AMS Capital Markets and Portfolio Theory

Transcription

1 AMS Caital Markets and Portfolio Theory I Leture 2 - Fixed Inome Seurities and the Term Struture of Interest Rates Robert J. Frey Researh Professor Stony Brook University, Alied Mathematis and Statistis frey@ams.sunysb.edu We over the markets in whih ash is loaned and borrowed and disuss the relationshi between the term of a debt and the interest rate aid. This material, with some extensions, is based on Chaters 3 and 4 of Luenberger s text. February 02, Fixed Inome Seurities Debt Markets Ë Finanial Instruments - also alled seurities, are ommitments. There are, in effet, ontrats, usually in some standardized form. Ë Finanial Markets - are organized exhanges for trading in finanial instruments. Ë Liquidity Ë Priing Ë Fixed Inome Seurities - are romises to ay a well-defined stream of ash in the future in return for ash today. Ë Fixed Inome Markets - define the market for money Tyes of Fixed Inome Markets Ë Savings Aounts Ë Demand Deosits - e.g., standard savings aounts. Ë Timed Deosits and Certifiates of Deosit - held for a fixed term; early withdrawal, when ermitted, involves enalties. Ë Money Market Ë Commerial Paer - unseured loans of one year or less. Ë Banker s Aetane - a bank guarantee on a future ayment; an be sold now at disount. Ë Eurodollar Deosits and Certifiates of Deosit - denominated in US$ but held outside the US. Ë Sovereign Debt Seurities ("Treasuries in the US) Ë Bills - 13, 26 and 52 week maturities; sold at disount at aution. Ë Notes - 1 to 10 years, 6 month ouon, return of fae amount at end of term; sold at aution at disount or remium. Ë Bonds - similar to notes but with maturities > 10 years and they are allable. Ë Stris - ouons and final return of fae value stried into searate zero ouon bonds.

2 2 ams-q01-le-02-.nb Ë Other Bonds Ë Muniial Bonds - general obligation versus revenue bonds; exemt for federal and loal taxes. Ë Cororate Bonds - features inlude all rovisions, sinking funds, debt subordination. Ë Mortgages Ë Preayment Ë Balloon Payment Ë Fixed Rate, Adjustable Rate, Hybrids Ë Annuities - a ontrat that ays the annuitant money eriodially aording to a redetermined shedule or formula Valuation Formulas Ë Consol - rie, frequeny er year m, ouon, and annual nominal rate r = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = ÅÅÅÅÅÅÅÅÅÅÅÅ H1 + r ê ml k r ê m k=1 Examle - Semi-annual ayments of $100, urrent nominal annual rate for six-months is 5%. = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 100 ÅÅÅÅÅ 0.05 ê 2 = 4000 Ë Finite Streams - n years n m = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = ÅÅÅÅÅÅÅÅÅÅÅÅ H1 + r ê ml k r ê m C1-1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ H1 + r ê ml n m G k=1

3 ams-q01-le-02-.nb 3 Think of the differene between two onsols, the first long starting right now and the seond short starting at the exiry of the first. Then, disount the seond bak to the resent Examle - 30-year mortgage, 200,000, monthly ayments, 6% fixed annual rate. = ÅÅÅÅÅÅÅÅÅÅÅÅ r ê m C1-1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ H1 + r ê ml n m G fl = rh1 + rln m ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ H1 + rl n m - 1 r ê m = 0.06 ê 12 = 0.005, n m = 30 ä 12 = 360 yr = 30; = ä ä ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ = v = ; y = 12; ar = 0.06; n = yr y; ir = ar ê y; mt = ir H1 + irln v ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ H1 + irl n Ë Bonds +f n m = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H1 + r ê ml + f ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ k H1 + r ê ml n m k=1 = ÅÅÅÅÅ r C1-1 f ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ H1 + r ê ml n m G + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅ H1 + r ê ml n m Ë Given the ouon, its frequeny, the bond s fae and its term, if we are given the interest rate, then we an omute the rie; if we are given the rie, then we an omute the IRR. The IRR of a bond is usually termed its yield to maturity or more ommonly just yield. Ë If the yield is higher than the ouon rate, then the bond is selling at a disount, < f; otherwise, if the yield is lower than the ouon rate, then the bond is selling a remium, > f. When the rie equals the fae value, the bond is selling at ar. Ë Bid and Ask Pries - quoted. e.g., as 100:17 whih means /32 % of fae. Ë Pries are without arued interest = (days sine last ouon / days in ouon eriod) ä ouon. The alendar onventions that are assoiated with the bond must be onsidered in any of these time alulations. Ë Bonds are rated by how reditworthy and senior their debt is. Lower ratings means a higher rate but an inreased robability of default.

4 4 ams-q01-le-02-.nb Ë Investment Grade Bonds - high or medium grade. Ë Junk Bonds - seulative. Examle - 5% ouon rate, monthly ouon, $1,000 fae, 20 year term. There is no arued interest due. Ë Quoted at 102:23 - yield is %; the solution in Mathematia is below. n = 12 20; mt = ê 12; Ä lo ê 32 f = Join mo - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄ É o } ÇÅ n 100 ~ o, Table@ ê 12, 8k, 1, n - 1<D, mt< ÖÑ ; df = TableAH1 + r ê 12L -k, 8k, 0, n<e; FindRoot@df.f, 8r, 0.05 ê 12<D {r -> } Ë Quoted at 5.5% - rie is $ and would be quoted at 93:30; ontinuing the results above, the rie is omuted below. f = Join@Table@ ê 12, 8k, 1, n - 1<D, mt<d; df = TableAH ê 12L -k, 8k, 1, n<e; f.df Examle - Continuous ash flow bond. Ë We omute the ontinuous bond by allowing the omounding frequeny to go to. Ä Ä f r n m ÉÉ Simlify Limit ÄÄÄÄÄÄÄÄÄ ÇÅ ÇÅ m H1 + y ê ml -k + f H1 + y ê ml -n m, m -> k =1 ÖÑ ÖÑ = e-n y f HH-1 + e n y L r + yl ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄ y Ë Alying this result to the 30-year bond above we get a retty good aroximation. ê. 8n -> 30, y -> 0.11, r -> 0.10, f -> 1000< Duration Ë General Form - Time weighted, normalized PV. D = t 0 PVHt 0 L + t 1 PVHt 1 L + + t k PVHt k L + t n PVHt n L ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ PV Ë Maaulay Duration - Seial ase of duraion when the yield-to-maturity is used to omute the resent values. For a onstant ouon bond with yield y we have n m k=1 HkêmL n f ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ H1+yêmL D = k H1+yêmL ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ n m

5 ams-q01-le-02-.nb 5 Examle - 30 year, $10,000, semi-annual, 10% bond at ar. f = ; yr = 30; y = 2; ar = 0.10; n = yr y; ir = ar ê y; t = Table@j, 8j, 0.5, 30, 0.5<D; = f ir; y = 0.10; = ; f = Join@Table@, 8j, 1, n - 1<D, 8 + f<d; df = TableAH1 + y ê yl -k, 8k, 1, n<e; HtfL.df ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Ë Modified Duration d ÅÅÅÅÅÅÅÅ d y = - 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1 + y ê m D 1 D M = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ 1 + y ê m D d ÅÅÅÅÅÅÅÅ d y = -D M Examle - The modified duration for the bond immediately above is ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ 1+0.1ê2 = Continuing on with the Mathematia ode above, the atual hange in rie is df = TableAH ê yl -k, 8k, 1, n<e; Hf.dfL Ë Duration and Prie Sensitivity D > -D M D y Examle - Consider the same 30-year bond above, if the yield hanges from 10% to 11%, then the aroximate hange in rie is D > -D M D y = ä ä 0.01 = Ë Duration and Zero Couon Bond - A zero ouon bond is a bond that is sold at a disount and redeemed at the end of its term for its fae amount. There are no intervening ouon ayments. Ë Many ash flows an be modeled as a ortfolio of zeroes. Ë The term of a zero ouon bond and its (Maaulay) duration are the same.

6 6 ams-q01-le-02-.nb Ë Portfolio Duration - If we assume that all of the yields in a ortfolio of bonds are equal then the duration of the ortfolio is the dollar-weighted mean of the durations of its omonents Immunization Frequently, a ortfolio of fixed inome instruments is onstruted to math a series of future ash obiligations. Ë A ortfolio of zero ouon bonds that mathes the antiiated flows would aomlish this. It isn t always ratial, beause zeroes are only ommon with Treasuries and are not available with arbitrary maturities. Ë You ould math the resent value of the bond ortfolio with the resent value of the obligations. You would fund the obligations through an aroriate ombination of ouons and ortfolio sales. This only works if market yields don t hange. Ë A more ommon aroah is math both the PV and duration of the ortfolio to the PV and duration of the obligations it s designed to fund. This is, in effet, using a first-order Taylor series aroximation to math the ash flows. K PV O D O O O = K 1 2 n O D 1 1 D 2 2 D n n j k Ë Immunized ortfolios must be rebalaned eriodially as yields drift and the existing first-order aroximation beomes less and less valid. i x 1 x 2 ª x n y z { Convexity Ë Just as duration is related to the first derivative of rie with reset to yield, we an imrove the fit of an immunized ortfolio by emloying a seond-order aroximation. Convexity is defined as d 2 ÅÅÅÅÅÅÅÅÅÅÅÅ = C d y 2 Ë Convexity an be used to imlement ortfolio immunization more effetively by mathing the PV and both its first and seond derivatives of the obligations and funding ortfolio. Ë Convexity an also be used to imrove the aroximation for hanges in rie as a funtion of hanges in yield. D > -D M D y + ÅÅÅÅÅ 1 C HD yl2 2 Examle - Consider the same 30-year bond above. The onvexity is ClearAll@yD; Remove@yD; df = TableAH1 + y ê yl -k, 8k, 1, n<e; HH y,y Hf.dfLL ê. y -> 0.11L ê If the yield hanges from 10% to 11%, then the aroximate hange in rie is D > -D M D y + 1 ÅÅÅÅÅ 2 C HD yl2 = This loser to than the exat hange, , omuted above than the duration-only aroximation.

7 ams-q01-le-02-.nb Term Struture of Interest Rates The Yield Curve Ë The yield urve shows the yield on a fixed inome investment as a funtion of maturity. Although the urve hanges over time, it usually has a shae suh as that shown below. These are atual data shown for what is alled onstant maturity Treasuries. Treasuries are soveriegn debt of the United States CMT Treasuries 82005, 01, 28< Rate Years Years Yield Soure: htt:// Ë The yields shown above are for a omounding frequeny equivalent to the subjet bond instrument; e.g., monthly for one month debt and semi-annual for most longer term instruments. Ë Yield urves are usually soken about in terms of level, sloe and urvature. If you think about the urve as being aroximated by three oints: a short-term oint (say one month), a mid-term oint (say 3 or 5 years) and a longterm oint (say 10 to 20 years) Sot Rates and the Sot Curve Ë It s lear from the shae of a tyial yield urve that ash flows tend to ommand higher rates as they move out in time. The yield atures this as an average but it is often more useful to have a seifi rate for eah term. Ë This is onfirmed by the yields on zero ouon bonds. Arbitrage arguments suggest that sine any deterministi ash flow an be reresented as a ortfolio of zeroes, the yield on a seifi ash flow element ought to be the same as that of the orresonding zero. Ë The most ommon method of building a sot urve from market ries is to Ë Math sot rates to any available zeroes.

8 8 ams-q01-le-02-.nb Ë Bootstra rates from available other bond ries. Examle - Consider a ase in whih we have a one year zero with known yield y 1 and rie 1. We also have a twoyear onstant ouon bond with an annual ouon with ouon 2, fae value f 2, yield y 2 and rie 2. Let s t denote the annual sot rate omounded annually for a ash flow ourring t years out. Clearly, s 1 = y 1. Now to omute s 2 we have the following non-linear equation in one unknown. 2 = 2 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ f 2 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅ 1 + s 1 H1 + s 2 L 2 If this relationshi were not true, then there would be arbitrage oortunities. (What are they?) Ë Real market data reflet small suly and demand imbalanes and numerial instabilities. There will also be ortions of the urve where the sot rate will be either over- or under-determined. This means that a sot urve derived from market ries and yields must be smoothed in order to aount for these diffiulties. Ë Obviously, the omutation deends uon the frequeny of omounding. With Treasuries the usual omounding frequeny is the atual term of the ash flow for time sone year and less and semi-annual for time greater than one year. In using or analyzing a sot urve it is imortant to know just what onventions have been adoted. Ë Plotting the sot rate as a funtion of time defines the sot urve, analogous to the yield urve. The sot urve is the most omlete desrition of the time value of money in finanial markets Forward Rates Ë A forward rate is an interest rate on money to be borrowed between two dates in the future under terms that are agreed to today. Ë Assuming an arbitrage free interest rate market, the relationshi bewteen a forward rate and the sot urve is H1 + s j L j = H1 + s i L i H1 + f i, j L j-i, for i < j If this were not true weould loan money with term j and borrow with term i and borrow at the forward rate (or the reverse) and lok in a risk free rofit. Ë With time in years, the above exression is for annual omounding. For omounding with annual frequeny m H1 + s j ê ml m j = H1 + s i ê ml m i H1 + f i, j ê ml mhj-il, for i < j where the i and j are exressed in years. (Note this is different from the onvention followed in the text in whih the exonents i and j are exressed as omounding eriods). We an use Mathematia to solve for f i, j FullSimlify ASolveAH1 + sj ê ml m j == H1 + si ê ml m i H1 + fijl mhj-il, 8fij<EE lo lo mm -> -1 + oofij i i m + si -i m y i m + sj j y m 1 ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄ y H-i +jl m o o jj ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄ z j ÄÄÄÄÄÄÄÄ ÄÄÄÄÄÄÄÄÄÄÄÄÄ z z } } kk m { k m { { oo nn ~ ~ 3 - Modelling Interest Rates Models of Term Struture Ë Exetaions Hythesis - assumes that the sot urve (and hene the forward rate surfae) reresents a foreast of future interest rates. Although this is true to some degree, Ë The fat that the sot urve almost always sloes uward uts us in the osition of asserting that rates on average tend to go u, and Ë In ratie, histori analysis of forward rates reveal that they are not very good foreasts.

9 ams-q01-le-02-.nb 9 Ë Liquidity Preferene Hyothesis - assumes that, all else being equal, investors refer to tie u their money for shorter eriods rather than longer ones. This also is true to some degree, but ignores an investor s exetation of the future movement of rates. Ë Market Segmentation Hyothesis - assumes that different maturities of debt servie omletely different market segments. One again, there is a valid oint here, but the resene of arbitrage relationshis often allow a ash flow to be onstruted more than one way, so the fat that there are different market segments is not as strong a statement as it first seems Exetation Dynamis Ë Short rates are the forward rates sanning a single eriod. Using annual omounding for simliity we have Ë The forward rates follow a similar relationshi H1 + s k L k = H1 + r 0 L H1 + r 1 L H1 + r k L H1 + f i, j L j-i = H1 + r i L H1 + r 1 L H1 + r j L, for i < j Ë Invariane Theorem - If interest rates evolve aording to the exetations hyothesis, then (assuming annual omounding) a sum of money invested for n years will grow by H1 + s n L n indeendent of the investment and reinvestment strategy as long as all funds remain fully invested. Argument: We saw from the relationshi between sot and forward rates, that, for examle, investing in a two year zero and investing in a one year note followed by reinvestment in another one year note are equivalent. If we aly this argument indutively we get the first (sot) relationshi above. Given the relationshi between sot and forward rates, we get the seond (forward) relationshi above Fisher-Weil or Quasi-Modified Duration Ë Quasi-Modified Duration - is an extension of modified duration whih using a resent value based on the urrent sot urve alulates the imat of a arallel shift in the urve uon rie. HlL = t K1 + s t + l ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ m t e! d Hl ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ d l = -D Q H0L Ë Immunization using quasi-modified duration is more robust than that using modified duration beause it onsiders the resent struture of the sot urve on hanges in rates. O -m t