Chapter 4

# Chapter 4 - Present Value 0 A dollar paid to you one year...

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Unformatted text preview: 8/26/2010 Present Value 0 A dollar paid to you one year from now is less valuable than a dollar paid to you today _ . Why? ‘E *rccimé is; woven mom Wee ﬂ m ammﬂa - A dollar deposited today can earn interest and become \$1 x (1+i) one year from today. i” c.,»'; " \IOU 09y? ﬂea/ﬂea: :13 #135: 2*." WM? if»; 8/26/2010 iim’ex View wt Discounting the Future g; LetifJO Inon’e year \$100X(1+ 0.10)=\$110 Intwoyears \$110X(1+0.10)=\$121 0r10()X(1+0.10)2 Inthree years \$121 X (1 + 0.10): \$133 0r100X(1+0.10)3 r—EXQQT” a”. _ q s fﬂf , at 7142’ Am #nziiu (“new 4‘ 114,} Q» t j . i?\ R t; be" \ e “P! x I Egg} w "A 1(\ a bi‘b J< G M w _ ‘ ‘ .5 _.~ I . u; « ‘I _ l \ m a 5e. 0%: ﬂeck“: 7‘?“ .u t .a.-.mgw,~.;:4.~w urn». - kw“. _ _ _,A _ 7- v“ . WV . We u".}3»m-:Z£-.-sz."x-e'yflqbk.L. w 41" mm "a Simple Present Value e; m . an“ PV = today's (preseﬁéVa CF = future cash ﬂow (payment) “er-“’- [fr- d” i = the interest rate _ : XweCF (1+ir . ' \ NJ; {java e {EM a \$ ?“ 4- 6,. w . a. j . may“; 3 A {’3‘}? a: 3? :5 h — g- ‘ a 55? 51., :‘3. ‘ g": s; I i . a “a a ‘ c 'e‘ ﬂ; i? . J j 9' ' '4} if“: M giwgé “3? 5 "’L 8/26/2010 Time Line CF41 Wit Cannot directly compare payments scheduled in different points in the time line D» \$100 \$100 \$100 \$100 __ Year 0 1 2 n PV 100 1001(1+i) 1001(1 +i)2 1001(‘1 +i)n t . E If.) Instruments 0 Simple Loan w H o Fixed Payment Loan Llamartfagaﬁﬁxm. : . 0 Coupon Bond ~ ' ‘ ‘ 0 Discount Bond \ . "i Li " , ‘ 8/26/2010 to Matu 5M Ghee. . The interest rate that equates the present value of cash ﬂow payments received from a debt instrument with its value today A . , a (-3,. > . !_ , _ r. 9 S f 1 mt .V: - Simple Loan r" v. x; . ‘ ‘ ‘wﬂy’kﬁﬁ Fifty???“ PV = amount borrowed = \$100 CF = cash ﬂow in one year = \$110 1’: = number ofyears = 1 \$110 (I +i)‘ (I + 1') \$100 = \$110 (1+i):\$110 Hts-EA \$100 -% -1 i=0.10=10% tat For simple loans, the simgle interest rate equals the m_________ _._____a _. . \$100: yield to maturity ' “*“ \$7 =7. 29% r: 8/26/2010 Fixed Payment Logan?“ z h \ I -, m a - L. . L}; OJ, “5:: '- r 3: i r \. w: a r", ‘ r l, l ; a t ’ U“ .2} ., k i i? . t ~ , L w a a a i ,v The same cash ﬂow payment every period throughout the life of the loan LV : loan value gore \$3?“ a PP = ﬁxed yearly payment n = number of years until maturity vaﬂ+i2+i3+m+i 1+; (1+i) (1+1) (1+i)” ) . t . If?” a . ., ,7 g a Em" wow ” ’0; pm} «m f 90%“ KO nearer ;, x . ‘ \‘ .+ 1“ ‘15 J2 Pang; ‘r‘m Qty CECE-“(‘2 {3 PM? \ rug A... gr} g‘x ‘ - 1‘ . a a.» Anew-1': 9 Wmeéﬁaé MLW” ﬂue-M .xv‘ g 2. a 3 Using the same strategy used for the ﬁxed-payment loan: P 3 price of coupon bond C = yearly coupon payment F I face value of the bond 71 = years to maturity date P£~(—:—_+%+L_3+.. +L MF— l+l (1+1) (1+1) ,‘x s. winem- gjt’iaa. tangential I L; p: ‘ MA _. w 8/26/2010 Table 1 Yields to Maturity on a 10%- Coupon-Rate Bond Maturing in Ten Years (Face Value = \$1,000) - When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate 0 The price of a coupon bond and the yield to maturity are negatively related - The yield to maturity is greater than the coupon rate when the bond price is below its face value Consol or Pgrpetuity Surname o A bond with no maturity date that does not repay principal but pays fixed coupon payments forever P = C / is PC = price of the consol C = yearly interest payment ic = yield to maturity of the consol can rewrite above equation as this : it 2 C / PC For coupon bonds, this equation gives the current yield, an easy to calculate approximation to the yield to maturity 8/26/2010 Discount Bond For any one year discount bond . F -13.; P I? :31 l : C 5 Pt inverse; 9. F = Face value of the discount bond Pf current price of the discount bond The yield to maturity equals the increase in price over the year divided by the initial price. As with a coupon bond, the yield to maturity is negatively related to the current bond price. '3 -x . \r ,3» f‘nﬂ" a: rm.” .35 g 53}; \. U Q .. 3‘ i“? iJ.,r€E‘gl‘ \i’wl W , J eaglikjerag/ thigh .‘J ‘ 5.; _. 22% \ '5“; Rate of Return {was is; = r2 ‘ , ML“ \gwx, agguwﬂg an inner? wart-ii WE'RE; The payments to the owner plus the change in value A -- t 3(5) expressed as a ﬁaction of the purchase price ‘32:: RH = E + 9—H: “ P: P, R RET = return ﬁom holding the bond from time t to time t+ l R = price of bond at time t 15“+1 : price of the bond at time 2‘ + 1 C = coupon payment E = current yield : in. PI 13 _ . . HIP ‘ = rate of capital gain = g , '. fr: M r ~"' ;wa§:g{}m - ‘ (j 1:. pom 2:: {:é’ﬁb’ri' "t ‘ r: ,, (5:? d: Rate of Re Rates 0 The return equals the yieid to maturity only if the holding period equals the time to maturity - A rise in interest rates is associated with a fa]! in bond prices, resulting in a capital loss if time to maturity is longer than the holding period a The more distant a bond’s maturity, the greater the size of the percentage price change associated with an interest-rate change vx i-‘rOUJ do ibis inn t cliwé‘eif «Hue, ?’ 3% {gm magmas: «1—»...3 ml? . Eyb’ :Ldgﬁiﬂ C‘skhrg Q: tn: d i‘iSQinte "new" its“: P‘f ‘ ‘ ‘1 i , a r“ '3“ j ,|_ Jr :__ \\GQQ\$§¢\ ‘ ’ t;- i‘“ n W— nun? " J L.“ a x“ - ‘ ' ﬂ . __ ‘ , I\ _Pgwod 2ft increases “raj; égng< 22 1 \ Evil 1 1 M: =00 not; Pig: UTE {— “L—‘Fﬁ :5 Girl 5?? " Ni Rate of Return and Interest Rates (cont’d) - The more distant a bond’s maturity, the lower the rate of return the occurs as a result of an increase in the interest rate o Even if a bond has a substantial initial interest rate, its return can be negative if interest rates rise 8/26/2010 I? was r n: 8/26/2010 Table 2 One-Year Returns on Different-Maturity 10%-Coupon-Rate Bonds When Interest Rates Rise from 10% to 20% Interest-Rate Risk 0 Prices and returns for long-term bonds are more voiatiie than those for shorter—term bonds 0 There is no interest-rate risk for any bond whose time to maturity matches the holding peﬁod 8/26/2010 Real and Nominal Interest Rates 0 Nominal interest rate makes no allowance for inflation 0 Real interest rate is adjusted for changes in price level so it more accurately reflects the cost of borrowing - Ex ante real interest rate is adjusted for expected changes in the price level - Ex post real interest rate is adjusted for actual changes in the price level Fisher Equation ) ‘ l a“, n ‘ 5M3 f It , Wk me. it} rel“ lid'lfliﬁilts’f‘an , 3" , g - a ._ . e - d “gap mfg?“ ﬂag-Vi 1—54-77: ire?“ M V is up r‘i‘i‘rew ﬂax??? Elf-r? i6%?\‘)ﬂ \ J v; V: w.“ 5‘ z” = nominal interest rate fr = real interest rate ﬂ'e = expected inﬂation rate When the real interest rate is low, there are greater incentives to borrow and feiver incentives to lend. The real interest rate is a better indicator of the incentives to ‘ml‘lairion borrow and lend. ‘ \1/ xi» Nv‘ﬁn 1r : 2630 ‘3,” Gag bwﬁi‘; QMOUW‘ET i)»; Qinj‘Q—r 53.3 r: 1 in“ L 4921 ‘0 LC; & agar. 10 8/26/2010 FIGURE 1 Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953—2008 interest Rate (as) 16 32 B ﬂaming} Rate \ 4 . 0 —~ ----------- —— ‘ ——————————————————— -— g. -—— \ Estimated Real Rate —¢ 1955 1860 1965 1970 1975 1930 \$85 1390 1995 2000 2005 20?“ Sources: Nominal rates from www.federalreservegovjreleasesiH15. The real rate is constructed using the procedure outlined in Frederic S. Mishkin, “The Real Interest Rate: An Empirical Investigation,"Carnegie- Rachester Conference Series on Public Poficy 15 (1981): 151—200. This procedure involves estimating expected inflation as a function of past interest rates, inflation, and time trends and then subtracting the expected inflation measure from the nominal interest rate. , -— v’a \ V \ w x 1135f, f i a . 2 is ﬁle ‘ 3* ‘s c r .i it=13£f¥§irguﬂqaiﬁwyf ﬁg; “‘3”; W’s" "’ ‘10:}; g 34,“: J3 .sm“ a.an ﬁx!" a" , 1, i» r w Ex; 5 , “a a .3 m1,» 2 w z v 3Q? MM) 11 ...
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Chapter 4 - Present Value 0 A dollar paid to you one year...

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