Other things remaining equal, the value of cash flows in future _me periods will decrease as lecture notes for the present value of an annuity will be PV(A,r,n)
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THETIMEVALUEOFMONEY
AswathDamodaran
2Intui ¨ Therearethreereasonswhyadollartomorrowisworthlessthanadollartoday future,thelessthatcashflowwillbevalued. ¨ Otherthingsremainingequal,thevalueofcashflowsinfuture Discoun ¨ Themechanismforfactoringintheseelementsisthediscountrate.Thediscountrateisarateatwhichpresentandfuturecashflowsaretradedoff.Itincorporates
¨ Ahigherdiscountratewillleadtoalowervalueforcashflowsin thefuture. ¨ Thediscountrateisalsoanopportunitycost,sinceitcapturesthe • DiscounPresentValuePrinciple1
¨ Cashflowsatdi fferentpointsin ¨ Themechanismforfactoringintheseelementsisthediscountrate.Thediscountrateisarateatwhichpresentandfuturecashflowsaretradedoff.Itincorporates
¨ Ahigherdiscountratewillleadtoalowervalueforcashflowsin thefuture. ¨ Thediscountrateisalsoanopportunitycost,sinceitcapturesthe • Discoun4 $ 100$ 100$ 100$ 100 Figure 3.1: A Time Line for Cash Flows: $ 100 in Cash Flows Received at the End of Each of Next 4 years
Cash Flows
Year 6CashFlowTypesandDiscoun ¨ Therearefivetypesofcashflows-
• simplecashflows,• annui • MostassetsrepresentcombinaI.SimpleCashFlows
¨ Asimplecashflowisasinglecashflowinaspecifiedfuture¨ Therearefivetypesofcashflows-
• simplecashflows,• annui
¨ Between1926and2013, stocksontheave ragemadeabout9.55%ayear,whilegovernmentbondsonaverage madeabout4.93%aye arandT.Bills earned3.53%ayear.
¨ Ifyourholdingperiodisoneyear,thedifferencein
end-of-periodvaluesissmall: 9HoldingPeriodandValue
10ConceptCheck
¨ Mostpensionplansallowindividualstodecidewheretheirpensionsfundswillbeinvested-stocks,bondsormoneymarketaccounts.
¨ Wherewouldyouchoosetoinvestyourpensionfunds?
a. Predominantlyorallequityb. Predominantlyorallbondsandmoneymarketaccountsc. AMixofBondsandStocks¨ Willyouralloca a. Yesb. No 11 TheFrequencyofCompounding
¨ Thefrequencyofcompoundingaffectsthefutureandpresentvaluesofcashflows.Thestatedinterestratecandeviatesignificantlyfromthetrueinterestrate-
compounding,worksoutto- Effec 2 2 -110.25%Monthly10%12(1+r/12) 12 -110.47%Daily10%365(1+r/365) 365
-110.5156%ConII.Annui ¨ Anannuityisaconstantcashflowthatoccursatregularintervalsforafixedperiodof AAAA ||||01234 13 PresentValueofanAnnuity
¨ Thepresentvalueofanannuitycanbecalculatedbytakingeachcashflowanddiscoun PV of an Annuity = PV(A,r,n) = A
1 - 1 (1+r) n r 14 Example:PVofanAnnuity
¨ Thepresentvalueofanannuityof$1,000attheendofeachyearforthenextfiveyears,assumingadiscountrateof10%is-
¨ Thenota PV of $1000 each year for next 5 years = $1000
1 - 1 (1.10) 5 .10 = $3,791 15 Annuity,givenPresentValue
¨ Thereverseofthisproblem,iswhenthepresentvalueisknownandtheannuityistobees¨ This,forinstance,istheequa Annuity given Present Value = A(PV,r,n) = PV
r 1 - 1 (1+r) n 16 Compu ¨ Supposeyouborrow$200,000tobuyahouseona30-yearmortgagewithmo nthlypayments.Theannualpercentage rateontheloanis8%.Themonthlypaymentsonthisloan,withthepaymentsoccurringattheendofe achmont h,canbecalculatedusingthisequa Monthly Payment on Mortgage = $200,000
0.0067
1 - 1 (1.0067) 360
=$1473.11 17 FutureValueofanAnnuity
¨ Thefuturevalueofanend-of-the-periodannuitycanalsobecalculatedasfollows- ¨ Thisistheequa FV of an Annuity = FV(A,r,n) = A
(1+r) n - 1 r 18 AnExample
¨ Thus,thefuturevalueof$1,000attheendofeachyearforthenextfiveyears,attheendofthefishyearis(assuminga10%discountrate)-
¨ Thenota anannuitywillbeFV(A,r,n). FV of $1,000 each year for next 5 years = $1000
(1.10) 5 - 1 .10 = $6,105 19 Annuity,givenFutureValue
¨ ifyouaregiventhefuturevalueandyouarelookingforanannuity-A(FV,r,n)intermsofnotaAnnuity given Future Value = A(FV,r,n) = FV
r (1+r) n - 1 20 Applica ¨ Assumethatyouwanttosendyournewbornchildtoaprivatecollege(whenhegetstobe18yearsold).Thetui 18 years)=$127,537 ¨ Ifyouneedtosetasidealumpsumnow,theamountyouwould needtosetasidewouldbe- 18 =$31,916 ¨ Ifsetasideasanannuityeachyear,starApplica ¨ Assumethatyouwereearning$40,000/yearbeforeenteringprogramandthattui ¨ Assumethatyouwillwork30yearsasergradua¨ Assumethatyouhaveathirty-yearmortgagefor$200,000thatcarriesaninterestrateof9.00%.Themortgagewastakenthreeyearsago.Sincethen,assumethatinterestrateshavecomedownto7.50%,andthatyouarethinkingofrefinancing.Thecostofrefinancingisexpectedtobe2.50%oftheloan.(Thiscostincludesthepointsontheloan.)Assumealsothatyoucaninvestyourfundsat6%.
¨ MonthlySavingsfromrefinancing=$1,609-$1,398=$211 23
Refinancing:TheTradeOff
¨ Ifyouplantoremaininthishouseindefinitely,
27
=$9,532 ¨ RefinancingCost=2.5%of$200,000=$5,000¨ TotalRefinancingCost=$9,532+$5,000=$14,532¨ NetEffect=$33,815-$14,532=$19,283:Refinance
24
Follow-upQues impactyourdecision? 25
ValuingaStraightBond
¨ Youaretryingtovalueastraightbondwithafiseenyearmaturityanda10.75%couponrate.Thecurrentinterestrateonbondsofthisrisklevelis8.5%.
15 1186.85
¨ Ifinterestratesriseto10%,
15 $1,057.05 Percentagechangeinprice=-10.94%
¨ Ifinterestratefallto7%,
15 $1,341.55 Percentagechangeinprice=+13.03%
¨ Thisasymmetricresponsetointerestratechangesiscalled convexity. 26
BondPricingProposi ¨ Thelongerthematurityofabond,themoresensiPrice Changes as a function of Bond Maturities
Bond Maturity
% Change in Price -15.00% -10.00% -5.00% 0.00% 5.00% 10.00%
15.00%
20.00%
151530
% Change if rate drops to 7% % Change if rate increases to 10% 27
BondPricingProposi ¨ Thelowerthecouponrateonthebond,themoresensiBond Price Changes as a function of Coupon Rates
Coupon Rate
% Price Change -20.00% -15.00% -10.00% -5.00% 0.00% 5.00% 10.00%
15.00%
20.00%
25.00%
0%5%10.75%12%
% Change if rate drops to 7% % Change if rate increases to 10% 28
III.GrowingAnnuity
¨ Agrowingannuityisacashflowgrowingataconstantrateforaspecifiedperiodof 0123
A(1+g)
2 Figure 3.8: A Growing Annuity
A(1+g)
3 A(1+g)
nA(1+g) n 29
PresentValueofaGrowingAnnuity
• Thepresentvalueofagrowingannuitycanbees • Inthatspecificcase,thepresentvalueisequaltothe PV of an Annuity = PV(A,r,g,n) = A(1+g)
1 - (1+g) n (1+r) n (r-g) 30
TheValueofaGoldMine
¨ Considertheexampleofagoldmine,whereyouhavetherightstothemineforthenext20years,overwhichperiodyouplantoextract5,000ouncesofgoldeveryyear.Thepriceperounceis$300currently,butitisexpectedtoincrease3%ayear.Theappropriatediscountrateis10%.Thepresentvalueofthegoldthatwillbeextractedfromthisminecanbees PV of extracted gold = $300*5000*(1.03)
1 - (1.03) 20 (1.10) 20 .10-.03 =$16,145,980 31
Present Value of Extracted Gold as a function of Growth Rate $5,000,000 $10,000,000 $15,000,000 $20,000,000 $25,000,000 $30,000,000 $35,000,000 $40,000,000 $45,000,000 $50,000,000 0%1%2%3%4%5%6%7%8%9%
10%11%12%13%14%15%
Growth Rate in Gold Prices
Present Value of Extracted Gold
32
IV.Perpetuity
¨ Aperpetuityisaconstantcashflowatregularintervalsforever.Thepresentvalueofaperpetuityis- ¨ Forevermaybeatoughconceptforhumanbeings
PV of Perpetuity =
A r 33
ValuingaConsoleBond
¨ Aconsolebondisabondthathasnomaturityandpaysafixedcoupon.Assumethatyouhavea6%couponconsolebond.Thevalueofthisbond,iftheinterestrateis9%,isasfollows-
ValueofConsoleBond=$60/.09=$667
34
V.GrowingPerpetui ¨ Agrowingperpetuityisacashflowthatisexpectedtogrowataconstantrateforever.Thepresentvalueofagrowingperpetuityis-
where 1 PV of Growing Perpetuity =
CF 1 (r-g) 35
ValuingaStockwithGrowingDividends
¨ IntwelvemonthsleadingintoJanuary2014,ConEdpaiddividendspershareof$2.52. ¨ Itsearningsanddividendshadgrownat2%ayear
¨ Therateofreturnrequiredbyinvestorsonstocksof
equivalentriskwas7.50%. ¨ Withtheseinputs,wecanvaluethestockusinga
perpetualgrowthmodel: 36
ValueandGrowth!
quotesdbs_dbs14.pdfusesText_20
TheFrequencyofCompounding
¨ Thefrequencyofcompoundingaffectsthefutureandpresentvaluesofcashflows.Thestatedinterestratecandeviatesignificantlyfromthetrueinterestrate-
compounding,worksoutto-Effec 2 2 -110.25%Monthly10%12(1+r/12) 12 -110.47%Daily10%365(1+r/365) 365
-110.5156%ConII.Annui ¨ Anannuityisaconstantcashflowthatoccursatregularintervalsforafixedperiodof AAAA ||||01234 13 PresentValueofanAnnuity
¨ Thepresentvalueofanannuitycanbecalculatedbytakingeachcashflowanddiscoun PV of an Annuity = PV(A,r,n) = A
1 - 1 (1+r) n r 14 Example:PVofanAnnuity
¨ Thepresentvalueofanannuityof$1,000attheendofeachyearforthenextfiveyears,assumingadiscountrateof10%is-
¨ Thenota PV of $1000 each year for next 5 years = $1000
1 - 1 (1.10) 5 .10 = $3,791 15 Annuity,givenPresentValue
¨ Thereverseofthisproblem,iswhenthepresentvalueisknownandtheannuityistobees¨ This,forinstance,istheequa Annuity given Present Value = A(PV,r,n) = PV
r 1 - 1 (1+r) n 16 Compu ¨ Supposeyouborrow$200,000tobuyahouseona30-yearmortgagewithmo nthlypayments.Theannualpercentage rateontheloanis8%.Themonthlypaymentsonthisloan,withthepaymentsoccurringattheendofe achmont h,canbecalculatedusingthisequa Monthly Payment on Mortgage = $200,000
0.0067
1 - 1 (1.0067) 360
=$1473.11 17 FutureValueofanAnnuity
¨ Thefuturevalueofanend-of-the-periodannuitycanalsobecalculatedasfollows- ¨ Thisistheequa FV of an Annuity = FV(A,r,n) = A
(1+r) n - 1 r 18 AnExample
¨ Thus,thefuturevalueof$1,000attheendofeachyearforthenextfiveyears,attheendofthefishyearis(assuminga10%discountrate)-
¨ Thenota anannuitywillbeFV(A,r,n). FV of $1,000 each year for next 5 years = $1000
(1.10) 5 - 1 .10 = $6,105 19 Annuity,givenFutureValue
¨ ifyouaregiventhefuturevalueandyouarelookingforanannuity-A(FV,r,n)intermsofnotaAnnuity given Future Value = A(FV,r,n) = FV
r (1+r) n - 1 20 Applica ¨ Assumethatyouwanttosendyournewbornchildtoaprivatecollege(whenhegetstobe18yearsold).Thetui 18 years)=$127,537 ¨ Ifyouneedtosetasidealumpsumnow,theamountyouwould needtosetasidewouldbe- 18 =$31,916 ¨ Ifsetasideasanannuityeachyear,starApplica ¨ Assumethatyouwereearning$40,000/yearbeforeenteringprogramandthattui ¨ Assumethatyouwillwork30yearsasergradua¨ Assumethatyouhaveathirty-yearmortgagefor$200,000thatcarriesaninterestrateof9.00%.Themortgagewastakenthreeyearsago.Sincethen,assumethatinterestrateshavecomedownto7.50%,andthatyouarethinkingofrefinancing.Thecostofrefinancingisexpectedtobe2.50%oftheloan.(Thiscostincludesthepointsontheloan.)Assumealsothatyoucaninvestyourfundsat6%.
¨ MonthlySavingsfromrefinancing=$1,609-$1,398=$211 23
Refinancing:TheTradeOff
¨ Ifyouplantoremaininthishouseindefinitely,
27
=$9,532 ¨ RefinancingCost=2.5%of$200,000=$5,000¨ TotalRefinancingCost=$9,532+$5,000=$14,532¨ NetEffect=$33,815-$14,532=$19,283:Refinance
24
Follow-upQues impactyourdecision? 25
ValuingaStraightBond
¨ Youaretryingtovalueastraightbondwithafiseenyearmaturityanda10.75%couponrate.Thecurrentinterestrateonbondsofthisrisklevelis8.5%.
15 1186.85
¨ Ifinterestratesriseto10%,
15 $1,057.05 Percentagechangeinprice=-10.94%
¨ Ifinterestratefallto7%,
15 $1,341.55 Percentagechangeinprice=+13.03%
¨ Thisasymmetricresponsetointerestratechangesiscalled convexity. 26
BondPricingProposi ¨ Thelongerthematurityofabond,themoresensiPrice Changes as a function of Bond Maturities
Bond Maturity
% Change in Price -15.00% -10.00% -5.00% 0.00% 5.00% 10.00%
15.00%
20.00%
151530
% Change if rate drops to 7% % Change if rate increases to 10% 27
BondPricingProposi ¨ Thelowerthecouponrateonthebond,themoresensiBond Price Changes as a function of Coupon Rates
Coupon Rate
% Price Change -20.00% -15.00% -10.00% -5.00% 0.00% 5.00% 10.00%
15.00%
20.00%
25.00%
0%5%10.75%12%
% Change if rate drops to 7% % Change if rate increases to 10% 28
III.GrowingAnnuity
¨ Agrowingannuityisacashflowgrowingataconstantrateforaspecifiedperiodof 0123
A(1+g)
2 Figure 3.8: A Growing Annuity
A(1+g)
3 A(1+g)
nA(1+g) n 29
PresentValueofaGrowingAnnuity
• Thepresentvalueofagrowingannuitycanbees • Inthatspecificcase,thepresentvalueisequaltothe PV of an Annuity = PV(A,r,g,n) = A(1+g)
1 - (1+g) n (1+r) n (r-g) 30
TheValueofaGoldMine
¨ Considertheexampleofagoldmine,whereyouhavetherightstothemineforthenext20years,overwhichperiodyouplantoextract5,000ouncesofgoldeveryyear.Thepriceperounceis$300currently,butitisexpectedtoincrease3%ayear.Theappropriatediscountrateis10%.Thepresentvalueofthegoldthatwillbeextractedfromthisminecanbees PV of extracted gold = $300*5000*(1.03)
1 - (1.03) 20 (1.10) 20 .10-.03 =$16,145,980 31
Present Value of Extracted Gold as a function of Growth Rate $5,000,000 $10,000,000 $15,000,000 $20,000,000 $25,000,000 $30,000,000 $35,000,000 $40,000,000 $45,000,000 $50,000,000 0%1%2%3%4%5%6%7%8%9%
10%11%12%13%14%15%
Growth Rate in Gold Prices
Present Value of Extracted Gold
32
IV.Perpetuity
¨ Aperpetuityisaconstantcashflowatregularintervalsforever.Thepresentvalueofaperpetuityis- ¨ Forevermaybeatoughconceptforhumanbeings
PV of Perpetuity =
A r 33
ValuingaConsoleBond
¨ Aconsolebondisabondthathasnomaturityandpaysafixedcoupon.Assumethatyouhavea6%couponconsolebond.Thevalueofthisbond,iftheinterestrateis9%,isasfollows-
ValueofConsoleBond=$60/.09=$667
34
V.GrowingPerpetui ¨ Agrowingperpetuityisacashflowthatisexpectedtogrowataconstantrateforever.Thepresentvalueofagrowingperpetuityis-
where 1 PV of Growing Perpetuity =
CF 1 (r-g) 35
ValuingaStockwithGrowingDividends
¨ IntwelvemonthsleadingintoJanuary2014,ConEdpaiddividendspershareof$2.52. ¨ Itsearningsanddividendshadgrownat2%ayear
¨ Therateofreturnrequiredbyinvestorsonstocksof
equivalentriskwas7.50%. ¨ Withtheseinputs,wecanvaluethestockusinga
perpetualgrowthmodel: 36
ValueandGrowth!
quotesdbs_dbs14.pdfusesText_20
-110.5156%Con
¨ Anannuityisaconstantcashflowthatoccursatregularintervalsforafixedperiodof ¨ Thepresentvalueofanannuitycanbecalculatedbytakingeachcashflowanddiscoun ¨ Thepresentvalueofanannuityof$1,000attheendofeachyearforthenextfiveyears,assumingadiscountrateof10%is- ¨ Supposeyouborrow$200,000tobuyahouseona30-yearmortgagewithmo nthlypayments.Theannualpercentage rateontheloanis8%.Themonthlypaymentsonthisloan,withthepaymentsoccurringattheendofe achmont h,canbecalculatedusingthisequa ¨ Thus,thefuturevalueof$1,000attheendofeachyearforthenextfiveyears,attheendofthefishyearis(assuminga10%discountrate)- ¨ Assumethatyouwanttosendyournewbornchildtoaprivatecollege(whenhegetstobe18yearsold).Thetui ¨ Assumethatyouwereearning$40,000/yearbeforeenteringprogramandthattui ¨ Assumethatyouhaveathirty-yearmortgagefor$200,000thatcarriesaninterestrateof9.00%.Themortgagewastakenthreeyearsago.Sincethen,assumethatinterestrateshavecomedownto7.50%,andthatyouarethinkingofrefinancing.Thecostofrefinancingisexpectedtobe2.50%oftheloan.(Thiscostincludesthepointsontheloan.)Assumealsothatyoucaninvestyourfundsat6%. ¨ RefinancingCost=2.5%of$200,000=$5,000¨ TotalRefinancingCost=$9,532+$5,000=$14,532¨ NetEffect=$33,815-$14,532=$19,283:Refinance ¨ Youaretryingtovalueastraightbondwithafiseenyearmaturityanda10.75%couponrate.Thecurrentinterestrateonbondsofthisrisklevelis8.5%. ¨ Agrowingannuityisacashflowgrowingataconstantrateforaspecifiedperiodof • Thepresentvalueofagrowingannuitycanbees ¨ Considertheexampleofagoldmine,whereyouhavetherightstothemineforthenext20years,overwhichperiodyouplantoextract5,000ouncesofgoldeveryyear.Thepriceperounceis$300currently,butitisexpectedtoincrease3%ayear.Theappropriatediscountrateis10%.Thepresentvalueofthegoldthatwillbeextractedfromthisminecanbees ¨ Aconsolebondisabondthathasnomaturityandpaysafixedcoupon.Assumethatyouhavea6%couponconsolebond.Thevalueofthisbond,iftheinterestrateis9%,isasfollows- ¨ Agrowingperpetuityisacashflowthatisexpectedtogrowataconstantrateforever.Thepresentvalueofagrowingperpetuityis-PresentValueofanAnnuity
PV of an Annuity = PV(A,r,n) = A
1 - 1 (1+r) n r 14 Example:PVofanAnnuity
¨ Thenota
PV of $1000 each year for next 5 years = $1000
1 - 1 (1.10) 5 .10 = $3,791 15 Annuity,givenPresentValue
¨ Thereverseofthisproblem,iswhenthepresentvalueisknownandtheannuityistobeesAnnuity given Present Value = A(PV,r,n) = PV
r 1 - 1 (1+r) n 16 Compu
Monthly Payment on Mortgage = $200,000
0.0067
1 - 1 (1.0067) 360
=$1473.11 17 FutureValueofanAnnuity
¨ Thefuturevalueofanend-of-the-periodannuitycanalsobecalculatedasfollows- ¨ Thisistheequa
FV of an Annuity = FV(A,r,n) = A
(1+r) n - 1 r 18 AnExample
¨ Thenota
r (1+r) n - 1 20 FV of $1,000 each year for next 5 years = $1000
(1.10) 5 - 1 .10 = $6,105 19 Annuity,givenFutureValue
¨ ifyouaregiventhefuturevalueandyouarelookingforanannuity-A(FV,r,n)intermsofnotaApplica
Applica
Refinancing:TheTradeOff
¨ Ifyouplantoremaininthishouseindefinitely,
27
=$9,532
Follow-upQues
ValuingaStraightBond
1186.85
¨ Ifinterestratesriseto10%,
15 $1,057.05 Percentagechangeinprice=-10.94%
¨ Ifinterestratefallto7%,
15 $1,341.55 Percentagechangeinprice=+13.03%
¨ Thisasymmetricresponsetointerestratechangesiscalled convexity. 26
BondPricingProposi
Bond Maturity
% Change in Price -15.00% -10.00% -5.00% 0.00% 5.00% 10.00%
15.00%
20.00%
151530
% Change if rate drops to 7% % Change if rate increases to 10% 27
BondPricingProposi
Coupon Rate
% Price Change -20.00% -15.00% -10.00% -5.00% 0.00% 5.00% 10.00%
15.00%
20.00%
25.00%
0%5%10.75%12%
% Change if rate drops to 7% % Change if rate increases to 10% 28
III.GrowingAnnuity
A(1+g)
2 Figure 3.8: A Growing Annuity
A(1+g)
3 A(1+g)
nA(1+g) n 29
PresentValueofaGrowingAnnuity
PV of an Annuity = PV(A,r,g,n) = A(1+g)
1 - (1+g) n (1+r) n (r-g) 30
TheValueofaGoldMine
PV of extracted gold = $300*5000*(1.03)
1 - (1.03) 20 (1.10) 20 .10-.03 =$16,145,980 31
Present Value of Extracted Gold as a function of Growth Rate $5,000,000 $10,000,000 $15,000,000 $20,000,000 $25,000,000 $30,000,000 $35,000,000 $40,000,000 $45,000,000 $50,000,000 0%1%2%3%4%5%6%7%8%9%
10%11%12%13%14%15%
Growth Rate in Gold Prices
Present Value of Extracted Gold
32
IV.Perpetuity
¨ Aperpetuityisaconstantcashflowatregularintervalsforever.Thepresentvalueofaperpetuityis- ¨ Forevermaybeatoughconceptforhumanbeings
PV of Perpetuity =
A r 33
ValuingaConsoleBond
ValueofConsoleBond=$60/.09=$667
34
V.GrowingPerpetui
PV of Growing Perpetuity =
CF 1 (r-g) 35
ValuingaStockwithGrowingDividends
¨ IntwelvemonthsleadingintoJanuary2014,ConEdpaiddividendspershareof$2.52. ¨ Itsearningsanddividendshadgrownat2%ayear
¨ Therateofreturnrequiredbyinvestorsonstocksof
equivalentriskwas7.50%. ¨ Withtheseinputs,wecanvaluethestockusinga
perpetualgrowthmodel: 36
ValueandGrowth!
quotesdbs_dbs14.pdfusesText_20