1、Introduction to the Measurement of Interest Rate Riskby Frank J. Fabozzi,Copyright 2007 John Wiley however, in this case it is easier to use and demonstrates the point about interest rate risk).,Full Valuation Approach to Measuring Interest Rate Risk,Full Valuation Approach for Measuring Portfolio I

2、nterest Rate Risk,The full valuation approach measures bond values at different interest rate levels and when various bonds are combined together it also can be used for valuing a bond portfolio. An advantage of the full valuation approach is that it can be used for parallel and non-parallel interes

3、t rate movements for bonds with different maturities. Exhibit 2 shows the basic full valuation approach to changing interest rates for a bond portfolio with a parallel shift in the yield curve.,Full Valuation Approach for Measuring Portfolio Interest Rate Risk,Parallel and Non-Parallel Shifts in Int

4、erest Rates,Rate,Maturity,Original Curve,Parallel Shift,Non-Parallel Shift,Full Valuation Approach for Interest Rate Risk (Non-Parallel Shift in Yields),Exhibit 3 shows the basic full valuation approach to evaluating changing interest rates for a bond portfolio with a nonparallel shift in the yield

5、curve. This approach of examining the change in price and yield works fine; however, it is quite time consuming and it would be useful to have a single measure that could express the amount of interest rate risk for a single bond or a bond portfolio without having to compute the full valuation of ea

6、ch bond. Thats where the duration measure plays a role.,Full Valuation Approach for Interest Rate Risk (Non-Parallel Shift in Yields),Measuring Interest Rate Risk: Approximate Percentage Price Change,The formula for approximating the percentage change for a 100 basis point change (duration) in yield

7、 is: Price (100 bps decline) Price (100 bps increase) 2 X Initial Price X Change in Yield Example in Chapter 2 resulted in an average of 10.44 basis points change for a 100 basis point change in market yield. This 10.44 is the duration of the bond. This means that for a +/- 1% change in market rates

8、, this bond would change in price by -/+ .1044%. A 50 basis point change would be .1044%/2 or .0522% A 10 basis point change would be .1044%/10 or .001044%,Bond Price/Yield Relationship,The characteristics of a bond that affect its price volatility are (1) maturity, (2) coupon rate, and (3) presence

9、 of any embedded options. The shape of the price/yield relationship for an option-free bond is convex. The price sensitivity of a bond to changes in the required yield can be measured in terms of the dollar price change or percentage price change (Exhibit 4).,Bond Price/Yield Relationship,Price Vola

10、tility Characteristics of Option-Free Bonds,The price of a bond changes in opposite direction to the movement in the bonds yield.,Price Volatility Characteristics of Option-free Bonds Non-linear shape,As yield increases, the price of a bond decreases However, this relationship is not linear, the sha

11、pe is convex.,General Properties Concerning the Price Volatility of Option-Free Bonds,Although bond prices move in the opposite direction from the change in yield, the percentage price change is not the same for all bonds. For small changes in yield, the percentage price change for a given bond is r

12、oughly the same, whether the yield increases or decreases. For large changes in yield, the percentage change is not the same for an increase in yield as it is for a decrease in yield. For a given large change in yield, the percentage price increase (with falling rates) is greater than the percentage

13、 price decrease (with increasing rates).,General Bond Properties,Property 1. In response to a given change in yield, the percentage change in the value of all bonds is not the same. This is because the convexity of all bonds is not the same. Longer maturity, for example, increases convexity. Lower c

14、oupon increases convexity.,General Bond Properties #1,General Bond Properties (contd),Property 2. For a very small change in the yield, the percentage gain and loss is approximately the same. If the yield change is very close to the original yield, the price-volatility relationship is close to symme

15、tric. The curve is approximately symmetric close to any one point.,General Bond Properties (contd),Property 3. For a large change in the yield, the the percentage gain and loss is not the same. Farther away from the original yield, the price-volatility relationship is not symmetric. The curve is not

16、 symmetric over a large range.,General Bond Properties (contd),Property 4. For a given change in yield, the effect on price of a decrease in yield is greater than for an increase in yield. If the yield goes down 1%, then the increase in price is greater than the decrease in price from the yield goin

17、g up 1%. The slope of the curve is strictly increasing. The convexity of the curve is the reason for this effect.,General Bond Properties #3 and 4,Less Convex Bond,Highly Convex Bond,General Bond Properties,The explanation for properties 3 and 4 come from the convex shape of the price/yield relation

18、ship. As can be seen in the graph below, (Y4 Y3) is a larger change in yield than (Y2 Y1,) yet the price change, (P4 P3), is smaller than (P2 P1).,Logic Behind Properties 3 and 4,As yield increases (decreases), the slope of the price/yield curve decreases (increases). As yield increases, bond prices

19、 fall, but this is tempered by the decline in slope of the curve.,General Bond Properties from the Example,Property 1: Compare the curvature of 1, 10 and 30-year bonds. Property 2: When the yield changes up or down 1%, the effect on price is similar (-8.7% versus 10.3%). Property 3: When the yield c

20、hanges up or down 5%, the effect on price is quite different (-33% versus 77%). Property 4: In either case, the effect of a decrease in yield is greater than the effect of an increase in yield.,Plot of the Price-Yield Relationship Convexity Varies by Maturity,Convexity or Slope of the Price/Yield Cu

21、rve is Important,Yield,Price,Smaller slope,Greater slope,Original Yield,Price Volatility of Bonds with Embedded Options,The price of a bond with an embedded option consists of two components: Value of an option-free bond Value of the embedded option The most common types of embedded options are call

22、 (or prepay) options and put options. Call bonds have a region of the price-yield relationship that displays negative convexity.,Price-Yield for a Callable Bond,Part 1: The price-yield for an option-free bond is convex:,Price,Yield,Price-Yield for a Callable Bond,Part 2: A call option has the follow

23、ing effects on the price of the option-free bond: At high interest rates, the call option has almost no value (very unlikely to be exercised). The price behaves at this point like an option-free bond As interest rates decrease, the call option takes on negative value because it is more likely to be

24、called. The price behaves at this point differently than an option-free bond,Price-Yield for a Callable Bond,Price-yield for a call option: At lower yields, the call option has increasing negative value to the investor this will result in a negative convexity region in the price-yield relationship.,

25、Yield,Price,0,Price-Yield for a Callable Bond,Combining these, the price - yield relationship for a callable bond is shown in Exhibit 12 and below:,y*,Price,Yield,Option-free bond,Callable bond,Area of negative convexity,Price-Yield for a Callable Bond,At high interest rates, the prices of callable

26、and option-free bonds are approximately the same. As interest rates decrease, the price of a callable bond increases at a slower rate than does an option-free bond, due to the increasing negative value of the call option.,Convexity for a Callable Bond,Negative convexity means that for a large change

27、 in interest rates, the amount of the price appreciation is less than the amount of the price depreciation. Option-free bonds exhibit positive convexity. Positive convexity means that for a large change in interest rates, the amount of the price appreciation is greater than the amount of the price d

28、epreciation. A callable bond exhibits positive convexity at high yield levels and negative convexity at low yield levels where high and low yield levels are relative to the issues coupon rate. At low yield levels (low relative to the issues coupon rate), the price of a putable bond is basically the

29、same as the price of an option-free bond because the value of the put option is small; as rates rise, the price of a putable bond declines, but the price decline is less than that for an option-free bond.,Price-Yield for a Callable Bond,Convexity Option-free bonds have positive convexity through the

30、ir range Below y* (in Exhibit 12) callable bonds have negative convexity This means that (contrary to an option-free bond) the absolute change in price is less as interest rates decline than when they rise.,Exhibit 12. Negative and Positive Convexity of a Callable Bond,Yield,Price,Option-free bond:

31、positive convexity,Callable bond at the call range: negative convexity,Valuing a Callable or Putable Bond,A great advantage in analyzing bonds with embedded options is that it is possible to separate the value and characteristics of the bond from those of the option. Callable (putable) bond price =

32、straight bond price (+) call (put) option price Putable bonds have a lower positive convexity than an option-free bond,Price/Yield Relationship for a Putable Bond,Convexity of Putable Bonds,Yield,Price,Option-free bond,Putable bond: smaller positive convexity,Convexity of Mortgage-Backed Bonds,Yield

33、,Price,Mortgaged-backed bonds,Higher prepayment risk at low interest: negative convexity,Lesser prepayment hence “extension” risk: negative convexity,Duration The Other Approach to Analyzing Bond Price Volatility,Duration is measure of the approximate price sensitivity of a bond to interest rate cha

34、nges. It is the approximate percentage change in price for a 100 basis point change in interest rates. It is the first (linear) approximation of the percentage price change. To improve the duration approximation, an adjustment for “convexity” will be made. This is known as the duration/convexity app

35、roach.,Calculating Duration Fabozzis Equation (1),Duration = (V- - V+) / 2(V0)(y) Where y = change in yield in decimal V0 = initial price V- = price if yields decline by y V+ = price if yields increase by y Example y = .002, V0 = 134.6722, V- = 137.5888, and V+ = 131.8439 Duration = (137.5888 131.84

36、39)/2(134.6722)(.002) = 10.66,Duration,Duration - It can be thought of as the price sensitivity of a bond (or a portfolio of bonds) to changes in interest rates. A zero coupon bond that matures in n years has a duration of n years A coupon bearing bond maturing in n years has a duration of less than

37、 n years, because the holder receives some of the cash payments prior to year n.,Duration,Duration is a first approximation of a bonds price or a portfolios value to interest rate changes. To improve the estimate provided by duration, a convexity adjustment can be used. Using duration combined with

38、a convexity adjustment to estimate the percentage price change of a bond to changes in interest rates is called the duration/convexity approach to interest rate risk measurement. Duration does a good job of estimating the percentage price change for a small change in interest rates but the estimatio

39、n becomes poorer the larger the change in interest rates.,Calculating Duration,In calculating duration, it is necessary to shock interest rates (yields) up and down by the same number of basis points to obtain the values when rates change. In calculating duration for option-free bonds, the size of t

40、he interest rate shock is unimportant for reasonable changes in yield.,Calculating Duration,In Chapter 2, bond duration was introduced as: (price if yield declines price if yield rises) divided by (2 times the initial price times the change in the yield) which is equation (1) in Chapter 7. The durat

41、ion of a bond is estimated as follows:,Duration for Bonds with Embedded Options,For bonds with embedded options, the problem with using a small shock to estimate duration is that divergences between actual and estimated price changes are magnified by dividing by a small change in rate in the denomin

42、ator of the duration formula; in addition, small rate shocks that do not reflect the types of rate changes that may occur in the market do not permit the determination of how prices can change because expected cash flows may change. For bonds with embedded options, if large rate shocks are used the

43、asymmetry caused by convexity is encountered; in addition, large rate shocks may cause dramatic changes in the expected cash flows for bonds with embedded options that may be far different from how the expected cash flows will change for smaller rate shocks.,Modified and Effective Duration,Modified

44、duration is the approximate percentage change in a bonds price for a 100 basis point change in yield assuming that the bonds expected cash flows do not change when the yield changes. In calculating the values to be used in the numerator of the duration formula, for modified duration the cash flows a

45、re not assumed to change and therefore, the change in the bonds price when the yield is changed is due solely to discounting at the new yield levels. Effective duration is the approximate percentage change in a bonds price for a 100 basis point change in yield assuming that the bonds expected cash f

46、lows do change when the yield changes. Modified duration is appropriate for option-free bonds; effective duration should be used for bonds with embedded options.,Modified and Effective Duration,Modified and Effective Duration,The difference between modified duration and effective duration for bonds

47、with an embedded option can be quite dramatic. Macaulay duration is mathematically related to modified duration and is therefore a flawed measure of the duration of a bond with an embedded option. Interpretations of duration in temporal terms (i.e., some measure of time) or calculus terms (i.e., fir

48、st derivative of the price/yield relationship) are operationally meaningless and should be avoided.,Error in Estimating Price Based on Duration Only,R0,Tangent line,Actual Price,Error in estimating price based only on duration,Error,Yield,Price,Implications of Tangent Line Duration Approximation,Dur

49、ation is good for estimating the impact of small interest rate changes. It is not as accurate for large interest rate movements. The duration estimate is less accurate, the more convex the bond price/yield relationship. The tangent line (duration approximation) always underestimates the actual price

50、. This only works for option-free bonds.,Interpretation of Duration,Dont make this out to be rocket science While duration is the first derivative of the bonds price/yield function, it is simply a measure of the approximate percentage price change for a 100 basic point change in interest rates. It i

51、s possible for two bonds to have the same duration, but different convexity and to behave differently across large interest rate changes.,Portfolio Duration,The duration for a portfolio is equal to the market-value weighted duration of each bond in the portfolio. In applying portfolio duration to es

52、timate the sensitivity of a portfolio to changes in interest rates, it is assumed that the yield for all bonds in the portfolio change by the same amount. The duration measure indicates that regardless of whether interest rates increase or decrease, the approximate percentage price change is the sam

53、e; however, this is not a property of a bonds price volatility for large changes in yield.,Portfolio Duration,Portfolio duration is pretty straightforward it is simply the weighted average of the bonds in the portfolio. Many practitioners simply do a weighted average of the duration of each bond in

54、the portfolio; however, this assumes that there is a parallel shift in the yield curve. Fabozzi recommends the full valuation approach of calculating the dollar price change for a given number of basis points for each bond in the portfolio and summing up the price changes and dividing by the initial

55、 market value of the portfolio.,Convexity,None of the previous price-volatility measures capture the non-linearity or curvature of that relationship. These measures are only approximate. These measure are only good locally. We need an additional factor to capture the curvature of the relationship.,P

56、ricing Error and Convexity,Price,Yield,Duration,Pricing error due to Convexity,Convexity Adjustment,A convexity adjustment can be used to improve the estimate of the percentage price change obtained using duration, particularly for a large change in yield. The convexity adjustment is the amount that

57、 should be added to the duration estimate for the percentage price change in order to obtain a better estimate for the percentage price change. The same distinction made between modified duration and effective duration applies to modified convexity adjustment and effective convexity adjustment.,Conv

58、exity Adjustment for a Bond with an Embedded Option,For a bond with an embedded option that exhibits negative convexity at some yield level, the convexity adjustment will be negative.,Convexity Adjustment,The formula for the convexity adjustment to the percentage price change is: C times (change in the yield) squ