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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’
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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 HtsEA $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.» Anew1': 9
Wmeéﬁaé MLW” ﬂueM .xv‘ g 2. a 3
Using the same strategy used for the ﬁxedpayment 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%
CouponRate 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? wartii 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 interestrate 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
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 ‘ ' 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 OneYear Returns on DifferentMaturity
10%CouponRate Bonds When Interest Rates
Rise from 10% to 20% InterestRate Risk 0 Prices and returns for longterm bonds are
more voiatiie than those for shorter—term
bonds 0 There is no interestrate 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?“ ﬂagVi
1—5477: ire?“ M V is up
r‘i‘i‘rew ﬂax??? Elfr? 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
(ThreeMonth 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
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 Spring '08
 STAHL
 Economics

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