Primer on Interest Rate Risk

Robert Serena

May 2004

About the author

Bob has over 25 years of Risk Management experience across the Insurance (P&C, Health, Life and Annuity, Reinsurance, and Consulting),Banking (Commercial and Investment), Energy (Independent Power Producers, Regulated Utilities, Integrated Oil & Gas, and Consulting), andManufacturing industries (Military Contractor).

Additionally, he has broad experience across a range of functional areas, including Quality Assurance, Software Engineering, System Design andImplementation, Claims, Financial and Medical Underwriting, Financial Reporting and Valuation, Pricing and Product Development, Asset LiabilityManagement, Retirement Planning, and Risk Management – Enterprise Risk, Strategic Risk, Market Risk, Credit Risk, Insurable Risk,Regulatory/Compliance Risk, and Operational Risk.

Bob is a native of Connecticut, and holds a BS degree in Electrical Engineering from Rice University and a MS degree in Operations Research fromthe University of New Haven. He also holds several professional designations, including Fellow in the Society of Actuaries (FSA), CharteredFinancial Analyst (CFA), Financial Risk Manager (FRM), and Chartered Property Casualty Underwriter (CPCU).

Bob is a resident of Boerne, Texas with his wife and two children.

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Robert Serena, FSA, CFA, FRM, CPCU

Risk Associated with Fixed Income Securities (1 of 3)

Market or Interest Rate Risk• The change in the present value (PV) of an individual cash flow or fixed income security due to a change in interest rates.• Present values and interest rates are inversely related…when rates increase, present values go down…when rates decrease,

present values go up.

Reinvestment Risk• The total return of a fixed income security is composed of (1) Coupon payments, (2) Capital Gains, and (3) Interest earned on

intermediate cash flows (i.e. bond coupons and pre-payments of principal).• If interest rates have decreased between the time of original issue and the time of the cash flow payment, the investor will

have to reinvest the cash flows at lower rates, thereby decreasing the 3rd return component above. So, interest rate movementshave opposing effects on components (2) and (3) above.

• Increasing rates cause a decrease in market values (i.e. capital losses), but present the investor with higher reinvestment rates.Decreasing rates cause an increase in market values (capital gains), but present the investor with lower reinvestment rates.

Timing/Call Risk• Bond indentures may contain a provision that allows the issuer to retire, or “call”, all or part of the issue before the maturity

date. Typically, the issuer seeks to retain this right in expectation of being to able to refinance an existing bond at a lowerinterest rate.

• The presence of a call provision presents 3 difficulties to the investor, as follows: (1) The cash flow pattern of a callable bond isnot predictable, (2) An issuer is apt to call a bond when interest rates are below initial levels, giving rise to reinvestment risk forthe investor, and (3) The capital appreciation potential of a callable bond is potentially limited, if the call value (i.e. the return ofprincipal to the investor) is less than the current market value of the bond.

• Several types of fixed income securities have call risk, including long-dated Treasuries and Agency bonds (Fannie Mae),corporate and municipal bonds, and most mortgage-backed securities. In return for selling this valuable option to the issuer, aninvestor will typically pay a lower price for the bond/receive a higher yield.

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Risk Associated with Fixed Income Securities (2 of 3)

Credit Risk/Default Risk• The risk that the issuer of a fixed income security either (1) Completely defaults on the obligation (unable to make timely interest and

principal payments on the security) or (2) Experiences a credit downgrade, resulting in a decrease in the market value of the security.• Credit quality is assessed by means of quality rating, provided by rating agencies like Moody, S&P, Duff & Phelps, McCarthy, and Fitch.• Investors more commonly face the second scenario above, or a decrease in the security’s market value due to an increase in the

credit spreads (i.e. the total yield, which is the sum of the risk-free rate and the credit spread, increases when spreads increase).

Yield Curve/Maturity Risk• This is the “interest rates” version of the basis risk that one finds with commodities (e.g. locational basis differential for natural gas,

quality differential for crude oil), or the risk that yields at different maturities change at different rates.

Inflation/Purchasing Power Risk• The risk that a fixed income investor will lose purchasing power because the rate of inflation is higher than the rate of interest paid on

a bond’s coupons.

Marketability/Liquidity Risk• Liquidity is the ease with which an investor can liquidate a fixed income position at or near its true value. This risk is quantified in the

form of the bid-ask spread on a security.• The size of the spread for a fixed income security is inversely proportional to the trading volume for that security, and the higher the

spread, the greater the liquidity risk.

Exchange Rate/Currency Risk• If the cash flows on a fixed income security are denominated in a currency other than the investor’s home currency, the investor faces

the risk that the foreign currency depreciates against his/her home currency between the time of purchase and the time when thecash flows are repatriated to the home currency.

• Also, the investor faces interest rate risk in the foreign currency, as well as the foreign exchange risk.

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Risk Associated with Fixed Income Securities (3 of 3)

Volatility Risk• This affects only those fixed income securities that have embedded optionality, such as callable and putable

bonds, mortgage backed securities, etc.• As the underlying (interest rates) exhibit greater volatility, the value of the embedded options become greater,

thereby changing the value of the underlying security.

Political/Legal Risk• The risk that the value of a fixed income security changes as a result of the actions of a governmental body.• Two examples are (1) If the Federal Government decreased the marginal income tax rate, thereby making

Municipal Bonds less attractive or (2) If the tax-status of a fixed income security was changed from tax-exempt totaxable.

Event Risk• The risk that an issuer’s ability to make timely payments on a bond is impaired by the occurrence of some external

event.• For example, (1) If a manufacturing firm experienced a natural disaster or industrial accident that impaired its free

cash flow or (2) The takeover of one firm by another impaired the free cash flow of the firm being acquired.

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Time Value of Money

• Interest rates are the “prices” associated with borrowing/lending money for different periods of time. Interestmay be compounded in many different ways, ranging from annually to continuously.

• The general formula for calculating the present value (PV) of a cash flow paid out 1 year into the future (i.e. Mcompounding periods), where the interest rate (I) is compounded M times per year, is given by 𝑃𝑉 =

𝐶𝐹1

1+𝐼

𝑀 ×100

𝑀 .

• To span the spectrum with the above formula, the formula can range from (1) Annual compounding (M=1),

with the 𝑃𝑉 =𝐶𝐹1

1+𝐼

100

1 to (2) Continuous Compounding, where the number of compounding periods is infinite

(literally, interest is compounded every second of every day, with 𝑀 → ∞). The present value of a cash flow

paid 1 year from now formula is given by the formula 𝑃𝑉 = 𝐶𝐹1 × 𝑒 −𝐼/100

• This is a general formula for converting from rates compounded M times per year to either (1) An annual

effective rate (M=1) or (2) A continuously compounded rate 1 +𝑖

𝑀𝑋100

𝑀= 1 + 𝑖/100 1 = exp(−𝑖/100)

• A bond pays $1,000 1 year from today with an interest rate of 9%:

• Compounded annually 𝑃𝑉 =1000

(1+.09)= $917.43

• Compounded semi-annually 𝑃𝑉 =1000

(1+.09

2)2= $915.73

• Compounded monthly𝑃𝑉 =1000

(1+.09

12)12

= $914.23

• Compounded continuously 𝑃𝑉 = 1000 𝑋 𝑒−(.09) = $913.93

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Interest Rate Sensitivity Measures (1 of 2)

• The cornerstone of the valuation of any fixed income asset is the Taylor Series expansion. For a fixedincome security, the price change of a bond given a small change in yields, 𝑓𝑟𝑜𝑚 𝑦0 𝑡𝑜 𝑦0 + ∆𝑦, can

be expressed as follows: 𝑃1 = 𝑃0 + 𝑓′ 𝑦0 ∆𝑦 +1

2𝑓′′ 𝑦0 (∆𝑦)2+ additional terms . Following are

definitions of the key terms:

• 𝑓 𝑛 () are the various derivative moments of the main price function, given by 𝑓 𝑦 .• The term 𝑃0 is the value of the bond taken at the initial yield 𝑦0, and the term 𝑃1is the value of

the bond taken at the new yield 𝑦1, or 𝑦0 + ∆𝑦 .• The remaining terms in the Taylor Series expansion are higher order derivatives combined with

∆𝑦 raised to increasing powers.

• Building on the above terminology, following are the definitions of various interest rate sensitivitymeasures:• Price Value of a Basis Point (PVBP) – also known as the Dollar Value of a Basis Point (DV01).

• Defined as the average of (1) The absolute value of the change in the present value of thebond’s cash flows when the yield rate increases by 1 basis point and (2) The value of thechange in the present value of the bond’s cash flows when the yield rate decreases by 1 basispoint.

• The greater the PVBP/DV01 of a particular bond, the greater the price volatility of that bond.• To illustrate the calculation for a cash flow received at time T using continuous compounding:𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑉𝑎𝑙𝑢𝑒 = 𝑓 𝑦0 + 1 𝑏𝑝 − 𝑓(𝑦0) = 𝐶𝐹𝑇 × 𝑒−𝑦0𝑇 𝑒−.0001𝑇 − 1

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Interest Rate Sensitivity Measures (2 of 2)

Duration• Dollar Duration (analogous to delta for an option the first-order or linear component of a bond’s sensitivity to interest rates.

• The Duration (given by D) for a single cash flow is equal to the maturity of the cash flow. For example, the duration of a $1,000cash flow paid 6 months from today is .5 years.

• The Dollar Duration for a single cash flow is the product of the Duration and the original present value of the cash flow before theinterest rate change.

• The Dollar Duration is the negative of the first derivative of the price function, given by the formula:𝜕𝑃

𝜕𝑦=

𝜕 𝐶𝐹𝑇×exp(−𝑦×𝑇)

𝜕𝑦=

− 𝑇 × 𝐶𝐹𝑇 × exp −𝑦 × 𝑇 = −𝐷 × 𝑃0• Limitations of using duration as an interest rate sensitivity measure:

• The use of duration alone in estimating price changes is accurate only for a small range around the initial yield, where the pricechanges are approximately linear (i.e. same absolute price change given an increase in yields of X bps versus a decrease in yieldsof X bps). As the yields move further away from the initial yield, the price/yield curve becomes more “non-linear”, necessitatingthe use of the second order term (i.e. convexity) in the Taylor series approximation.

• In a portfolio of bonds, the portfolio duration is simply a weighted average of individual bond durations and represents theapproximate % change in the portfolio’s value when the yield changes. But this assumes that the entire yield curve shifts in aparallel fashion. This type of assumption is typically unrealistic.

Convexity• Convexity (given by C) is analogous to gamma for an option=>it is the second-order or quadratic component of a bond’s price

sensitivity.

• Defined as𝜕2𝑃

𝜕𝑦2=

𝜕2(𝐶𝐹𝑇×exp −𝑦×𝑇 )

𝜕𝑦2= 𝑇2 × 𝐶𝐹𝑇 × exp −𝑦 × 𝑇 = C × 𝑃0

• Convexity can be interpreted in a couple of ways (1) As the second derivative of the bond price function, or the “curvature” of thebond price/yield graph and (2) The second term in the Taylor series approximation to the full price of a bond, the first term beingrelated to the duration.

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Illustration of bond pricing

• The graph and accompanying data below illustrates the non-linear relationship between the price of a bond and underlying yields.

• The absolute price change is larger given an X bps decrease in yields than an X bps increase in yields. This can be seen on the graph by noting that theprice curve is steeper as yields decrease below the par yield (5.00%) than it is for yields above 5.00%.

• This can also be seen on the data table below the graph.• The last row in the table gives market value changes in the bond when going from the par yield (5.00%) to the reference yield. For example, the

absolute change in Market Value is greater when yields decrease 200 bps from 5.00% to 3.00% ($297.55) than when yields increase by 200 bps from5.00% to 7.00% ($211.88).

• The price curve flattens out as yields increase above the par yield.

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