portfolio." Applications Pension funds use immunization to lock in current market rates, when they are attractive, over a specified planning horizon, and to fund a future stream of pension benefit payments to retirees. Banks and thrift (savings and loan) associations immunize in order to manage the relationship between assets and liabilities, which affects their capital requirements. Insurance companies construct immunized portfolios to support guaranteed investment contracts, structured financial instruments which are sold to institutional investors. How portfolios are immunized Immunization theory assumes that the yield curve is flat, and that interest rate changes are parallel shifts up or down in that yield curve. Cash flow matching Conceptually, the easiest form of immunization is cash flow matching. For example, if a financial company is obliged to pay 100 dollars to someone in 10 years, it can protect itself by buying and holding a 10-year, zero-coupon bond that matures in 10 years and has a redemption value of $100. Thus, the firm's expected cash inflows would exactly match its expected cash outflows, and a change in interest rates would not affect the firm's ability to pay its obligations. Nevertheless, a firm with many expected cash flows can find that cash flow matching can be difficult or expensive to achieve in practice. Once, that meant that only institutional investors could afford it. But the advent of the Internet and the personal computer relieved much of this difficulty. Dedicated portfolio theory is based on cash flow matching and is being used by personal financial advisors to construct retirement portfolios for private individuals. Withdrawals from the portfolio to pay living expenses represent the stream of expected future cash flows to be matched. Individual bonds with staggered maturities are purchased whose coupon interest payments and redemptions supply the cash flows to meet the withdrawals of the retirees. Mathematically, this can be expressed as follows. Let the net cash flow at time be denoted by , i.e.: where and represent cash inflows and outflows or liabilities respectively. Assuming that the present value of cash inflows from the