Investments03

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

## This note was uploaded on 02/04/2010 for the course ECON 106v106xv taught by Professor Weill during the Spring '10 term at UCLA.

Ask a homework question - tutors are online