10-Inclass

Bond Durations (Yield to Maturity = 8% APR; Semiannual Coupons)

BKM 16.*

Duration is AdditiveThe duration of a portfolio of securities is the weighted average of the durations of the individual securities with the weights reflecting the proportion invested in each.Example: Let 25% of a portfolio be invested in a bond with a duration of 5 and let 75% of the portfolio be invested in a bond with a duration of 10.Dp = (0.25 * 5) + (0.75 * 10) = 8.75 years

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Why is Duration a Big Deal?Simple summary statistic of effective average maturity

Measures sensitivity of bond price to interest rate changesMeasure of bond price volatilityMeasure of interest-rate risk

Useful in the management of riskYou can match the duration of assets and liabilitiesOr hedge the interest rate sensitivity of an investment

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ExampleConsider a 3-year 10% coupon bond selling at $1078.7 to yield 7%. Coupon payments are made annually. Whats the duration of this bond? If yields increase to 7.10%, how does the bond price change (duration rule & PV formula)?

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ExampleModified duration of this bond:

If yields increase to 7.10%, how does the bond price change?The percentage price change of this bond is given by:

= 2.5661 .0010 = .2566 %

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ExampleWhat is the predicted change in dollar terms?

New predicted price: $1078.7 2.768 = $1075.932

Actual dollar price (using PV formula): N=3; PMT=100; FV=1000; I/Y =7.1; PV=$1075.966Good approximation!

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Duration and Interest Rate SensitivityExample: A 30-year bond that pays an annual coupon of 8%. The current YTM is 9%. Then the duration should be 11.37 years. The current bond price is $897.26. If YTM changes to 9.1%, we predict that the bond price should change by:DP = - D P Dy/(1+y) = -11.37897.260.001/1.09 = - $9.36 (Decrease)Whats the price change if we use the annuity formula?The bond price with 9.1% interest equals $887.98. The difference of $9.28 is quite close to $9.36 predicted by duration formula.

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Duration is a Local ConceptSuppose that in the last example the YTM changed to 10%. Whats the price change predicted by the duration formula?DP = -11.37897.260.01/1.09 = - $93.59.

Whats the price change predicted by the annuity formula?DP = 811.46 - 897.26 = - $85.80.

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Duration is a Local ConceptNow the divergence is more substantial. This points to an important limitation of the duration formula. Duration is a local concept, and its value changes as the YTM changes. Why?

ConvexityThe relationship between bond prices and yields is not linear.Duration rule is a good approximation for only small changes in bond yields.Bonds with greater convexity have more curvature in the price-yield relationship.

Bond Price Convexity: 30-Year Maturity, 8% Coupon; Initial YTM = 8%

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ConvexityMeasures how much a bonds price-yield curve deviates from a straight lineSecond derivative of price with respect to yield divided by bond price

Allows us to improve the duration approximation for bond price changes

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ConvexityRecall approximation using only duration:

New bond price

The predicted percentage price change accounting for convexity:

New bond price

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Numerical Example with ConvexityConsider the bond in Figure 16.3, with a 30-year 8% coupon bond selling at par value ($1,000), to yield 8%. We can find that the modified duration is 11.26, and the convexity is 212.4.If the yield increases from 8% to 10%, the bond price will fall to $811.46, a decline of 18.85%.The duration rule indicates that

After correcting for convexity

The convexity of the bond is 164.106.

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Numerical Example with ConvexityWhat if yields fall by 2%?If yields decrease instantaneously from 8% to 6%, whats the percentage price change of this bond?

Note that predicted change is NOT SYMMETRIC.

Convexity of Two Bonds

Investors Like ConvexityBonds with greater curvature gain more in price when yields fall than they lose when yields rise.The more volatile interest rates, the more attractive this asymmetry.Bonds with greater convexity tend to have higher prices and/or lower yields, all else equal.

There are two passive bond portfolio strategies. Both strategies see market prices as being correct, but the strategies have very different risks.Indexing: have the same risk-reward profile as the bond market index to which it is tiedImmunization: seek to establish a virtually zero-risk profileActive managementInterest rating forecastingIdentification of relative mispricing within the fixed-income marketPassive Management VS. Active Management

Bond Index FundsMajor bond indexes:Barclays Capital Aggregate Bond Index, Salomon Broad Investment Grade Index, etc.Bond indexes contain thousands of issues, many of which are infrequently traded.Difficult to purchase the securities at a fair market priceBond indexes turn over more than stock indexes.Bonds are continually dropped from the index as their maturities fall below certain level (i.e. 1 year), and new bonds are added when they are issued.Therefore, bond index funds hold only a representative sample of the bonds in the actual index.

ImmunizationImmunization is a way to control interest rate risk. Widely used by pension funds, insurance companies, and banks, since these institutions often have a mismatch between asset and liability maturity structures.For example, banks liabilities are short-term deposits, but their assets are long-term loans or mortgages.When interest rate increase unexpectedly, banks can suffer serious decreases in net worth.

Result: Value of assets will track the value of liabilities whether rates rise or fall.

Immunization An ExampleAn insurance company must make a payment of $19,487 in 7 years. The market interest rate is 10%, so the present value of the obligation is $10,000. The companys portfolio manager wishes to fund the obligation using 3-year zero-coupon bonds and perpetuities paying annual coupons. How can the manager immunize the obligation?

Immunization: the duration of the portfolio of assets equal the duration of the liability

Immunization An ExampleCalculate the duration of the liabilityone single payment: duration = 7 years

Calculate the duration of the asset portfolioduration of the zero-coupon bond = 3 yearsduration of the perpetuity is 1.1/0.1 = 11 yearsassume the fraction in the zero is w, then the portfolio duration = w * 3 + (1 - w) * 11

Immunization An ExampleFind the asset mix that sets the duration of assets equal to the 7-year duration of liabilitiesw * 3 + (1 - w) * 11 = 7, implying w = 0.5

Fully fund the obligation.The manager must purchase $5,000 of the zero and $5,000 of the perpetuity.Face value of zero = $5,000 * (1.1)3 = $6,655

Immunization An ExampleSuppose that 1 year has passed, and the interest rate remains at 10%. The portfolio manager needs to reexamine her position. Is the position still fully funded (i.e., value of the asset = value of the obligation) ? Is it still immunized?

We first need to calculate the PV of the asset and the obligation.

PV of the obligation: FV = $19,487; I/Y = 10; PMT = 0; N = 6; CPT PV = $11,000

PV of the asset: Zero: PV = 6,655/ (1.1)2 = 5,500Perpetuity : Get paid $500, and remain worth 5,000

Value of the asset = Value of the obligation fully funded

Immunization An ExampleWe next need to find the asset mix that sets the duration of assets equal to the duration of liabilities

Immunization: w * 2 + (1 - w) * 11 = 6, w = 5/9

The manager now must invest a total of $11,000*(5/9)= $6,111.11 in the zero.

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