## Risk Management

**Introduction**

U.S. Treasury securities are commonly used to manage interest rate risk. This is because the market is very large, with many participants, and the U.S. Treasury has a AAA credit rating (though S&P, one of the ratings agencies, downgraded it to AA+ as a result of the debt ceiling crisis of August 2011) making Treasuries a low risk investment. Furthermore, the market’s high liquidity helps to reduce transaction costs, meaning there is very little “friction” in trading Treasuries, especially when bid/offer spreads are so tight. In fact, U.S. Treasury securities are so popular that they are considered the benchmarks interest rates in the U.S. and the global capital markets.

Over the years, the industry has created a number of tools to measure interest rate risk. This article explains some of the more popular ones.

Though not required, it may be helpful to read “Price and Yield Calculations” before reading this article.

**Price-Yield Curve**

There is a mathematical relationship between Treasury prices and yields. When interest rate rises (yield rises), price falls. Conversely, when interest rate falls (yield falls), prices rises. This relationship is generally depicted using a Price-Yield Curve.

The price-yield curve is different for every Treasury security. Its shape is dependent on the terms of the issue, namely, its coupon and maturity. The settlement date is also an important factor. The sensitivity of prices to changes in yield is the basis for many of the risk measurements used for Notes and Bonds.

Note that the price yield relationship is non-linear, but rather a convex curve. So the interest rate risk metrics are different depending on where you are on the curve. Many of the interest rate risk measurements are useful for small interest rate changes. For larger changes, the risk measurements need to be recalibrated.

**Risk Measurements**

There are three popular risk measurements: PV01, Duration, and Risk. Though similar, each is used differently by different types of market participants.

The following table contains calculated risk measurements for the Benchmark Treasury securities quoted on August 2, 2011 for settlement on August 3, 2011. Benchmark issues are the most recently auctioned issues of a given maturity. (the 2-, 3-, 5-, 7-, and 10-year Treasury Notes and the 30-year Treasury Bond).

Displayed are the coupon and maturity for each issue as well as the quote in percent (see the article on “Invoice Price” for an explanation of price quoting conventions). The decimal price is the Quote converted to a decimal number. All other fields are calculated using bond math: Yield, PV01, Risk (dP/dY), Duration, and Modified Duration.

**PV01**

PV01 is an acronym for the Price Value for a 01 change in yield. This measures the impact on price of a 0.01% (1 Basis Point or 1 BP) change in yield.

For example: the yield on the 5Y Treasury Note is 1.218%. If you add 01 basis point (0.01%) to this, the yield becomes 1.219%. With this newly calculated yield, we could then calculate a new price using the Price-Yield equation for Notes and Bonds. This new price would be lower than the original (remember, when yield goes up, prices come down). The difference between the original price and the newly calculated price is PV01: the change in price for an 01 basis point change in yield.

Returning to the 5Y Treasury, we see that PV01 is 0.048643. So if interest rates were to go up by 0.01%, the price would fall by this decimal figure. If the yield were to increase by 0.02%, then price would fall by twice this amount. Conversely, if yield were to decrease by 0.01%, price would rise by 0.048643.

As you can see, PV01 is a good price sensitivity measure. This risk tool should be used for small movements in yield (a few basis points up or down). Remember, the Price-Yield Curve is not linear; for larger movements, the impact on price will be slightly distorted, and it is probably better to redo the Price-Yield calculation.

**Risk (dPdY)**

Though Risk and PV01 have similar goals, they involve different methods. PV01 takes a 01 change in yield, looks at two points on the Price-Yield Curve, and calculates the change in price. Risk is the derivative of price with respect to yield at the tangent of the Price Yield Curve. Using calculus notation, this is dP/dY. Risk measures the instantaneous change in price for an infinitesimal change in yield.

By convention, Risk is also quoted in basis points. As you can see from the table above, Risk is fairly close to 100 times PV01.

For most practical purposes, Risk and PV01 can be used interchangeably. Many traders, however, prefer using Risk because it is, theoretically, more accurate than PV01.

**Duration**

The Duration of a Note or Bond is defined as the time-weighted average of the cash flows.

Each cash flow is weighted using the time from the settlement date. Let’s take 10-year Treasury Note as an example (calculating from the issuance date on 5/15/2011). The first coupon is multiplied by 1, the second coupon by 2, and so forth, until the final coupon and principal are multiplied by 20 (10 years times 2 semi-annual periods per year). The present value of these “time-weighted” cash flows is divided by the un-weighted price.

The Duration measurement, as the name implies, involves time. Generally speaking, Duration attempts to measure the average time of all cash flows. The duration of the 10-year Treasury Note is 8.483 years. Putting aside the math, this means that all the cash flows (coupons and principal) are equivalent to having one cash flow 8.483 years out from issuance. Even though we’re dealing with a 10-year security, some of the cash flows (coupon payments) come in earlier. Thus the “Duration” of the 10-year Treasury Note is really less than 10 years.

Before getting into the practical use of Duration, we need to explain Market Risk and Re-investment Risk.

**Market Risk**

As with any “fixed income” security, when interest rates change in the marketplace, the prices moves. Usually, the longer the maturity, the more sensitive the security is to interest rate changes. Notice in the table above that Risk (dP/dY) is higher for those Treasury securities with longer maturities. When general interest rate levels fall, Note and Bond prices rise; conversely when interest rates rise, Note and Bond prices fall. This is Market Risk: the risk that interest rate changes pose to price.

**Re-investment Risk**

Notes and Bonds pay interest (coupons) semi-annually until maturity. If we were to evaluate the total return on a Note or Bond, we’d need to calculate what we’d earn “re-investing” these coupons at some interest rate. Assuming interest rates are constant, the coupons will be re-invested at the yield (yield to maturity).

But what if interest rates fall? In this case the coupons will have to be re-invested at a lower rate, and the total return will be lower. The opposite is also true: if interest rates rise, the coupons will be re-invested at a higher rate, and the total return will be higher.

**Duration and Cash Flows**

Market Risk and Re-investment Risk act like balancing forces on interest rate changes. When interest rates fall, prices rise (because of Market Risk) but returns on re-invested coupons fall (because of Re-investment Risk). The converse is also true.

As the weighted average of all the cash flows, Duration has a special property. Mathematically, it is as if all the cash flows can be represented by one, equivalent cash flow at the duration date. So the Price-Yield calculations will generate the same results for this equivalent cash flow as they will for the series of cash flows of the original Note or Bond.

Let’s use this visual analogy: three children are sitting on one side of a seesaw at varying distances from the fulcrum. A larger adult may be sitting on the opposite side of the seesaw. The adult’s weight doesn’t necessarily have to equal the weight of the three children to balance the seesaw. The adult just needs to move up (towards the fulcrum) or down (away from the fulcrum) along the seesaw until it balances. By adjusting her distance from the seesaw, she is changing her downward force, which equals weight times distance from the fulcrum. As long as her weighted average distance is the same as that of the three children, the seesaw balances on the fulcrum.

Let’s apply this example to Duration. The one cash flow at duration (the adult, similar to distance from the fulcrum, but instead using time in years) is equivalent to a series of cash flows of a Note or Bond (children) at various dates.

Duration is sometimes called “Macaulay’s Duration” in recognition of its creator, 20th century Economist Frederick Macaulay.

**Duration and Investment Horizon**

The Investment Horizon is the time for which an investor expects to hold a security. Remember, an investor doesn’t need to hold the security until maturity; securities can be resold in the secondary market.

Most investors have a plan for how they want to invest and when they’ll need money. Take, for example, parents saving to send their child to college. The investment horizon is known—the parents will need the money when the son or daughter is about 18. They save and invest with this future date and a target amount in mind.

It is common for an institutional investor to have a shorter investment horizon than the maturity of the security he is buying. Again referring to the table above, the 5-year Treasury Note is yielding 1.218%, whereas the 30-year Treasury Bond is yielding 3.910% — much more. The investor may choose to buy the 30-year Bond over the 5-year Note (in order to earn a higher return) and sell it when he needs the money at his investment horizon. Here he is “chasing” the yield. But the risk of the 30-year Bond (18.331) is higher than that of the 5-year Note (4.865).

In addition to Market Risk, this investor is also exposed to Re-invesment Risk because all of the coupons he would receive will have to be re-invested for the duration of his investment horizon.

**Duration and Interest Rate Immunization**

Now we have enough background information to explain the usefulness of Duration.

Duration has the following “magical” property: if the Duration is equal to the investment horizon, then the Market Risk will be equal to the Re-invesment Risk for small changes in interest rates. Since the rise (or fall) of prices at the investment horizon is equal to the fall (or risk) of the returns on the re-invested coupons, the Treasury security is considered to be “immunized” against interest rates. If interest rates rise or fall, there are no changes to the total return up to the investment horizon because Market Risk and Re-investment Risk will cancel one another out.

As you can see, this is a powerful tool for investment managers because they can construct portfolios of securities that are protected from interest rate risk. The process of managing assets against interest rate risk is commonly referred to as “interest rate immunization,” “bond immunization,” or simply “immunization.”

**Modified Duration**

Modified Duration is a variant of Duration. Its definition:

**Modified Duration = Duration / (1 + Yield/2)**

Mathematically, it is also linked to Risk (dP/dY)

**Modified Duration = dP/dY / Price**

Since dP/dY represents a small change in price with respect to yield, then (dP/dY)/Price represents a percentage change in price with respect to yield. This is because it is equivalent to dP/Price (the percentage change in price) divided by dY.

Traders, more interested in percent changes in price than gross changes in price, will use Modified Duration instead of Risk (dP/dY) or PV01.

**Summary**

There are a number of tools used by market participants to manage interest rate risk. Traders tend to favor PV01 and Risk because they are concerned with instantaneous changes in interest rates and how they can impact their positions and Profit and Loss (P&L). They tend to hold positions intraday, or at most for very short periods of time.

Buyside investors, who typically hold securities for a much longer term, tend to use Duration. They want to protect their portfolios from interest rate movements aligned with their investment horizons.

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