Bond Duration. Floyd Vest

Transcription

1 Bond Duration Floyd Vest It is well known that when arket interest rates change, the price of a bond, or the share prices in a bond fund, changes. Bond duration is widely used to estiate the change in bond price, or the percentage change in price, for a given change in arket interest rates. We will call the following exaple, the Easy Bond Duration Rule: For exaple, if a bond has a duration of 2 years, the price would fall about 2 when interest raised one percentage point. For a bond with price, interest payents (coupons) C at the end of each coupon period, value at aturity M, with years to aturity, and interest rate per coupon period of i, C 1 1+ i () + M 1 (1) the bond duration D = 1+ i () =1 = D1 years. (See Ho uen Ng, The Duration of a Bond, UMA Module 99685, in the bibliography for this course, for a derivation of the above forula. The author of the article left it to the reader to develop a closed for forula for bond duration.) Derivation of a closed for forula for bond duration. We will derive a closed for forula for bond duration D starting with D1, which is the nuerator of the above expression: D1 = C 1 =1 1+ i D1 = C ) (1) + () Consider 1 D1= ( 1 ) 2 1 (1) + () + M 1 1+ i (2) + () 1 (2) (3) (3) (), +, + M 1 1 ( 1) () + ()-,- + M 1 +1 () Bond Duration Spring,

2 Then consider 1 D1 1+ i D1 = ( 1 1+ i ) i i ( 1 (1+ i) + = C ) i -, i Therefore we have 1+ i (2) D1 = i where 1 1 = D = D1 years i + - C 1, 1+ i ( C () 1+ i + M i - ), C 1 (1+ i) ( ) i, i 1+ i. +1 () + M 1 1+ i +1 ()(C + Mi), () M 1 1+ i Exaple 1 (Ho uen Ng, Exercise 10) Consider as an exaple Bond A with C = 10 per year, = 5 years, Value at aturity = M = 100, i = interest rate per year = 0.06, and rice = (See Forula 7 below for pricing the bond.) Calculating with Forula 2, D1 = D1 = D = D1 = (1.06)5 ( ) 1.06, - 6 (5) (0.06) ( ) = years, as in Ho uen Ng. A ore convenient forula for bond duration. A ore convenient closed for forula for duration for an annual pay bond, referred to as MacD is (3) MacD = 1+ i (1+ i) + C M i years, where M = value at aturity, i C ( M (1+ i) 1) + i C = coupon, i = yearly interest rate (yield), = the nuber of years to aturity. (See wikipedia.org/wiki/bond_duration.) +1 () Bond Duration Spring,

3 Exaple 2. Calculating for the above Bond A with C = 10 per year, = 5 years, M = 100, i = interest rate per year = 0.06, gives ( ) MacD = = 4.23 years as above. 10 ( 100 ( )5 1) MacD for a seiannual coupon bond. For a seiannual coupon bond with i = seiannual discount rate, = the nuber of seiannual periods to aturity, r = annual coupon rate in percent for ( r + MacD = ) 1+ i i 100(1+ i) i r ( 2 ( and is easured in years. i) 1) +100i -, Exaple 3 (fro Wikipedia) Consider a two year seiannual coupon bond with the annual coupon rate r = 20, r = 10 in percent for, 4 annual yield, i = 0.02, = ( ) + 4( (0.02))( 1+ MacD = (( ) 4 1) +100(0.02) ) 2, - = years. We can calculate Duration for this bond fro the basic definition with C = 10, M = 100, i = 0.02, for four seiannual periods, and = : 1 D = ) , + 2(100) = years. ( Notice that this bond has a price at a 4 yield of , and it pays 20 per year coupon rate based on M = 100, where C = 10 per six onths, which is 10 of M. The bond was bought above the aturity value of 100 and thus yields only 4. See the Side Bar Notes for discussion of this bond and the large decline fro 20 to 4. See Forula 7 below for pricing the bond. Forulas for estiating. Ho uen Ng derived the following Forulas 4 and 5 for using bond duration to estiate the change in price, denoted Δ, and the percentage change in price, denoted, fro a change in interest rate i, denotedδi. (4) D 1+ i i. (5) For sall i, DΔi. Bond Duration Spring,

4 (6) Fro Sharpe (p. 386), (i +1) D 1+ i. Exaple 4. Consider Bond A above with C = 10 per year, M = 100, = 5 years, i = 0.06 per year, price = , and D = years. We calculate the percentage change in when rates rise to 7 per year using Forulas 4, 5, and 6. Fro Forula 4, Δ Di = (116.85)(4.235)(0.01) = 4.67 (loss) 1+ i For this rise in interest rate of one percentage point fro 6 to 7, the price of the bond drops by approxiately The new price is approxiately = Using Δ 4.67, 4.67 = 0.04 = 4 as the percentage drop in price, for a one percentage point increase in i fro 6 to 7. Using Forula 4, D 1+ i i gives 4.67 (0.01) = = Fro Forula 5, DΔi = ( 4.235)(0.01) = = 4.2. Fro Forula 6, (1+ i) D 1+ i = = 0.04 = 4 (loss). For the yearly Bond A, the actual accurate values for change in price and percentage change, with C = 10, = 5, = , M = 100, i = 0.06, new i = 0.07, are: 1 ( )5 The new rice 1 = ( )5. 1 = Δ = = 4.55 loss. = 4.55 = = 3.89 loss in price It appears that Forula 6 fro Sharpe ay be the ore accurate. Duration for a bond utual fund. Duration as an average is reported for bond utual funds. For exaple, for a bond fund, on 5/13/2011, it was reported that Average Maturity of 23.4 years. Average Duration of 12.9 years. It is also worth noting: Share rice 5/12/ Expense Ratio 0.22 Yield 4.81 Average annual perforance for the last five years 6.60 Average annual perforance for the last ten years 6.78 Average annual perforance since inception (3/1/94) 7.36 Bond Duration Spring,

5 We could apply Forula 4 above to get the change, and percent change, in share price (NAV) for a raise in arket interest rate of one percentage point fro 4.81 to an average of Fro Forula 4, 12.9 (0.01) = = 12.3 loss in NAV for a one percentage point increase in interest rates (0.01) = 1.50 (loss in NAV) The new NAV = Investors often study the volatility of bond fund share prices by tracking the changes over tie: To see how far and how fast they can drop, and To see how long it takes to recover forer highs. For exaple, for the above bond fund researched on 5/12/2011, the all tie high NAV = (8/20/2010), and the all tie low NAV = 8.72 (11/07/1994). For bond funds, dividends can be paid onthly and reinvested, so for volatility, one should study the supplied graph giving growth in the last ten years of 10,000 invested with dividends reinvested. Roughly, tie to recover a high: May 03 to Feb. 04, June 05 to Aug. 06, June 08 to Dec.08, Feb 09 to July 09. There was an unrecovered drop starting Nov This is a lot less volatile than the stock arket fro 1999 to Bond calculations on the TI 83/84. For bond calculations on the TI 83/84, consider Forula 7 below, which gives a forula for the price of a bond with coupons C, interest rate per coupon period i, the nuber of periods to aturity N, and value at aturity F. The Forula 8 for V is found in the Appendix to the TI83/84 anual and can be used to calculate the price of a bond. We rewrite the V Forula 8 as Forula 9 and copare it to Forula 7. N 1 (1+ i) (7) = C + F(1+ i) N i (8) V = MT FV i ( 1 (1+ i) MT N i (9) V = MT (1+ i) N 1 1 FV i 1+ i This says that to calculate the price of a bond with the TI83/84 TVM Solver, enter C = MT positive, M = F = FV as negative, enter i as percent in I, enter C/Y as 1, select MT:END. N Bond Duration Spring,

6 For the yearly Bond A above, where C = 10, = 5, M = 100, and i = 0.06, we give the Code and coentary: 2 nd Finance Enter to select the TVM Solver 5 Enter for N 6 Enter for I (-)10 Enter for MT (-)100 Enter for FV 1 Enter for /Y Select t:end Enter to highlight V Alpha Solve and read Then for the new price when i increases to 0.07: Code and coentary: to highlight I 7 Enter for I highlight V Alpha Solve and read Bond Duration Spring,

7 Exercises 1. Given a three year annual pay 8 bond, with C = 80, and M = 1000 = : (a) Calculate bond duration D by Forula 1. (A bond like this one with = M and i = C is refereed to as selling at par.) M (b) Calculate D by Forula 3. (c) Apply the Easy Bond Duration Rule in the first paragraph of this article, to a rise in interest rates, and then to a drop in interest rates. (d) Apply Forulas 4 and 6 to calculate and Δ. Tell what happens when rates rise fro 8 to 9, when rates drop by one percentage point. (e) What do notice about the calculation of MacD? State and prove a new MacD forula for annual pay bonds sold at par. (f) Calculate the actual price of the bond and the actual Δ and. Copare. 2. Given a five-year seiannual pay bond with annual yield y = 4, and seiannual rate i = 2, with = M = 100, and C = 2. Note that r = 2 in percent for. 2 (a) Calculate MacD, put on units. (b) Apply the Easy Bond Duration Rule to a rise in price, for a drop in interest rates of one percentage point. (c) Apply Forulas 4 and 6 to calculate and Δ. Tell what happens when rates rise two percentage points, when they fall two percentage points. (d) Calculate the actual price of the bond and the actual Δ and. (e) Copare the answers to (d), and the approxiations in (c). (f) State and derive a new Forula for MacD, for a seiannual pay bond at par, where C = Mi and = M. (g) Calculate the duration for this bond fro the basic definition as done in Exaple Consider a three-year zero coupon bond that pays 100 at aturity in three years, at an annual interest rate of 6. (a) Calculate the duration fro Forula 1. Use the Copound Interest Forula (which is part of Forula 9) for the price. What do you observe? (b) State and prove a theore about the duration of a zero coupon bond. (c) Calculate the actual price of the bond and the actual Δ and. 4. rove the Theore: For any coupon bond, its duration is less than the aount of tie to aturity. Bond Duration Spring,

8 5. rove the Theore: For two bonds with everything the sae except yield, the one with the lower yield will have the greater duration. Use Forula In this article, we derived a forula for the su of a sequence that sees to be part geoetric and part arithetic. (See Forula 1.) (a) Derive a forula for the su of a sequence where the first ter is 1, the second ter is 2, the coon difference is 1, with n ters. (b) Derive a forula for the su of an arithetic sequence with first ter (a + d), second ter (a + 2d), etc. for n ters. 7. rove that S = r = r(1 r ) + =1 (1 r) 2 forula applies. r =1 1 r. Nae one application to which the 8. Derive a closed for forula for S = ( +1)ar. n1 = rove that = R(1+ I) (1+ r) = R 1 (1+ y) (, where y = r I =1 y 1+ I applications to which this forula applies.. Nae two 10. For the long ter bond fund discussed above, there was a drop in NAVs fro on 2/4/87 to a low of 9.56 on 4/18/87, about 2.5 onths. Estiate the Δi, the rise in interest rates. Bond Duration Spring,

9 Side Bar Notes: Fro vanguard.co Duration, a easure of the sensitivity of bond and bond utual fund prices to interest rate oveent. For exaple, if a bond has a duration of 2 years, the price would fall about 2 when interest rates rose about one percentage point. On the other hand, the bond s price would rise by about 2, when interest rate falls by one percentage point. TI81 rogra for Bond Duration and rice rogra 9: BOND DURATION AND RICE Disp INUT N, NUMBER OF COUONS Input N Disp INUT C, AMOUNT OF COUON Input C Disp INUT M, MATURITY VALUE Input M Disp INUT I, SEMIANNUAL INTEREST RATE Input I N M(1+I)^(-)N S M(1+I)^(-)N 1 T Lbl 1 T C(1+I)^(-)T+S S C(1+I)^(-)T+ 1+T T if T>N Goto 2 Goto 1 Lbl 2 Disp DURATION Disp S Disp RICE Disp Bond Duration Spring,

10 Bond risks. Bonds have interest rate risks where if interest rate rise, the price of the bond can drop. Also bonds have credit risks. Bonds issued by corporations are rated by Standard oor s and Moody s on the basis of the corporation s ability to pay dividends and value at aturity in a tiely way. The highest rated bonds are AAA or Aaa respectively. A rating of BBB or Baa is for a bond that is likely to be paid. Ratings of BB/Ba and lower are speculative bonds and investents are best anaged by experts in bonds. Bonds are also rated CCC/Ca. Bonds behind in their interest payents are rated even lower. (See and click on Ratings Definitions. See and click Understanding credit ratings, The ABCs of Rating Scales. See etc.) The bond in Exaple 3. The bond seeingly had an annual rate of 20 that dropped to 4. How could there be such a large drop, and such a high coupon rate? erhaps it was a low rated bond to begin with and paid a high 20 rate. Soething ay have happed and the bond was a credit risk and thus would sell at a discount, but it sold at a preiu of where par was 100. (Do the calculations.) erhaps the quality of the bond increased and then sold at a preiu. The copany ay have been bought out by a very strong copany. erhaps the original bond investent was venture capital and the product developent proved successful, and the owner of the bond sold it at a 30 profit. It ay have been that this bond originally had a nuber of years to aturity and interest rates gradually dropped fro 20 to 4. But this doesn t explain the high 20. erhaps it is not an actual historical bond, but an exaple ade up by a wikipedia author. How uch does it cost to be financially independent? That is to be in the upper 25 percent in incoe and not have to work, but live off yield and interest. To be in the upper 25 percent according to IRS files for 2010, the estiated required incoe is at least 72,377 per year. In 2011, living off a portfolio on interest fro 50 in five-year Treasury notes, and on yield fro 50 in S 500 stocks, paying a piddling 1.77 per year requires 4,089,096. The required assets haven t always been this large because the portfolio often earned a better rate of return. (Scott Burns, Denton Record Chronicle, July 10, (To see a coplete 1986 to 2011 table of the Life of Riley Index, visit Scott Burns at Would you like to be a illionaire? You will be by the tie you retire, if you plan and invest responsibly. Shopping for Long Ter Bond Funds. Go to Click ersonal investors, Research Bonds and Stocks, Click Other Fund Failies, Click Mutual fund screener, Fill in: Mutual funds, Bond Fund, Taxable. Add Long Ter Bond Fund. Add All fund failies. Click on Get results. Click for 5-year total return in descending order. There are six funds left. One is a load fund and it is not clear if the load is subtracted to calculate Total Return. Bond Duration Spring,

11 References Ho uen Ng, The Duration of a Bond, 1989, UMA Module (in this bibliography). Sharpe, Willia F. and Gordon J. Alexander, Investents, Fourth Edition, rentice-hall, For theores on changes in bond prices, see Vest, Floyd, Bond ricing Theores, 2011 (in this bibliography). For inforation on bonds and bond duration, see or Bond Duration Spring,

12 Answers to Exercises (a) D = =1 ( 1 + () ) (3)., D = D1 = = 2.78 years (b) MacD = 2.78 years. (c) If interest rate rises to 9, the price of the bond drops by (d) By Forula 4, 2.78 (0.01) = = Δ ( )(1000) = When rates fall fro 8 to 7, the price rises by about and i (e) For an annual bond at par, MacD = i ( 1 (1+ i) ) +. 1 ( )3 (f) = 1000 at 8. At 9, rice = Δ = = loss. = ( ) 3 = (a) For this seiannual bond, ( 4 100( ) MacD = (0.02) ( 2 ( = years. 0.02)4 1) +100(0.02) - ), (b) For a one percentage point drop in annual interest rates, there is a rise in the price of the bond. (c) Since duration is in years where y is the annual yield, Forula 4: D ( 1+ y y ). = (0.01) = = For a two percentage point rise, the bond price drops by 2(1.87) = The price of the bond drops by (f) For a seiannual pay bond at par with i = the seiannual rate, = M, C = im, and r 2 in percent for = 100i, MacD = 1+ i 2i 1 1 (1+ i) 2 years where is in years. Bond Duration Spring,

13 3. (a) For the zero coupon bond with M = 100, C = 0, = 3 years, i = 6 per year, D = (3) = 3 years. 100( ) 3 4. Theore: For any coupon bond, with coupon C, tie to aturity, value at aturity M, and interest rate per period i, the bond duration D, D. roof: In Forula 1, for all,, so C 1 1+ i =1 So D = = =1 =1 C 1 1+ i () + M 1 1+ i C 1 1+ i + M 1 1+ i () = =. =1 () ( C 1 1+ i (). =1 C 1 1+ i () + M 1 1+ i () 10. Fro Forula 5, the Δi, rise in interest rate, was approxiately 0.47 of a percentage point. Bond Duration Spring,

14 Teachers Notes For the basics of atheatics of finance, copound interest, and annuities, see asting, and Luttan. For the basics of bonds, see Vest, The Matheatics of Bond ricing and Interest Rate Risk, HiMA ull-out, Consortiu 59. For ore interesting probles on bond duration, see Ho uen Ng, all in this course. Bond Duration Spring,