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**Managing Bond Portfolios**

Chapter 11 Managing Bond Portfolios Describes the financial instruments traded in primary and secondary markets. Discusses Market indexes. Discusses options and futures. McGraw-Hill/Irwin Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. 1

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11.1 Interest Rate Risk 11-2

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**Interest Rate Sensitivity**

Inverse relationship between bond price and interest rates (or yields) Long-term bonds are more price sensitive than short-term bonds Sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases 2. There are some exceptions to #2 because deep discount bonds can have lower duration at longer maturities. This is pretty much a math quirk and won’t be true for most traded bonds. 11-3

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**Interest Rate Sensitivity (cont)**

A bond’s price sensitivity is inversely related to the bond’s coupon Sensitivity of a bond’s price to a change in its yield is inversely related to the yield to maturity at which the bond currently is selling An increase in a bond’s yield to maturity results in a smaller price decline than the gain associated with a decrease in yield If interest rates increase and I have a high coupon bond, I am getting more current income to reinvest at the new higher rates so my bond’s price is not affected as much as a bond with a lower coupon and vice versa. At higher yield rates the present values of the more distant cash flows are reduced by more than the present value of the nearer in time cash flows. Thus at higher yield rates the near term cash flows make up a higher percentage of the bond’s value. With more of the value based on near term cash flows, the bond will have lower price volatility. This is an artifact of convexity. As the bond price interest rate relationship is curvilinear a given increase in interest rates results in a different percentage price change than the same decrease in rates. 11-4

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**Summary of Interest Rate Sensitivity**

The concept: Any security that gives an investor more money back sooner (as a % of your investment) will have lower price volatility when interest rates change. Maturity is a major determinant of bond price sensitivity to interest rate changes, but It is not the only factor; in particular the coupon rate and the current ytm are also major determinants. Do not equate lower price volatility with lower interest rate risk. Interest rate risk is reduced by minimizing the difference between the duration of the bond portfolio and the investor’s investment horizon and not necessarily by reducing the duration. 11-5

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**Change in Bond Price as a Function of YTM**

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Duration Consider the following 5 year 10% coupon annual payment corporate bond: 1 2 3 4 5 $100 $1100 Because the bond pays cash prior to maturity it has an “effective” maturity less than 5 years. We can think of this bond as a portfolio of 5 zero coupon bonds with the given maturities. The average maturity of the five zeros would be the coupon bond’s effective maturity. We need a way to calculate the effective maturity. 11-7

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**Duration Duration is the term for the effective maturity of a bond**

Time value of money tells us we must calculate the present value of each of the five zero coupon bonds to construct an average. We then need to take the present value of each zero and divide it by the price of the coupon bond. This tells us what percentage of our money we get back each year. We can now construct the weighted average of the times until each payment is received. 11-8

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Duration Formula Wt = Weight of time t, present value of the cash flow earned in time t as a percent of the amount invested CFt = Cash Flow in Time t, coupon in all periods except terminal period when it is the sum of the coupon and the principal ytm = yield to maturity; Price = bond’s price Dur = Duration The amount invested is the price 11-9

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**Calculating the duration of a 9% coupon, 8% ytm, 4 year annual payment bond priced at $1033.12,**

$ 83.33 77.16 71.45 $801.18 8.06% 7.47% 6.92% 77.55% 0.0806 0.1494 0.2076 3.1020 $1,033.12 100.00% yrs Duration = years 11-10

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**Using Excel to Calculate Duration**

Excel can be used to calculate a bond’s duration. Usage notes: The dates should be entered using the formulas given If you don’t know the actual settlement date and maturity date, set the 6th term in the duration formulae to 0 as shown and pick a maturity date with the same month and day as the settlement date and the correct number of years after the settlement date. The par is not needed 11-11

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**More on Duration Duration increases with maturity**

A higher coupon results in a lower duration Duration is shorter than maturity for all bonds except zero coupon bonds Duration is equal to maturity for zero coupon bonds All else equal, duration is shorter at higher interest rates 11-12

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**More on Duration The duration of a level payment perpetuity is**

Note that for a perpetuity the coupon rate does not affect the duration. 11-13

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**Figure 11.2 Duration as a Function of Maturity**

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**Duration/Price Relationship**

Price change is proportional to duration and not to maturity DP/P = -D x [Dy / (1+y)] D* = modified duration D* = D / (1+y) DP/P = - D* x Dy D = Duration The minus sign in this equation reminds us that if interest rates go up, prices go down and vice versa. Although the text simplifies this you have to be careful using modified duration. It is used for instruments that have non-annual cash flows as follows: “Modified duration” = DurationMod Purpose: For bonds & loans with non-annual payments DurationMod = DurationAnnual / (1 + rperiod); where rperiod = periodic interest rate, typically semiannual for a bond The predicted price change using modified duration is %ΔPPr= -DurationMod * ΔrAnnual ; Notice using modified duration allows one to plug in the annual rate change rather than the change in semi-annual rates. 11-15

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**11.2 Passive Bond Management**

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Interest Rate Risk Interest rate risk is the possibility that an investor does not earn the promised ytm because of interest rate changes. A bond investor faces two types of interest rate risk: Price risk: The risk that an investor cannot sell the bond for as much as anticipated. An increase in interest rates reduces the sale price. Reinvestment risk: The risk that the investor will not be able to reinvest the coupons at the promised yield rate. A decrease in interest rates reduces the future value of the reinvested coupons. The two types of risk are potentially offsetting. Price risk is not present if you hold the bond to maturity. The risks are offsetting because if interest rates rise, sale price will fall but the reinvestment income will be higher and vice versa. If we could choose just the right amount of price volatility to offset the change in the future value of the reinvestment income we could eliminate interest rate risk. We are then said to be ‘immunized.’ 11-17

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Immunization Immunization: An investment strategy designed to ensure the investor earns the promised ytm. A form of passive management, two versions Target date immunization Attempt to earn the promised yield on the bond over the investment horizon. Accomplished by matching duration of the bond to the investment horizon Even though it is a passive strategy it does require trading over the investment horizon. Note that a well defined horizon is required. 11-18

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**Terminal Value of an Immunized Portfolio over a 5 year Horizon**

This works because if you match the duration to the investment horizon then the price volatility is just enough to offset the change in future value of the coupons if interest rates move. 11-19

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**Figure 11.3 Growth of Invested Funds**

This graph can be used to illustrate that after an interest rate increase the bond price drops but the future value of the invested funds will be higher due to the higher interest rate and at an investment horizon = duration = 5 years the two changes exactly offset. 11-20

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**Immunization Net worth immunization**

The equity of an institution can be immunized by matching the duration of the assets to the duration of the liabilities. 11-21

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Figure 11.4 Immunization 11-22

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**Cash Flow Matching and Dedication**

Cash flow from the bond and the obligation exactly offset each other Automatically immunizes a portfolio from interest rate movements Not widely pursued, too limiting in terms of choice of bonds May not be feasible due to lack of availability of investments needed Dedication is simply multi-period cash flow matching. Note this can be done with a set of zeros of different maturities or with coupon bonds. The STRIPS mentioned in the prior chapter can be useful for this purpose. 11-23

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**Problems with Immunization**

May be a suboptimal strategy Does not work as well for complex portfolios with option components, nor for large interest rate changes Requires rebalancing of the portfolio periodically, which then incurs transaction costs Rebalancing is required when interest rates move Rebalancing is required over time May be suboptimal if you have a rate forecast and are willing to take a position on which way rates will move. This is actually an important point. If you think rates will fall you want a duration longer than your investment horizon. If you are right, the portfolio will earn more than the promised ytm. If you think rates will increase you want a duration shorter than your investment horizon. Again, if you are right, the portfolio will earn more than the promised ytm. Note that this goes against conventional wisdom that states you are hurt in bonds by rising interest rates. (Real returns may be reduced if the rate increase is due to inflation, but not nominal returns.) 11-24

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11.3 Convexity 11-25

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**The Need for Convexity Duration is only an approximation**

Duration asserts that the percentage price change is linearly related to the change in the bond’s yield Underestimates the increase in bond prices when yield falls Overestimates the decline in price when the yield rises 11-26

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**Pricing Error Due to Convexity**

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**Convexity: Definition and Usage**

Where: CFt is the cash flow (interest and/or principal) at time t and y = ytm The prediction model including convexity is: 11-28

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Bond Price Convexity Note that convexity is a good thing if you are long in bonds. The duration prediction is always pessimistic. 11-29

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Convexity of Two Bonds Which bond would an investor prefer to hold? Bond A because its higher convexity implies its price will rise more than B when rates fall and A’s price will fall by less than B when interest rates rise. 11-30

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**Prediction Improvement With Convexity**

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**11.4 Active Bond Management**

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**Swapping Strategies Substitution swap**

Exchanging one bond for another with very similar characteristics but more attractively priced Intermarket spread swap Exploiting deviations in spreads between two market segments Rate anticipation swap Choosing a duration different than your investment horizon to exploit a rate change. Rate increase: Choose D > Investment horizon Rate decrease: Choose D < Investment horizon 2. If your charts tell you the spread between Baa and Aaa is too wide and will narrow you should short the Aaa and buy the Baa until the spread corrects. You will capture the drop in price on the Aaa and the increase in price on the Baa. This is a typical hedge fund strategy and was a common strategy for Long Term Capital Management Hedge fund. 11-33

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**Swapping Strategies Pure yield pickup**

Switching to a higher yielding bond, may be longer maturity if the term structure is upward sloping or may be lower default rating. Tax swap Swapping bonds for tax purposes, for example selling a bond that has dropped in price to realize a capital loss that may be used to offset a capital gain in another security 4. This one is not associated with a rate change, just want a higher promised yield 11-34

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Horizon Analysis Analyst selects a particular investment period and predicts bond yields at the end of that period in order to forecast the bond’s HPY 11-35

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