Economcs 435 The Fnancal Sysem (2/13/13) Insrucor: Prof. Menze Chnn UW Madson Sprng 2013
Fuure Value and Presen Value If he presen value s $100 and he neres rae s 5%, hen he fuure value one year from now s: $100 $100(0.05) 05) = $105 Ths also shows ha he hgher he neres rae, he hgher he fuure value. In general: FV = PV PV() = PV(1 ) And: FV PV = ( 1 4-2 )
Fuure Value and Compound Ineres Wha f you leave your $100 n he bank for wo years a 5% yearly neres rae? The fuure value s: In general $100 $100(0.05) 05) $100(0.05) 05) $5(0.05) 05) = $110.25 $100(1.05)(1.05) = $100(1.05) 2 FV n = PV(1 ) ) n PV = FV ( 1 ) n 4-3
Complcaons Wha f paymens, X, occur all he way along unl he end? unl he end? Wha f he neres rae,, s no consan? = ) )...(1 ) (1 (1... ) ) (1 (1 ) 1 ( 1 1 1 2 1 n n X X X PV Bu a me, one doesn know n nformaon so: nformaon so: = ) )...(1 ) (1 (1... ) ) (1 (1 ) (1 X 1 1 1 2 1 n n X X PV ε ε ε correced
Bond Bascs The mos common ype of bond s a coupon bond. 4-5 Issuer s requred o make annual paymens, called coupon paymens. The annual neres he borrower pays ( c ), s he coupon rae. The dae on whch he paymens sop and he loan s repad (n), s he maury dae or erm o maury. The fnal paymen s he prncpal, face value, or par value of he bond.
Bond Prces 1. Zero-coupon or dscoun bond Promse a sngle paymen on a fuure dae Example: Treasury bll 2. Fxed-paymen loan Sequence of fxed paymens Example: Morgage or car loan 3. Coupon bond perodc neres paymens prncpal repaymen a maury Example: U.S. Treasury Bonds and mos corporae bonds 4Consol 4. perodc neres paymens forever, prncpal never repad Example: U.K. governmen has some ousandng 6-6
Zero-Coupon Bonds U.S. Treasury blls (T-blls) are he mos sraghforward ype of bond. Each T-bll represens a promse by he U.S. governmen o pay $100 on a fxed fuure dae. No coupon paymens - zero-coupon bonds Also called pure dscoun bonds (or dscoun bonds) snce he prce s less han face value - hey sell a a dscoun. Prce of $100 face value zero-coupon bond 6-7 = $100 n (1 )
Zero-Coupon Bonds Assume = 5% Prce of a One-Year Treasury Bll 100 = = $95. 24 ( 1 0.05) Prce of a Sx-Monh Treasury Bll = 100 = $97.59 1/ 2 (1 0.05) 6-8
Zero-Coupon Bonds For a zero-coupon bond, he relaonshp beween he prce and he neres rae s he same as we saw on presen value calculaons. When he prce moves, he neres rae moves wh, n he oppose drecon. We can compue he neres rae from he prce usng he presen value formula. The prce of a one-year T-bll s $95. = ($100/$95) - 1 = 0.0526 = 5.26% 6-9
Fxed-Paymen Loans Home morgages and car loans are fxed-paymen loans. They promse a fxed number of equal paymens a regular nervals. Amorzed loans - he borrower pays off par of he prncpal along wh he neres for he lfe of he loan. Value of a Fxed Paymen Loan = FxedPaymen FxedPaymen L FxedPaymen (1 ) (1 ) 2 (1 ) n The sum of he presen value of he paymens. 6-10
Coupon Bonds The ssuer of a coupon bond promses o make a seres of perodc neres paymens (coupon paymens), plus a prncpal paymen a maury. Prce of Coupon Bond = P CB CouponPaymen CouponPaymen CouponPaymen... 1 2 (1 ) (1 ) (1 ) = n 6-11 FaceValue n ( 1 )
Consols Consols or perpeues, are lke coupon bonds whose paymens py las forever. The borrower pays only neres, never repayng py he prncpal. p The U.S. governmen sold consols once n 1900, bu he Treasury has bough hem all back. The prce of a consol s he presen value of all fuure neres paymens. Yearly Coupon Paymen P Consol = 6-12
Bond Yelds We know how o calculae bond prces gven an neres rae. We also need o be able o go n he oher drecon. Calculae he reurn o an nvesmen, mplc n he bond s prce. We wll combne nformaon abou he promsed paymens wh he prce o oban he yeld: ld A measure of he cos of borrowng and he reward for lendng. We wll use he erms yeld and neres rae nerchangeably. 6-13
Yeld o Maury The mos useful measure of he reurn on holdng a bond s called he yeld o maury: The yeld bondholders receve f hey hold he bond o s maury when he fnal prncpal p paymen s made. $5 $100 Prce of 1yr 5% Coupon Bond = (1 ) (1 ) 6-14 The value of ha solves he equaon s he yeld o maury.
Curren Yeld Example: 1 year, 5% coupon bond sellng for $99 5 Curren Yeld = = 0.0505, or 5.05% 99 Yeld o maury for hs bond s 6.06 percen found as he soluon o: 6-15 $5 $100 (1 ) (1 ) = $99
Holdng Perod Reurns The one-year holdng gperod reurn s he sum of he yearly coupon paymen dvded by he prce pad for he bond and he change n he prce dvded by he prce pad. = Yearly Coupon Paymen Prce Pad Change n Prce Prce of he of he Bond Bond 6-16 = Curren Yeld Capal Gan (as a %)
Daa on Treasury Noes and Bonds hp://onlne.wsj.com/mdc/publc/page/2_3020-reasury.hml
Daa on Treasury Blls On he run
Daa on Treasurys hp://fnance.yahoo.com/bonds h /b See also: hp://www.reasury.gov/resource-cener/daa-char-cener/neres-raes/pages/texvew.aspx?daa=yeld
Secondary Marke, Consan Maury 7 6 10 year consan maury Treasurys 5 4 3 2 1 0 Three monh Treasurys, sec. mk. 00 01 02 03 04 05 06 07 08 09 10 11 12 2/8
Real and Nomnal Ineres Raes The nomnal neres rae you agree on () mus be based on expeced nflaon (π e ) over he erm of he loan plus he real neres rae you agree on (r). = r π e Ths s called he Fsher Equaon. The hgher expeced nflaon, he hgher he nomnal neres rae. 4-21
Daa on Treasury Inflaon Proeced Secures (TIPS) hp://onlne.wsj.com/mdc/publc/page/2_3020-ps.hml
Nomnal vs. Real 7 6 Ten yeartreasurys, consan maury 5 4 3 2 1 0 Ten year adjused by exp'd nfl Ten year TIPS, consan maury -1 00 01 02 03 04 05 06 07 08 09 10 11 12 13
Consan Maury vs. On he Run 2.0 1.5 1.0 0.5 Ten year TIPS, consan maury 0.0-0.5 10 yr 1-1/81/8 TIPS, Jan. 2021-1.0 2010 2011 2012 9/21
6-25 Facors Tha Shf Bond Supply
Facors Tha Shf Bond Demand 6-26