Investments03

Investments03 - Time value of money FV PV annuities...

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Unformatted text preview: Time value of money. FV, PV, annuities, perpetuities Professor Pierre-Olivier Weill 1 Last time !"#$#%"$&'"#()*+,-#)(. / 01-"*'%$(1'2&34'()*+%)+*- !"#5"#6'3+)')1-'7*"%-'32'$#'$((-) / 0",-89$&+-8328,3#-:'$5;+(),-#) / <"(='$5;+(),-#) / >7)"3#'$5;+(),-#) 2 Outline 0",-'9$&+-'32',3#-: !+)+*-'?$&+-'@!?AB'C*-(-#)'?$&[email protected]?AB':"-&5 D"#6&-'7$:,-#)'(-%+*"): / EF$,7&-.'G-*3'%3+73#'H3#5 I+&)"7&-'7$:,-#)'(-%+*"): / J##+")"-('$#5'7-*7-)+")"-( 3 Future Value (FV) 7% t=0 PV=$1000 t=1 FV=$1000(1.07) =1070 =1000 principal $70 interest t=2 FV=$1000x(1.07)^2 =1144.90 =1000 principal 70 interest yr 1 70 interest yr 2 4.90 interest over interest 4 time approximate calculation = 1000(1+2x0.07) ("simple interest")=1140...pretty close when interest is small or investment period is short suppose t=100 yrs FV=1000 x (1.07)^100 = 867,716.32 simple interest = 1000 x (1+100 x 0.07) = 8,000 Future Value (FV) 0",-&"#-' K#9-()"#6'23*'$'("#6&-'7-*"35 / FV = PV (1+r) K#9-()"#6'23*',+&)"7&-'7-*"35( / L3,73+#5"#6.'"#)-*-()'3#'"#)-*-() / FV = PV (1+r) t / (1+r)t is the future value factor / D",7&-'"#)-*-() 5 7% Present Value (PV) time t=0 t=1 FV=100 0 t=2 one year PV(1.07) = 1000 => PV = 1000/1.07 = 934.55 two year PV(1.07)^2 = 1000 => PV = 1000/ (1.07)^2 = 873.44 6 Present Value (PV) !"#$%&'(&)"*+,-./&0 1 2*3,45-6,+*,7*5,#&&+,8*,"#9&/8,8*+.7:,3"86, .#,"#8&)&/8,).8&,*;,r, 8*,6.9&,<,FV #&=8,7&.)> 1 PV= FV /(1+r) ?5%8"(%&'(&)"*+,-./&0 1 PV= FV /(1+r) t @"/-*5#8,;.-8*),AB,()&/&#8,9.%5&,;.-8*)C0 1 1/(1+r) t 7 An example of pricing by arbitrage D6.8,"/,86&,()"-&,*;,.,/&-5)"870, 1 86.8,(.7/,<EFFF,"#,83*,7&.) 1 36&#,)BGH price = 873.44 = 1000/(1.07)^2 suppose a broker is willing to see the security at $500 borrow $500, buy the security (-)$500. In two years...security pays $1000 owe to bank $500(1.07)^2 = $572.45 make a profit of 427.55 because arbitrage, profit = 0, since all investment opportnities are equal 8 PV, FV, r, t are tied together. If you know 3, you can find the last one D6.8,"/,86&,7"&%+:,r > 1 !5((*/&,7*5,I#*3,86&,()&/&#8,9.%5&:,.#+,86&,, A;585)&C,9.%5&,.8,.,$"9&#,;585)&,8"4&0 D6.8,"/,86&,7"&%+,A*),86&,)&85)#C,> 2*3,4.#7,(&)"*+/:,t > 1 !5((*/&,7*5,6.9&,.,<EFF:,86.8,86&,"#8&)&/8, ).8&,"/,JH,(&),7&.)K, 1 2*3,%*#$,+*&/,"8,8.I&,8*,L&,.,4"%%"*#.")&,> 9 How long does it take to be a millionaire? what is r if i know PV, FV, and t PV(1+r)^t = FV r=(FV/PV)^(1/t) -1 example: (1000/873.44)^(1/2) - 1 = .07 what is t if i know PV, FV and r? t=log(FV/PV)/log(1+r) example: PV = 100, FV = 1,000,000, r=.05 10 t=log(1,000,000/100)/log(1+.05) = Zero Coupon Bond M1-*-'53'G-*3'%3+73#'H3#5('%3,-'2*3,N / K((+-5'"#'7*",$*:',$*=-)([email protected]*-$(+*:'H3#5(A / D)*"77"#6'32'%3+73#'H3#5(' @-F$,7&-'32'O2"#$#%"$&'-#6"#--*"#6PA' no interest payments, only principal inverse relationship between price and interest rate price more sensitive to interest rate with longer investment period C+&&')3'7$*Q R34-*'"#)-*-()'*$)-(' S"61-*'H3#5'7*"%-(Q 11 Zero Coupon Bond: a numerical example (Handout 1) _-$*(')3'I$)+*"): U YU^ X^ YUUU YUUU Y W T YU [U[QU[ XTYQWY \ZUQ[Z WVTQT] [W]QTV VY\QWU XYZQ[[ TUVQWT 12 Multiple Payments $70 t=0 t=1 $1070 time t=2 Future Value r=.07 future value C1 C2 t=1 70 t=2 70 X 1-.07 = 1070 = FV(at t=2) 74.90 1070 1144.90 Present Value PV = 70/1.07 + 1070/(1.07)^2 = 1000 13 Multiple Payments 0",-&"#!+)+*-'9$&+-'32'()*-$,'32'%$(1'2&34('L(0),C(1),...,C(T) : FV (T ) C(0)(1 r )T C*-(-#)'9$&+-'. C(1)(1 r )T 1 ... C (T ) PV C(0) C(1) 1 1 ... C (T ) (1 r ) (1 r )T 14 _"-&5.'K#)-*#$&'<$)-'32'<-)[email protected]"(%+((-5'"#'&$)-*'%&$((A Annuities PV = C/(1+r) + C/(1+r)^2 +...+ C/(1+r)^t = C/r [1-1/(1+r)^t] ex. I can make payemnts of $632/month 48 months starting next bank says r=1% per month PV = 632/.01 [1-1/(1+.01)^48] = 24,000 how much interest do i end up paying my banker 632 x 48 = 30,336 15 Annuities J##+")"-(. / C$:('$'2"F-5'%$(1'2&34B'CB'23*'T 7-*"35(B'starting next / !3*,+&$. PV = C ( 1 1/(1+r) T ) / r EF$,7&-.'M1"%1'%$*'%$#':3+'$223*5N / _3+'1$9-'#3'&$*6-'$,3+#)'32'%$(1Q / _3+'%$#'$223*5'`\WZ'7-*',3#)1Q / _3+'%$#'H3**34'$)'$#'"#)-*-()'*$)-'32'Y^'7-*',3#)1Q / _3+'4$#)')3'1$9-'7$"5')1-'&3$#'"#'2+&&'"#']V',3#)1Q 16 Perpetuities C-*7-)+")"-( / a-2"#")"3#. C$:('$'2"F-5'%$(1'2&34B'CB'-9-*:'7-*"35'23*-9-* / EF$,7&-.'%3#(3&'H3#5 / C*"%"#6.' PV = C / r EF$,7&-. / D+773(-')1$)',$"#)-#$#%-'32':3+*'6*$9-'%3()('`YUU' -9-*:':-$*B'23*-9-*Q / 01-'"#)-*-()'*$)-'"('T^'7-*':-$*Q / S34',+%1',3#-:'(13+&5':3+'&-$9-')1-')*+()--'32':3+*' 6*$9-N 17 Relationship between interest rate and price !3*'7-*7-)+")"-(N !3*'3)1-*'H3#5(N !3*'()3%=(N a3-('b-#'b-*#$#=- ,$))-*'23*'7*"%-(N 18 For next class <-$5"#6(.'TQZ8TQTB'SW L3#%-7)'c+-()"3#('3#')1-'4-H'(")- 19 ...
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This note was uploaded on 02/04/2010 for the course ECON 106v106xv taught by Professor Weill during the Spring '10 term at UCLA.

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