If you invest in bonds, currencies or fixed securities you know that changes in interest rates can quickly turn your dreams of profit into a nightmare. Imagine being a huge investment bank with billions of dollars on the line and you can understand how losses can really pile up!

So how do these major players hedge their interest rate risk? One available strategy is interest rate immunization. (For background reading, see Advanced Bond Concepts.)

Why Do Interest Rates Matter?
Bonds and other fixed securities have two main types of risk: interest rate risk and credit risk. To measure interest rate risk, we use a concept called duration, a measure of how sensitive the price of a bond is to changes in interest rates. The longer a bond's maturity, the greater its duration and yield swings. If interest rates go up, bond prices will go down. For a portfolio of bonds, this means that increases in interest rates lower the portfolio's value.

Figure 1
Copyright coupons are received), the present value of cash flows and the future value. However, even after analyzing countless scenarios through complex mathematical formulas, a large number of variables can react in unexpected ways.

What Is Interest Rate Immunization?
Interest rate immunization is a hedging strategy that seeks to limit or offset the effect that changes in interest rates can have on a portfolio or fixed security. Immunization strategies use derivatives and other financial instruments to offset as much risk as possible when it comes to interest rates. In order to immunize an investment or portfolio, you need to understand two things: duration and convexity. (For background reading, see Use Duration And Convexity To Measure Risk.)

Duration and Convexity: The Basics
Duration
Duration is the sensitivity of the price of a bond or bond portfolio to a change in interest rates. (In this article we operate under the assumption that only parallel shifts in the interest rate occur. In fact, rates can also twist, tilt or bend ). In general, as a bond's maturity increases, so will its duration. Duration is the same as maturity for zero-coupon bonds. The basic formula for duration is:

 Where:D = DurationP = Bond\'s market valuer = Interest rateδ and D are the Greek symbol delta, meaning "change".

It can be approximated to:

This shows that a bond with a duration of four years will decline in value 4% for each 1% increase in interest rates. (Remember that bond prices and interest rates move in opposite directions.) A bond with a duration of two years would decline by 2% for every 1% increase in interest rates. Thus, bonds with longer durations are more sensitive to fluctuations in interest rates. (For more on this, read Advanced Bond Concepts: Duration.)

If a bond has fixed cash flow payments, the Macaulay formula for duration is used. The formula is a weighted average of the cash flows.

 Where:Dm = Macaulay durationt = How often payment is receivedT = Number of periods until final maturity= Bond principal or paymentPV= Present value calculation

To calculate the duration of a bond portfolio, you can take a weighted average of each component's duration.

Example - Calculating the Duration of a Bond Portfolio
Suppose that a bond has a \$10,000 face value and a 6% coupon. The yield-to-maturity is 5% and it matures in three years. The bond thus pays \$600 a year for three years. The \$10,000 principal will be returned in three years as well. The duration is calculated as:

Graphically, duration can be represented as such:

Where:
D0r = Change in the interest rate
D0p / 0p = A change in a portfolio's value or investment's price

 Example - Calculating the Convexity of a Bond Portfolio ConvexityConvexity (seen as the blue curve in Figure 2 above) is the curve to which the duration line is tangent. It is the rate at which prices rise as yields fall, and is calculated in squared time (t+1). Modifying the duration equation we get:Where:C = Convexityt = how often payment is receivedT = Number of periods until final maturity= Bond principal or paymentPV= Present value calculationTo calculate the convexity of a bond portfolio, you can take a weighted average of each component\'s convexity. The formula illustrates how duration is affected by changes in interest. Low coupons and low yields, as well as longer durations, can increase convexity due to time being squared.

If a portfolio manager only pays attention to duration, the tangent line can be quite inaccurate if a bond is heavily curved either positively or negatively; in other words, the accuracy of duration modeling deteriorates with greater changes to interest rates. Portfolio managers tend to gravitate toward bonds with greater convexity. This is because the bond price will theoretically increase more as interest rates fall when compared to bonds with lower convexity.

Interest Rate Immunization Strategies
Any immunization strategy requires a certain return over a fixed time period in order to be successful. The aim of the strategy is to match the duration of assets to the duration of liabilities, with the two ultimately being offset by each other. In the case of fixed-income instruments, such as bonds, immunization seeks to limit changes to the price and reinvestment risk.

 Where:t = How often payment is receivedT = Number of periods until final maturity= Difference between assets and liabilities= Bond principal or paymente = Interest rate riskPV = present value calculation

We then take another derivative in order to expand , leaving us with:

Using this formula requires you to plug in convexity and duration values that you calculate based on the bonds used in the portfolio.

Immunization Strategy Drawbacks
What the immunization formula does not tell you is what bond investments create a scenario in which assets and liabilities offset. That's where the complicated nature of immunization rears its ugly head. In order to use the formula, you would first have to find the duration and convexity of the future liabilities, and then scour the markets for assets that you could use to zero out. This often requires access to some pretty sophisticated software, as well as access to in-depth bond market information. You would then plug in possible assets and run scenarios until a set of solutions presented itself.

The bottom line is that immunization involves a complicated set of calculations. For a portfolio comprised of government securities and high-grade bonds, calculations might be relatively simple, but in today's financial environment, companies are investing in a variety of hard-to-grasp financial instruments, such as CDOs and junk bonds. Bonds with embedded options tend to be more volatile, which makes duration even more difficult to calculate. (To learn about junk bonds, read Junk Bonds: Everything You Need To Know.)

What Type of Investor Uses This Strategy?
Immunization strategies are frequently employed by pension funds, institutional investors, insurance companies and other large investors who may have large liabilities or assets. Businesses of this size often have millions or billions of dollars in future liabilities, and investing in bonds in order to protect capital puts them at risk if interest rates increase. They can also afford to purchase the hardware and software required to calculate the multitude of different scenarios in which assets and liabilities can offset.

This sort of strategy is often too complicated or too resource-intensive for the average investor. The analysis of different durations, cash flows and liabilities is complicated, and the transaction costs associated with the purchase and sale of derivatives can be prohibitive.

Conclusion
Interest rates are notoriously difficult to predict, but the consequences of downplaying the risk a portfolio faces when it comes to shifts in interest rates far outweigh the complexities of analysis. Pension funds, insurance companies and institutional investors use this hedging strategy to reduce the likelihood of big losses.

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