NPV Calculations, Annuities, Perpetuities-Irena Jankovic-CFA, Prmia

Irena Jankovic, Faculty of Economics, Belgrade, November 2015Irena Jankovic, Faculty of Economics, Belgrade, November 2015

Faculty of Economics and SECCF Faculty of Economics and SECCF University of BelgradeUniversity of Belgrade

IMQFIMQF

Mathematics and Modeling for FinanceMathematics and Modeling for Finance

NPV, Annuities, NPV, Annuities, PerpetuitiesPerpetuities

Irena M. JankovicIrena M. Jankovic

Belgrade, November 2015Belgrade, November 2015


SUMMARYSUMMARY IntroductionIntroduction Time Value of MoneyTime Value of Money Interest RatesInterest Rates Future ValueFuture Value FV of an AnnuityFV of an Annuity Present ValuePresent Value PV of an Annuity and PerpetuityPV of an Annuity and Perpetuity Net Present ValueNet Present Value Internal Rate of ReturnInternal Rate of Return ConclusionsConclusions


Time Value of MoneyTime Value of Money

Present vs future consumptionPresent vs future consumption Do 1000Euros today have the same Do 1000Euros today have the same

value as 1000 Euros received one or value as 1000 Euros received one or more years from now?more years from now?

How to deal with cash flows occurring How to deal with cash flows occurring on different dates?on different dates?


Interest rateInterest rate

rr Rate of return that reflects the Rate of return that reflects the

relationship between differently dated relationship between differently dated cash flowscash flows

1.1. Required rateRequired rate2.2. Discount rateDiscount rate3.3. Opportunity costOpportunity cost


Investors perspectiveInvestors perspective

Real riskReal risk--free interest ratefree interest rate+Inflation premium+Inflation premium+Default risk premium+Default risk premium+Liquidity premium+Liquidity premium+Maturity premium+Maturity premium

= = rr

Nominal riskNominal risk--free interest free interest raterate


Future valueFuture value Relationship between Relationship between Present value Present value

(PV)(PV) which earns which earns a rate of return (r)a rate of return (r) and and its its Future Value (FV)Future Value (FV) which will be which will be received received NN years from nowyears from now

For N=1, For N=1, FVFV11=PV(1+r)=PV(1+r) Simple interest,Simple interest, FVFVNN=PV(1+Nr)=PV(1+Nr) Compound interest, interest on Compound interest, interest on

interest,interest, FVFVNN=PV(1+r)=PV(1+r)NN


Simple interestSimple interest Compound interestCompound interest

YearYear BalanceBalance YearYear BalanceBalance

00 PVPV 00 PVPV11 PV+r*PVPV+r*PV 11 PV+r*PVPV+r*PV22 PV+r*PV+r*PV+r*PV+r*PVPV 22 (PV+r*PV)+r*(PV+r*PV)+r*(PV+r*PV)(PV+r*PV)

NN FV=PV(1+Nr)FV=PV(1+Nr) NN FV=PV(1+r)FV=PV(1+r)NN


Example 1. (Excel)Example 1. (Excel)

Suppose you deposit 1000Eur in an Suppose you deposit 1000Eur in an account leaving it there for 10 years account leaving it there for 10 years at interest rate of 10% p.a. How at interest rate of 10% p.a. How much will you have at the end of 10 much will you have at the end of 10 years? Compare simple vs years? Compare simple vs compound interest calculationscompound interest calculationsresults and graph them.results and graph them.


Example 2. (Excel)Example 2. (Excel)Person is 58 years old and intends to retire Person is 58 years old and intends to retire

at age 63. She starts retirement account: at age 63. She starts retirement account: At the beginning of each year 0, 1, 2, 3, 4 At the beginning of each year 0, 1, 2, 3, 4

she makes deposit into account and she makes deposit into account and expects it will earn 10% interest per expects it will earn 10% interest per annum.annum.

After retirement she expects to withdraw After retirement she expects to withdraw $20,000 each year for 8 years.$20,000 each year for 8 years.

How much should she deposit each year How much should she deposit each year in the account?in the account?


Frequency of compoundingFrequency of compounding When interest is compounded several When interest is compounded several

times a year (biannual, quarterly, daily, times a year (biannual, quarterly, daily, hourly, every minute)hourly, every minute)

The periodic rate and the number of The periodic rate and the number of compounding periods must be compounding periods must be compatible.compatible.

FVFVNN=PV(1+r/m)=PV(1+r/m)mNmN As the number of compounding periods As the number of compounding periods

increases, the future value increasesincreases, the future value increases


Continuous compoundingContinuous compounding If the number of compounding periods If the number of compounding periods

per year becomes infiniteper year becomes infinite FVFVNN=PVe=PVerNrN

The more frequent compounding, the The more frequent compounding, the larger the FVlarger the FV

Used for portfolio return calculations Used for portfolio return calculations and options valuationsand options valuations

m7182818.2e



You deposit $1000 in a bank that pays You deposit $1000 in a bank that pays 5% p.a. How much will you have after a 5% p.a. How much will you have after a year if the compounding is done 1, 2, year if the compounding is done 1, 2, 10, 20, 50, 100, 150, 300, 800 times a 10, 20, 50, 100, 150, 300, 800 times a year or continuously? Graph the effect year or continuously? Graph the effect of changing compounding frequency.of changing compounding frequency.


Effective annual rateEffective annual rateDiscrete:Discrete:EAR= (1+r/m)EAR= (1+r/m)mm--11Continuous:Continuous:EAR= eEAR= err--11e.g. A $1 would at 5%p.a. in a year grow e.g. A $1 would at 5%p.a. in a year grow

to:to:1.1. m=1, 1(1+0.05)=$1.05, EAR=rm=1, 1(1+0.05)=$1.05, EAR=r2.2. m=2, 1(1+0.05/2)m=2, 1(1+0.05/2)22=1.050625,=1.050625,EAR=(1+0.05/2)EAR=(1+0.05/2)22--1=0.050625=5.0625%1=0.050625=5.0625%3.3. Continuous compounding, Continuous compounding, EAR=eEAR=e0.050.05--1=0.051271=5.1271% 1=0.051271=5.1271%


The future value of an annuityThe future value of an annuity

Annuity Annuity a finite set of level sequential a finite set of level sequential cash flowscash flows

1.1. Ordinary annuity (t=1)Ordinary annuity (t=1)2.2. Annuity due (t=0)Annuity due (t=0)3.3. Perpetuity Perpetuity a set of level nevera set of level never--ending ending

sequential cash flows (t=1)sequential cash flows (t=1)


Ordinary annuityOrdinary annuity

[ ]

+=++++++++=

rrAFV

rrrrAFVN

N

NNN

1)1(

)1()1(...)1()1( 0121


Example 4. (Excel)Example 4. (Excel)Suppose we have 5 separate deposits of Suppose we have 5 separate deposits of $1000 occurring at the end of the next five $1000 occurring at the end of the next five years. Find the future value of this annuity years. Find the future value of this annuity after the last deposit at t=5. r=5%p.a.after the last deposit at t=5. r=5%p.a.

Example 5. (Excel)Example 5. (Excel)Two years from now a client will receive Two years from now a client will receive first of three annual payments of $20,000. If first of three annual payments of $20,000. If he can invest them at 9%p.a., how much he can invest them at 9%p.a., how much will they worth after 6 years?will they worth after 6 years?


Present valuePresent value

Given a future Given a future cash flow that is cash flow that is to be received in to be received in period N, and an period N, and an interest rate per interest rate per period r, we can period r, we can calculate the calculate the present valuepresent value

PV=FVPV=FVNN(1+r)(1+r)--NN

PV=FVPV=FVNN(1+r/m)(1+r/m)--mNmN

PV=FVPV=FVNN//eerNrN



Suppose you own liquid asset that will Suppose you own liquid asset that will pay you $100,000 in 10 years from today. pay you $100,000 in 10 years from today. Given an 8% discount rate, what will the Given an 8% discount rate, what will the asset be worth 4 years from today, and asset be worth 4 years from today, and what will be its value today? What would what will be its value today? What would be the value today in the case of monthly be the value today in the case of monthly compounding?compounding?


The present value of an ordinary The present value of an ordinary annuityannuity

+=

++++++++=

rrAPV

rA

rA

rA

rAPV

N

NN

)1(11

)1()1(...

)1()1( 12

Mortgages, auto loans and retirement savings plans are classic examples of application of annuity formulas.



Pension fund manager anticipates that Pension fund manager anticipates that $1 million per year must be paid to $1 million per year must be paid to retirees starting at year 11. The benefits retirees starting at year 11. The benefits will be paid until t=39 for a total of 30 will be paid until t=39 for a total of 30 payments. What is a PV of pension payments. What is a PV of pension liability if the appropriate discount rate liability if the appropriate discount rate is 5% compounded annually?is 5% compounded annually?


The PV of a PerpetuityThe PV of a Perpetuity

rAPV

rr

ArA

rA

rAPV

tt

=

+=++++++=

=0

)1(1...

)1()1()1( 132

>

Some government bonds and preferred Some government bonds and preferred stocks are typical examples of assets that stocks are typical examples of assets that make level payments for an indefinite period make level payments for an indefinite period of time.of time.


Example 8.Example 8.A perpetual preferred stock pays A perpetual preferred stock pays quarterly dividends of $1000 forever. If quarterly dividends of $1000 forever. If the required rate of return is 12%p.a. on the required rate of return is 12%p.a. on this type of investment, how much this type of investment, how much would you pay for this stock?would you pay for this stock?

Example 9.Example 9.Consider a level perpetuity of $100 p.a. Consider a level perpetuity of $100 p.a. with its first payment beginning at t=5. with its first payment beginning at t=5. What is its PV today given a 5% What is its PV today given a 5% discount rate?discount rate?


Example 10.Example 10.How long it will take for an investment of How long it will take for an investment of 1,000,000 to double in value? The interest 1,000,000 to double in value? The interest rate is 8%p.a.rate is 8%p.a.

Example 11.Example 11.En is 22 years old and is planning retirement at age 63. She plans to save $2,000 per year next 15 years. She wants to have retirement income of 100,000 per year for 20 years starting at t=41. How much must she save each year from t=16 to t=40 in order to achieve her goal? She plans to invest her money in investment fund earning on average 8%p.a.


Net Present Value (NPV)Net Present Value (NPV)

A method for choosing among A method for choosing among alternative investmentsalternative investments

Present value of investment cash Present value of investment cash inflows (benefits) minus the present inflows (benefits) minus the present value of its cash outflows (costs)value of its cash outflows (costs)

Decisions concerning capital budgeting Decisions concerning capital budgeting and capital structureand capital structure


NPV rulesNPV rules1.1. Identify all cash inflows and outflows Identify all cash inflows and outflows

associated with the investmentassociated with the investment2.2. Determine appropriate discount rate for Determine appropriate discount rate for

particular project (e.g. WACC)particular project (e.g. WACC)3.3. Sum all present values=NPVSum all present values=NPV4.4. If NPVIf NPV>0, undertake investment>0, undertake investment5.5. If you are choosing between two projects If you are choosing between two projects

(mutually exclusive) undertake the one with (mutually exclusive) undertake the one with higher positive NPVhigher positive NPV


NPV formulaNPV formula

CFCFtt = net cash flow at time t= net cash flow at time t r = the discount rate or opportunity cost of r = the discount rate or opportunity cost of

capitalcapital N = the projected life of the investmentN = the projected life of the investment When NPV is positive, the investment adds When NPV is positive, the investment adds

value!value!

= +=N

tt

t

rCFNPV

0 )1(


Example 12. Example 12.

Management of the Corporation plans to Management of the Corporation plans to invest $1 million in R&D. Incremental net invest $1 million in R&D. Incremental net cash flows are forecasted to be $150,000 cash flows are forecasted to be $150,000 per year in perpetuity. Corporationper year in perpetuity. Corporations s opportunity cost of capital is 10%p.a. opportunity cost of capital is 10%p.a. Should shareholders accept the plan?Should shareholders accept the plan?What would be their decision if the What would be their decision if the opportunity cost of capital is 15%?opportunity cost of capital is 15%?


Example 13. (Excel)Example 13. (Excel)Company is planning joint Company is planning joint

venture that requires venture that requires investment of $13 million. investment of $13 million. Expected cash flows are Expected cash flows are given below. Discount rate given below. Discount rate calculated for this proposal calculated for this proposal is 12%p.a.is 12%p.a.

a)a) Calculate investmentCalculate investments NPV s NPV b)b) Make a recommendation to Make a recommendation to

the managementthe management

YearYear Cash flowCash flow

00 --13,000,00013,000,000

11 3,000,0003,000,000

22 3,000,0003,000,000

33 3,000,0003,000,000

44 3,000,0003,000,000

55 10,000,00010,000,000


Internal Rate of Return (IRR)Internal Rate of Return (IRR) A single number that represents the rate of A single number that represents the rate of

return generated by an investmentreturn generated by an investment The discount rate that makes NPV=0The discount rate that makes NPV=0 PV of outflows=PV inflowsPV of outflows=PV inflows In the study of bonds IRR is called Yield to In the study of bonds IRR is called Yield to

MaturityMaturity

0)1(

...)1()1( 2

21

10 =+++++++= N

N

IRRCF

IRRCF

IRRCFCFNPV


IRR ruleIRR rule

Accept project or investment for Accept project or investment for which the IRR is greater than the which the IRR is greater than the opportunity cost of capitalopportunity cost of capital

When the opportunity cost is equal to When the opportunity cost is equal to IRR, NPV=0IRR, NPV=0

Caution: When the project after initial Caution: When the project after initial investment has positive and negative investment has positive and negative cash flows, it can have more than one cash flows, it can have more than one IRR.IRR.


Example 14. Example 14.

Use Example 12 data and calculate IRR.Use Example 12 data and calculate IRR.

Example 15. Example 15. Bank gives a $1000 loan to a customer Bank gives a $1000 loan to a customer and makes a 5 year payment plan (300, and makes a 5 year payment plan (300, 200, 150, 600, 900). Each payment 200, 150, 600, 900). Each payment consists of the principal and the interest. consists of the principal and the interest. Find the IRR for this loan schedule.Find the IRR for this loan schedule.


NPV vs IRRNPV vs IRR NPV and IRR give the same accept or reject NPV and IRR give the same accept or reject

decision when the projects are independentdecision when the projects are independent When projects are mutually exclusive, we When projects are mutually exclusive, we

have to rank them. have to rank them. Rankings according to NPV and IRR may Rankings according to NPV and IRR may

differ when:differ when:1.1. The size of the projects differsThe size of the projects differs2.2. The timing of the projectsThe timing of the projects cash flows cash flows

differsdiffers


IRR and NPV for Mutually IRR and NPV for Mutually Exclusive Projects of Different Exclusive Projects of Different

SizeSizeProjectProject Investment Investment

at t=0at t=0Cash Flow Cash Flow at t=1at t=1

IRRIRR NPV at NPV at 8%8%

AA --10,00010,000 15,00015,000 50%50% 3,888.93,888.9

BB --30,00030,000 42,00042,000 40%40% 8,888.98,888.9


IRR and NPV for Mutually IRR and NPV for Mutually Exclusive Projects with Different Exclusive Projects with Different

Timing of Cash FlowsTiming of Cash FlowsProProjectject

CF0CF0 CF1CF1 CF2CF2 CF3CF3

00 0021,22021,22000

IRRIRR NPV at NPV at 8%8%

AA --10,00010,000 15,00015,000 50%50% 3,888.93,888.9BB --10,00010,000 00 28.5%28.5% 6,845.126,845.12

When you have this kind of conflicts, choose the project with higher NPV because it would lead to higher increase in shareholders wealth!!!

SUMMARYTime Value of MoneyInterest rateInvestors perspectiveFuture valueExample 1. (Excel)Example 2. (Excel)Frequency of compoundingContinuous compoundingExample 3. (Excel)Effective annual rateThe future value of an annuityOrdinary annuityExample 4. (Excel)Present valueExample 6. (Excel)The present value of an ordinary annuityExample 7. (Excel)The PV of a PerpetuityExample 8.Example 10.Net Present Value (NPV)NPV rulesNPV formulaExample 12. Example 13. (Excel)Internal Rate of Return (IRR)IRR ruleExample 14. NPV vs IRRIRR and NPV for Mutually Exclusive Projects of Different SizeIRR and NPV for Mutually Exclusive Projects with Different Timing of Cash Flows

NPV Calculations, Annuities, Perpetuities-Irena Jankovic-CFA, Prmia

Documents

Transcript of NPV Calculations, Annuities, Perpetuities-Irena Jankovic-CFA, Prmia