Introduction to Money & Banking Lecture notes 3/2012. Matti Estola
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1 Intoduction to Mone & Banking Lectue notes 3/22 Matti Estola Inteest and pesent value calculation Tansfoation equations between inteest ates
2 Inteest calculation Inteest ate (/t is the ate of etun of an investent: [ ( t ( t ] ( t ( t ( t ( t [ t t whee (t ( is the initial capital (t ( the ending tie oent capital and t = t-t the length of the investent peiod The easueent unit of inteest ate is thus /t = t - because t is easued in tie units; /t is the evenue flow with unit /t which is divided b the initial capital (t (. Eaple: X(t = ( X(t= ( t = (ea =. (/ea. ]
3 Intepetation of the inteest foula The following definition fo inteest ate is used in calculating the ate of etun of a non-inteest beaing asset like T-bill. [ ( t ( t ] ( t ( t ( t [ t ( t t Suppose ou pa 96 ( (= (t of a thee onth T-bill with noinal value of (. Then setting t = 3 (n we get = (-96/(3 96 =.39 (/n and setting t = /4 ( we get =.67 (/ = 6.7 (%/. The fist defines the aveage onthl ate of etun obtained duing the holding peiod thee onths of the T-bill and the latte defines the ate of etun of the T-bill in the case this investent could be epeated consecutivel 4 ties in a ea. Thus the latte ate of etun akes the investent in the T-bill copaable with annual investents without noticing the copound inteest calculation. Assuing (t = 96 = 39 (/n (t 3n = 96( (+.39 (/n 3(n=( and (t =96( (+.67(/(=2(. Thus we get =[(2-96( /]/96( =.67(/ b ounding soe decials. ].
4 Pesent value calculation in discete tie Let tie be divided in eas and suppose a 3 ea bond is bought in the beginning of its issuing. The bond gives 3 % annual coupon ate at the end of eve ea and pas back the noinal value at atuit. The inteest ate in the econo is denoted b (/t t =. The pesent value of its evenues is: P V (.3(/ t t ( t (.3(/ t t 2 ( t (.3(/ t t 3 ( t ( ( t 3 ( Notice that the easueent unit of P V is. We can test that if =.3 (/t then P V = ( and if <.3 (/t then P V > ( and vice vesa. Fo eaple if =.2 (/ then P V = ( and if =.4 (/ P V = ( We can intepet P V as the highest pice that can be paid of the bond so that it gives the sae intenal ate of etun as the aket inteest ate. On the othe hand if P V is known (fo eaple the aket pice of this bond we can solve fo foula (. In that case easues the intenal ate of etun of this bond. Let P V = 94(. Then we get fo ( b nueical ethods 3 solutions fo fo which we choose the eal (not iagina nube fo the ate of etun as: =.52 (/.
5 We can change the onthl inteest ate to the coesponding annual one b assuing that the sae elative change is eaned eithe in units of annual ate of etun o as a ontl ate duing 2 onths; i efes to the coesponding tie unit. ( 2 /( (/ ( (/ ( (/ ( ( 2 (/ (2 ( 2. Siilal sei-annual inteest ate can be obtained fo the annual one as follows: (/ ( (/ 2 ((/ 2 (2(/ 2 2 (/ 2 ((/ 2 ((/ 2 2 (/ 2 (/ 2 2 (/ 2 (/ 2. In these cases we have not ade the copound inteest coection.
6 In the above we applied the changes in tie units accoding to the tansfoation between two easueent units as is coon in phsics. Fo eaple we can tansfo tie and length units as follows: 2( 2(2 n 2 2 ( n 24 ( n ( k ( ( (. The above shows that quantit 2 (eas consists of the nube of easueent (2 ultiplied b the unit of easueent (ea. We can thus take the nube 2 in the easueent unit in font of the unit and ultipl the two nubes 2 and 2. We can siilal wite: 2 ($ / 3( kg 4 ($ / h 2($ / h 5 ( / h 5 ($ / kg ($ 3 ($ 5($ / h These eaples show that in ultipling and dividing two quantities the nubes of easueent ae ultiplied (divided and so ae the easueent units.
7 Copound inteest calculation in discete tie Let ( be the initial capital and ( the capital afte one ea. With copound calculation of the two ealie defined inteest ates and we get: ( 2 (. Solving / fo both equations and setting the equal we get the tansfoation equation between the two inteest ates as: ( 2 fo which we can solve: 2 ( o ( /2 whee quantities i i i = ae diensionless quantities i.e. quantities without a easueent unit that is pue nubes.
8 Eaple: 2.62 ( ( 2.75 ( ( 2.(.949 ( 2.68(.268 ( (.268. ( 2.68(. ( ( 2(.2 ( ( ( (/.2 (/ / e e e e n
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