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In finance, **interest rate immunization**, as developed by Frank Redington is a strategy that ensures that a change in interest rates will not affect the value of a portfolio. Similarly, immunization can be used to ensure that the value of a pension fund's or a firm's assets will increase or decrease in exactly the opposite amount of their liabilities, thus leaving the value of the pension fund's surplus or firm's equity unchanged, regardless of changes in the interest rate.

**Finance** is a field that is concerned with the allocation (investment) of assets and liabilities over space and time, often under conditions of risk or uncertainty. Finance can also be defined as the art of money management. Participants in the market aim to price assets based on their risk level, fundamental value, and their expected rate of return. Finance can be split into three sub-categories: public finance, corporate finance and personal finance.

**Frank Mitchell Redington** was a noted British actuary. Frank Redington was best known for his development of Immunisation Theory which specifies how a fixed income portfolio can be "immunised" against changing interest rates.

An **interest rate** is the amount of interest due per period, as a proportion of the amount lent, deposited or borrowed. The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, the compounding frequency, and the length of time over which it is lent, deposited or borrowed.

- Cash flow matching
- Duration matching
- Calculating immunization
- Immunization in practice
- Difficulties
- History
- See also
- References
- External links
- Recommended reading

Interest rate immunization can be accomplished by several methods, including cash flow matching, duration matching, and volatility and convexity matching. It can also be accomplished by trading in bond forwards, futures, or options.

In finance, the **duration** of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage change in price for a parallel shift in yields.

In finance, **bond convexity** is a measure of the non-linear relationship of bond prices to changes in interest rates, the second derivative of the price of the bond with respect to interest rates. In general, the higher the duration, the more sensitive the bond price is to the change in interest rates. Bond convexity is one of the most basic and widely used forms of convexity in finance.

Other types of financial risks, such as foreign exchange risk or stock market risk, can be immunized using similar strategies. If the immunization is incomplete, these strategies are usually called hedging. If the immunization is complete, these strategies are usually called arbitrage.

A **hedge** is an investment position intended to offset potential losses or gains that may be incurred by a companion investment. A hedge can be constructed from many types of financial instruments, including stocks, exchange-traded funds, insurance, forward contracts, swaps, options, gambles, many types of over-the-counter and derivative products, and futures contracts.

In economics and finance, **arbitrage** is the practice of taking advantage of a price difference between two or more markets: striking a combination of matching deals that capitalize upon the imbalance, the profit being the difference between the market prices. When used by academics, an arbitrage is a transaction that involves no negative cash flow at any probabilistic or temporal state and a positive cash flow in at least one state; in simple terms, it is the possibility of a risk-free profit after transaction costs. For example, an arbitrage opportunity is present when there is the opportunity to instantaneously buy something for a low price and sell it for a higher price.

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. That meant that only institutional investors could afford it. But the latest advances in technology have 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.

In finance, a **bond** is an instrument of indebtedness of the bond issuer to the holders. The most common types of bonds include municipal bonds and corporate bonds.

A more practical alternative immunization method is **duration matching**. Here, the duration of the assets is matched with 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.

Immunization starts with the assumption that the yield curve is flat. It then assumes that interest rate changes are parallel shifts up or down in that yield curve. Let the net cash flow at time *t* be denoted by *R _{t}*, i.e.:

* R _{t} = A_{t} - L_{t} ; t = 1,2,3,...,n*

*where At and Lt represent cash inflows (A _{t}) and outflows or liabilities (Lt)*

*We will assume that the present value of cash inflows from the assets is equal to the present value of the cash outlfows from the liabilities. Thus, we have:*

*P(i) = 0*^{ [1] }

*Immunization can be done in a portfolio of a single asset type, such as government bonds, by creating long and short positions along the yield curve. It is usually possible to immunize a portfolio against the most prevalent risk factors. A principal component analysis of changes along the U.S. Government Treasury yield curve reveals that more than 90% of the yield curve shifts are parallel shifts, followed by a smaller percentage of slope shifts and a very 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 changes. ^{[ citation needed ]}*

In finance, the **yield curve** is a curve showing several yields or interest rates across different contract lengths for a similar debt contract. The curve shows the relation between the interest rate and the time to maturity, known as the "term", of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S. Treasury securities for various maturities are closely watched by many traders, and are commonly plotted on a graph such as the one on the right which is informally called "the yield curve". More formal mathematical descriptions of this relation are often called the **term structure of interest rates**.

*Immunization, if possible and complete, can protect against term mismatch but not against other kinds of financial risk such as default by the borrower (i.e., the issuer of a bond). It might also be difficult to find assets with suitable cashflow structures that are necessary to ensure a particular level of overall volatility of assets to have a proper match with that of liabilities. *

*Once there is a change in interest rate, the entire portfolio has to be restructured to immunize it again. Such a process of continuous restructuring of portfolios makes immunization a costly and tedious task. *

*Users of this technique include banks, insurance companies, pension funds and bond brokers; individual investors infrequently have the resources to properly immunize their portfolios.*

*The disadvantage associated with duration matching is that it assumes the durations of assets and liabilities remain unchanged, which is rarely the case.*

*Immunization was discovered independently by several researchers in the early 1940s and 1950s. This work was largely ignored before being re-introduced in the early 1970s, whereafter it gained popularity. See Dedicated Portfolio Theory#History for details.*

**Discounting** is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee. Essentially, the party that owes money in the present purchases the right to delay the payment until some future date. The **discount**, or **charge**, is the difference between the original amount owed in the present and the amount that has to be paid in the future to settle the debt.

*The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return. The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. It is widely used, although often with adjustments and corrections, by options market participants.*

*In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are denoted by Greek letters. Collectively these have also been called the risk sensitivities, risk measures or hedge parameters.*

A **swap** is a derivative in which two counterparties exchange cash flows of one party's financial instrument for those of the other party's financial instrument. The benefits in question depend on the type of financial instruments involved. For example, in the case of a swap involving two bonds, the benefits in question can be the periodic interest (coupon) payments associated with such bonds. Specifically, two counterparties agree to exchange one stream of cash flows against another stream. These streams are called the *legs* of the swap. The swap agreement defines the dates when the cash flows are to be paid and the way they are accrued and calculated. Usually at the time when the contract is initiated, at least one of these series of cash flows is determined by an uncertain variable such as a floating interest rate, foreign exchange rate, equity price, or commodity price.

**Rational pricing** is the assumption in financial economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.

**Bond valuation** is the determination of the fair price of a bond. As with any security or capital investment, the theoretical fair value of a bond is the present value of the stream of cash flows it is expected to generate. Hence, the value of a bond is obtained by discounting the bond's expected cash flows to the present using an appropriate discount rate. In practice, this discount rate is often determined by reference to similar instruments, provided that such instruments exist. Various related yield-measures are then calculated for the given price.

**Fixed income analysis** is the valuation of fixed income or debt securities, and the analysis of their interest rate risk, credit risk, and likely price behavior in hedging portfolios. The analyst might conclude to buy, sell, hold, hedge or stay out of the particular security.

**Interest rate risk** is the risk that arises for bond owners from fluctuating interest rates. How much interest rate risk a bond has depends on how sensitive its price is to interest rate changes in the market. The sensitivity depends on two things, the bond's time to maturity, and the coupon rate of the bond.

*The following outline is provided as an overview of and topical guide to finance:*

**Municipal bond arbitrage**, also called municipal bond relative value arbitrage, municipal arbitrage, or just muni arb, generally consists of building a leveraged portfolio of high-quality, tax-exempt municipal bonds and simultaneously hedging the duration risk in that municipal bond portfolio by shorting the equivalent taxable corporate bonds. These corporate equivalents are typically interest rate swaps referencing Libor or BMA. Muni arb is a relative value strategy that seizes upon an inefficiency that is related to government tax policy; interest on municipal bonds is exempt from federal income tax. Because the source of this arbitrage is artificially imposed by government regulation, it has persisted for decades.

**Fixed-income attribution** is the process of measuring returns generated by various sources of risk in a fixed income portfolio, particularly when multiple sources of return are active at the same time.

*In mathematical finance, a replicating portfolio for a given asset or series of cash flows is a portfolio of assets with the same properties. This is meant in two distinct senses: static replication, where the portfolio has the same cash flows as the reference asset, and dynamic replication, where the portfolio does not have the same cash flows, but has the same "Greeks" as the reference asset, meaning that for small changes to underlying market parameters, the price of the asset and the price of the portfolio change in the same way. Dynamic replication requires continual adjustment, as the asset and portfolio are only assumed to behave similarly at a single point.*

**Dedicated portfolio theory**, in finance, deals with the characteristics and features of a portfolio built to generate a predictable stream of future cash inflows. This is achieved by purchasing bonds and/or other fixed income securities that can and usually are held to maturity to generate this predictable stream from the coupon interest and/or the repayment of the face value of each bond when it matures. The goal is for the stream of cash inflows to exactly match the timing of a predictable stream of cash outflows due to future liabilities. For this reason it is sometimes called cash matching, or liability-driven investing. Determining the least expensive collection of bonds in the right quantities with the right maturities to match the cash flows is an analytical challenge that requires some degree of mathematical sophistication. College level textbooks typically cover the idea of “dedicated portfolios” or “dedicated bond portfolios” in their chapters devoted to the uses of fixed income securities.

**Option-adjusted spread** (OAS) is the yield spread which has to be added to a benchmark yield curve to discount a security's payments to match its market price, using a dynamic pricing model that accounts for embedded options. OAS is hence model-dependent. This concept can be applied to a mortgage-backed security (MBS), or another bond with embedded options, or any other interest rate derivative or option. More loosely, the OAS of a security can be interpreted as its "expected outperformance" versus the benchmarks, if the cash flows and the yield curve behave consistently with the valuation model.

*↑ The Theory of Interest, Stephen G. Kellison, McGraw Hill International,2009*

*Stulz, René M. (2003).**Risk Management & Derivatives (1st ed.)*. Mason, Ohio: Thomson South-Western. ISBN 0-538-86101-0.

*Wesley Phoa,**Advanced Fixed Income Analytics*, Frank J. Fabozzi Associates, New Hope Pennsylvania, 1998. ISBN 1-883249-34-1

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