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Tutorial+16+solution

# Tutorial+16+solution - Quesﬁsn i(a Three year spot...

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Unformatted text preview: Quesﬁsn i (a) Three year spot rate (83) 53 can be calculated using the information provided with regard to forward rates. n+5r=n+54f4x0+030 ' 0+5P=n+5fx0+03 0+53)3=0+o.138)2 xn+0.0989) i 0+53)=U‘49933 (1+s3)=m445 53:14.45% (5) (b) Bond A’s current price using the spot rates: Po ~ R160 R] 60 RI 160 -———-——~+ +——————-— i+0.l76 (”0.16332 {1.5034433 P0 = 5135.05 + R117.28 + 12213.71 = R1, 021.10 ‘ (5) (c) Expected prices of Bond B if YTM increases by 3%: (1) Using modiﬁed duration omenim = 15%, increese by 3% to 18%, so Ay = (0.18 .- 0.15) = 0.03 t at m t; mm; PRC?) “(crux t 1 90 0.8696 28.26 18.25 2 90 0.2551 08.05 136.11 3 90 0.5515 5 55.13 122.52 4 1090 3.5215 623.21 2,492.04 R 828.10 2,884.14 D = 2,884.74 / 828.10 = 3.481 D* = DI (1+ y) = 3.4830 / {1+ 0.15) = 3.027 AP = (- 13*) x (A y) x P0 = (. 3.027) * (0.03) '* (3828.70) =.- 1115.25 Bond price will decrease by R1525 if there is a 3% increase in YTM: Predicted new price = 51828.70 - R7525 = 1353.45 (ii) Duration-with-convexity AP [Po = (—D‘* xAy) + {0.5 x convexity: (A362) AP I P0 = [- 3.027 x (0.03)} + {0.5 x 32.54 x (0.032} -‘- - 0.0852 or — 8.52% AP I Po 243.52% xRSZBJO = —RYO.61 » Expected price at 18% Y‘I‘M is R758.09 (= 828.?!) — 70.61) Duration is a good approximation for small changes in the yield, but it over ‘ (under) estimates price decrease (increase) for a large increase (decrease) in Y’I‘M. (10) (d) Theories of the term. stricture of interest rates Alexander, Sharp and Bailey (1992) CH 21 Theories of the term structure ofinterest rates The yield curve graphs the relationship between the yield to maturity (discount rate that equates the present value of all expected interest payments and the repayment of principal from a bond with the present bond price) and the time to maturity of a bond. , . The Unbiased Matias Theorem Outline: , - forward rate = average expectation of ﬁxture spot rate — change in future rate due to change in (i) real rate or (2) inﬂation - Upward sloping yield curves maturity strategyvs rollover strategy ~investors don’t know the futue spot rate but have expectations demand and supply ensure that e513 = 11,2 - ineqnilibiimexpectedfatnrespotrate=forwazdrate " (1 + 52) x (I ‘1' 52} = (1+5!) 1 (951;) — ExpectedRetumonamaturitystrategy =ExpectedReturnonaRollover Strategy - est.” = 51.1,: so (14%.? 3 (1 + 5.4)“ a: (1+ est”) — Chang‘ug spot rates and inﬂation The ﬁgg‘dig: Menace Theorem Outline: - investors want short-term securities , need extra reward for buying long term - price risk if a long4erm security is sold before maturity - higher risk associated with a maturity strategy, so compensation = higher return - example: a 2-year holding period investor will not choose maturity strategy it it has the same expected return as rollover strategy - only way to choose maturity strategy is if it offers a higher expected return than rollover strategy - thus expected spot rate (e513) must be lower than the forward rate (1'13) - ft—l = est-l + Lu; - the extra return is called the liquidity premium, since lag-1.: > O - genemnvi (1 +SO‘ > (14-52-1)HX(1+eSt-x.z) (30) Quesﬁon 2: Nifty and Gritty (<3) (b) . 821 .96 = Deﬁniﬁons The Mperiod spot rate can be thought of as the interest rate on zero—coupon bond whose fife extends from today through N future pmods. The N—penod forward rate M periods hence can be thought of as the inferesi rate, determined iodgg, on a zero—coupon bond that win come into existence M periods from today and mature N periods after M- [5] The two year spot rate FV (1+52)2 Loco ' 0+”? P: \$2 = spot rate (1 + ”)2 = 1 .000/821.96 32= J .2126 -1=10.3% 52 can also be caicuia’red using the implied forward rate from years I to 2. ‘ [4] The forward rate from {rear four to ﬁve 5 {45){0‘35} )4 (I+s.f s e ”‘5 't “'5: £12.)? 4:143:93 (1-115) [51 Price of the bond if held to maiuribg _ 120 1.120 P- + ____..____ {1+3}? “+52? { 120 1.120 r=———+ ——————' (“may (34-303): P = 109.29 + 92059 = :1 1029.88 (61 (é) Exptaining the difference The reason oouldbethatfhebondmarketisnot meat, and that iswhy the bondis trading below its fair value. However, the bond markets are generally eﬁcient and such unis-pricing will be corrected immediately. A more iikety explanation is that something else causes the difference. The bond has a caii provision, so it might be currently priced to take into consideration the coil provision. The caii will be in i year's time at a price of R] 100 {=R1000 x i107 I if the market expects that the bond wiii be caiied it woutd be priced as toitows: 1220 P: ‘i (“'51) 1.220 P=.__._.__.. (1+.098)‘ P=RI 111.11 é { r . no: . The assumption that the market is correct in pricing the bond on the bass of time- tacait rather than time-to maturity can be tested in two ways. The ﬁrst one is to think what mil happen it the bond is catied. The company Witt need to ﬁnance the purchase, Le. it will need Rt 100 In one year’s time. This amount wit! he need for one year only {assuming the inn has previded for repayment of capitai in two years time). Therefore the company witi need a one—year loan in one year' s time. the current rate for such ioan' IS the forward rate form year one to year two fiz- Ihis rate is given at 10.80% which is considerabiy below the 12% rate the company would be paying it it does not call the bond. An aitemative way to test the assumption whether its reasonabie toprice the bond on the basis of time-to—catt rather than timeto—maturity is to caicuiate the Weid-to-maturity of the bond. The YTM shouid be caicuiated using the “fair price' of the bond associated with holding it untﬂ its maturity date, not caii date. 31,029.33=;_1.2L_+ “20 (“mgr (1”er YTM = 10.27% This rate is considerably beiow the bond '5 coupon rate of 12%, so if is reasonable ’to expect that the bond wit! be called. ...
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