Appendix B: Yields and Yield Curves

Transcription

1 Pension Finance By Davi Blake Copyright 006 Davi Blake Appenix B: Yiels an Yiel Curves Bons, with their regular an generally reliable stream of payments, are often consiere to be natural assets for pension funs to invest in. In this appenix we consier some of the key yiel measures on bons. We also examine ifferent types of yiel curve. B.. Current yiel B. YIELDS he simplest measure of the yiel on a bon is the current yiel (or flat yiel, interest yiel, income yiel or running yiel). his is efine as: rc (B.) P where: rc P current yiel clean price. For example, if the clean price of the bon is an the coupon is 8.75, then the current yiel is: 8 75 rc (9 8%) B.. Yiel to maturity he yiel to maturity (or reemption yiel) is the most frequently use measure of the return from holing a bon. It takes into account the pattern of coupon payments, the bon s te to maturity an the capital gain or loss arising over the remaining life of the bon. he yiel to maturity is equivalent to the money-weighte rate of return or the internal rate of return on the bon (see Section A.5.).

2 46 Pension Finance he yiel to maturity () is calculate by solving the following equation: P N tc 8 5 S B S N tc 8 5 S t 0 t B S N tc 8 5 S where: B S (B.) P P AI irty bon price (clean price plus accrue interest since last coupon payment) B par value of bon semi-annual coupon payment yiel to maturity N tc number of ays between current ate t an next coupon payment ate c S number of coupon payments before reemption (if is the number of complete years before reemption, then S if there is an even number of coupon payments before reemption, an S if there is an o number of coupon payments before reemption). Equation (B.) uses to iscount the bon s semi-annual cash flows back to the ate of the next coupon payment (the te in curly brackets ) an then iscounts the present value at that ate back to the current ate (the te in square brackets [ ]) he will be the yiel to maturity

3 Yiels an Yiel Curves 47 if the value so achieve equals the irty price of the bon at ate t. In other wors, is the internal rate of return that equates the present value of the iscounte cash flows on the bon to the current irty price of the bon (if ate t is the current ate). he solution for cannot be foun analytically an has to be foun through numerical iteration using a computer or a programmable calculator. o illustrate, suppose a bon has a irty price of 96.50, an annual coupon payment of 8.75, an there is exactly one year before maturity. In this case, (B.) becomes: Since this is a quaratic equation there will be two solutions, only one of which gives a positive. he positive solution is 58%. B. YIELD CURVES In this section we shall examine the relationship between various yiel measures an bons that have ifferent maturities but are otherwise similar. he relationship between a particular yiel measure an a bon s maturity is calle the yiel curve (or te structure of interest rates) for that particular yiel measure. o construct a yiel curve correctly, only bons from a homogeneous group shoul be inclue: for example, only bons from the same risk class or with the same egree of liquiity. We woul therefore not expect a yiel curve to be constructe using both government an corporate bons, since these woul be from ifferent risk classes. We consier the following types of yiel curve: the yiel to maturity yiel curve, the coupon yiel curve, the par yiel curve, the spot yiel curve, the forwar yiel curve, the annuity yiel curve an the rolling yiel curve. B.. he yiel to maturity yiel curve he most familiar yiel curve is the yiel to maturity (YM) yiel curve. his is a plot of the yiel to maturity (erive from (B.) above) against te to maturity for a group of homogeneous bons. hree ifferent shapes of yiel curve are shown in Figure B.. here are several problems with the YM yiel curve. Implicit in the efinition of YM is the assumption that coupon payments are reinveste

4 48 Pension Finance Rising Yiel Humpe Falling e to maturity Figure B. Yiel to maturity yiel curves at the YM. As market rates of interest vary over time, it becomes ifficult to achieve this, a feature known as reinvestment risk. he only type of bon evoi of reinvestment risk is a zero-coupon or pure iscount bon. Another problem is that the YM yiel curve oes not istinguish between the ifferent payment patterns of low-coupon bons an highcoupon bons with the same maturity. With the latter, the payments are concentrate in the early years of their lives, while with the foer, they are concentrate in the later years. Yet this is not taken into account in the YM curve, which assumes a flat payments pattern. In other wors, the cash payments on the bon are not iscounte at the appropriate interest rate. For reasons such as these, bon analysts have evise a number of other types of yiel curve. B.. he coupon yiel curve he coupon yiel curve is a plot of the yiel to maturity against te to maturity for a group of bons with the same coupon. A typical set of coupon yiel curves is presente in Figure B., inicating that highcoupon bons trae at a iscount (have higher yiels) relative to lowcoupon bons, because of reinvestment risk an tax reasons. here is a chance that interest rates will fall uring the life of the bon an this reuces the reinvestment return from reinvesting the coupon payments

5 Yiels an Yiel Curves 49 % % 0% Yiel 9% e to maturity Figure B. Coupon yiel curves (which is a greater risk for high-coupon compare with low-coupon bons) an high-rate taxpayers prefer to have a return in the fo of capital gains rather than coupon income, since capital gains tax can be eferre (whereas income tax cannot). It is clear that yiel can vary quite consierably with coupon for the same te to maturity, an with te to maturity for ifferent coupons. In other wors, ifferent coupon curves not only have significantly ifferent levels, but may also have significantly ifferent shapes. herefore, the kins of istortion that can arise in the YM curve if no allowance is mae for coupon are obvious. As an alternative to the two-imensional representation epicte in Figure B., we can construct a three-imensional yiel plane of coupon against yiel to maturity against te to maturity (see Figure B.3). B..3 he par (or swap) yiel curve he par (or swap) yiel curve is a plot of the yiel to maturity against te to maturity for bons price at par. he par yiel is therefore equal to the coupon rate for bons price at or near par (since the YM for bons price at par is equal to the coupon rate). he par yiel curve is use to eteine the require coupon on a new bon that is to be issue at par.

6 430 Pension Finance Coupon Yiel e to maturity Figure B.3 Yiel plane Suppose that the current par yiels on bons that will mature in one, two an three years time are given respectively by 0, 0.5 an 0.75%. his suggests, for example, that a new two-year bon issue at par woul have to have a coupon of 0.5% an that for a three-year bon with annual coupons traing at par the following equality hols: ( 075) ( 075) 3 emonstrating that the YM an the coupon are ientical when a bon is traing at par. B..4 he spot (or zero-coupon) yiel curve he spot (or zero-coupon) yiel curve is a plot of spot yiels (or zerocoupon yiels) against te to maturity. Spot yiels satisfy the following equation (assuming annual coupons an the calculation is mae on a coupon payment ate so that AI 0): P t t B ( rs t ) t ( rs ) D t B D (B.3)

7 Yiels an Yiel Curves 43 where: rs t the spot or zero-coupon yiel on a bon with t years to maturity D t ( rs t ) t the corresponing iscount factor. In (B.3), rs is the current one-year spot yiel, rs, the current two-year spot yiel, an so on. he spot yiel for a particular te to maturity is the same as the yiel on a zero-coupon bon of the same maturity (hence the alternative name). he spot yiels can be erive from the par yiels as follows. he erivation is base on the interpretation of a bon as a composition of an annuity (which provies the coupon stream) an a zero-coupon bon (which provies the principal repayment) represente by (using (B.3) with P B 00 an 00 rp ): rp t D t 00 D 00 rp A 00 D (B.4) where rp is the par yiel for a te to maturity of years, the iscount factor D is the fair price of a zero-coupon bon with par value of an a te to maturity of years, an where: A t D t A D (B.5) is the fair price of an annuity of per year for years (with A 0 0 by convention). Substituting (B.5) into (B.4) an rearranging gives the expression: rp D A (B.6) rp for the -year iscount factor. For one-year, two-year an three-year par yiels given by 0, 0.5 an 0.75%, respectively, we get the following solutions for the iscount factors: D (0 05)(0 909) D (0 075)( ) D

8 43 Pension Finance It is easy to verify that these are the correct iscount factors. Substituting them back into (B.4), we get, respectively, for the one-year, two-year an three-year par value bons: 00 0 (0 909) (0 909) 0 5 (0 85) (0 909) 0 75 (0 85) 0 75 (0 7349) Now that we have foun the correct iscount factors, it is easy to calculate the spot yiels. From Equation (B.3): D ( rs ) implying rs 0 0% D ( rs ) 0 85 implying rs 0 6% D 3 ( rs 3 ) implying rs 3 0 8% An alternative proceure for calculating the spot yiels is to equate Equations (B.3) an (B.4) for each an solve for the unknown spot yiel rs. For example, when (an given that rp rs 0% an rp 0 5%), we have: ( 05) ( 0) 0 5 ( rs ) which solves for rs 0 6%. Similarly for 3 (an given that rp %), we have: ( 075) ( 075) ( 075) ( 0) ( 06) ( rs 3 ) 3 which solves for rs 3 0 8%. In (B.3) we are iscounting the t-year cash flow (coupon payment an/or principal repayment) by the corresponing t-year spot yiel. In other wors, rs t is the time-weighte rate of return on a t-year bon (see Section A.5.3). hus, the spot yiel curve is the correct metho for pricing or valuing any cash flow (whether regular or irregular) because it uses the appropriate iscount factors. his contrasts with the YM proceure, shown in (B.), in which all cash flows are iscounte by the same yiel to maturity.

9 B..5 he forwar yiel curve Yiels an Yiel Curves 433 he forwar (or forwar-forwar) yiel curve is a plot of forwar yiels against te to maturity. Forwar yiels satisfy: P ( 0rf ) ( 0rf )( rf ) B ( 0rf ) ( rf ) B where: t t i ( i rf i ) i ( i rf i ) (B.7) i rf i implicit forwar rate (or forwar-forwar rate) on a one-year bon maturing in year i. Comparing (B.3) an (B.7), we can see that the spot yiel is the geometric mean of the forwar yiels: his implies that: ( rs t ) t ( 0rf )( rf ) ( t rf t ) (B.8) ( t rf t ) ( rs t ) t ( rs t ) t D t D t (B.9) For the spot yiels given above, we can erive the implie forwar yiels from (B.9): 0 rf 0%, rf 0 53% an rf 3 9%. his means, for example, that, given the current spot yiels, the market is expecting the yiel on a one-year bon maturing in three years time to be.9%. he relationship between the par yiels, spot yiels an forwar yiels is given in able B.. his relationship is also shown in Figure B.4 (in the case of rising yiel curves) an Figure B.5 (in the case of falling yiel curves). able B. he Relationship Between Par Yiels, Spot Yiels an Forwar Yiels Year Par yiel (%) Spot yiel (%) Forwar yiel (%)

10 434 Pension Finance Forwar Spot Yiel Par e to maturity Figure B.4 Rising par, spot an forwar yiel curves he relationship between par yiels an spot yiels can be shown using the following example. Suppose that a two-year bon with cash flows of 0.5 at the en of year an 0.5 at the en of year is traing at par (i.e. has a par yiel of 0.5%) an hence a teinal value Yiel Par Spot Forwar e to maturity Figure B.5 Falling par, spot an forwar yiel curves

11 Yiels an Yiel Curves 435 of: 0 5 ( 053) where the first year s coupon payment (0.5) is investe at the one-year forwar rate for year (0.53%). o be regare as equivalent to this, a pure iscount bon (making a lump sum payment at the en of year with no year payment) woul require a rate of return of 0.6% (the spot yiel), i.e. for the same investment of 00, the maturity value woul have to be: 00 ( 06) 57 As another example, if we know the spot yiels, then we can calculate the coupon require on a new bon if it is to be issue at par using: 00 ( 0) 00 ( 06) ( 08) 3 his gives 0 75, the same as the three-year par yiel. he relationship between spot yiels an forwar yiels is shown in (B.8). If the spot yiel is the average return, then the forwar yiel can be interprete as the marginal return. If the marginal return between years an 3 increases from 0.53 to.9%, then the average return increases from 0.6% to ( 06) ( 9) (0 8%) he relationship between forwar yiels an par yiels can be explaine as follows. Suppose a three-year bon pays coupons equal to the corresponing forwar rates; such a bon is similar to a floating-rate note in the sense that the current forwar rates are the market s best expectation of what the future spot rates will be. his bon will trae at par, since (iscounting using spot rates): ( 0) ( 06) ( 08) 3 A corresponing fixe-income bon, which also traes at par, will pay a fixe annual coupon equal to the three-year par yiel, since (again iscounting using spot yiels): ( 0) 0 75 ( 06) 0 75 ( 08) 3

12 436 Pension Finance So the par yiel is the constant or flat yiel corresponing to a given set of forwar yiels. B..6 he annuity yiel curve he annuity yiel curve is a plot of annuity yiels against te to maturity. An annuity yiel is the implie yiel on an annuity where the annuity is value using spot yiels. In (B.4) above, we ecompose a bon into an annuity an a pure iscount bon. We use the spot yiel to price the iscount bon component. Now we are concerne with the pure annuity (or pure coupon) component. he value of the annuity component of a bon is given by: A t t ( rs t ) t D t (B.0) where rs t an D t are efine in (B.3) an A is efine in (B.5). But A, the fair price of an annuity of per year for years, is given by the stanar foula (see Equation (A.8)): A ra A ( ra ) (B.) where ra is the annuity yiel on a -year annuity. Suppose that we have a three-year bon with annual coupon payments of he value of the annuity component is given by (B.0): 0 75 A 3 ( 0) ( 06) 0 75 ( 08) 3 his implies that A 3 47 (i.e. 6.5/0.75). Solving for ra 3 in (B.) gives a three-year annuity yiel of ra %. he relationship between the spot an annuity yiel curves is shown in Figure B.6. With an upwar-sloping spot yiel curve, the annuity yiel is below the en-of-perio spot yiel ( ); with a falling spot yiel curve, the annuity yiel curve lies above it.

13 Yiels an Yiel Curves 437 Spot Yiel Annuity e to maturity Figure B.6 Spot an annuity yiel curves B..7 Rolling yiel curve he rolling yiel curve is a plot of rolling yiels against te to maturity (see Figure B.7). he one-year rolling yiel is the yiel on a bon when the holing perio is one year but the prices of bons are assume to remain constant Yiel e to maturity Figure B.7 Rolling yiel curve (one-year horizon)

14 438 Pension Finance uring the year. For example, an investor coul buy a ten-year bon, hol it for one year an receive the coupon, an then sell it for the current price of a nine-year bon with the same coupon. he rate of return on this investment woul be the one-year rolling yiel for a te to maturity of ten years. he one-year rolling yiel is given by: where: rr P P (B.) rr one-year rolling yiel on a -year bon P irty price on a -year bon P irty price on a --year bon with the same coupon (). For example, suppose that the price of a ten-year 0% bon is 07.5 an the price of a nine-year 0% bon is he one-year rolling yiel on a ten-year bon is: rr (8 86%)