Chapter 5

Time Value of Money II

Chapter 4 Introduction to Financial Management (Pope)

Core Formula and Variables

  • Fundamental equation representing the relationship between Future Value (FV) and Present Value (PV):

    • FV=PV(1+r)tFV = PV(1 + r)^{t}

    • PV=racFV(1+r)tPV = rac{FV}{(1 + r)^{t}}

    • r=racFVPVrac1t−1r = rac{FV}{PV}^{ rac{1}{t}} - 1

    • t=racextlnracFVPVextln(1+r)t = rac{ ext{ln} rac{FV}{PV}}{ ext{ln}(1 + r)}

Handling Multiple Cash Flows

Unequal Periodic Cash Flows

  • Each cash flow must be treated individually.

  • Process overview:

    • Calculate the future value or present value for each cash flow over the relevant number of periods.

    • Sum the individual present or future values to obtain the total cash flow stream.

Example: Auto Purchase Payment Options

  • Scenario:

    • Vehicle choice: Pay $15,500 cash now or make three payments of $8,000 immediately and $4,000 over the course of the next two years.

    • Given interest rate on savings account: 8%

  • Decision-making equation for Present Value (PV):

    • PV=extΣfromt=1exttoTracCt(1+r)tPV = ext{Σ from } t=1 ext{ to } T rac{C_{t}}{(1+r)^{t}}

Present Value (PV) of Multiple Cash Flows

  • General equations for calculating PV:

    1. PV=C<em>1rac1(1+r)0+C</em>2rac1(1+r)1+C<em>3rac1(1+r)2+…+C</em>Trac1(1+r)TPV = C<em>{1} rac{1}{(1+r)^{0}} + C</em>{2} rac{1}{(1+r)^{1}} + C<em>{3} rac{1}{(1+r)^{2}} + … + C</em>{T} rac{1}{(1+r)^{T}}

    2. PV=Cimesrac1−(1+r)−TrPV = C imes rac{1 - (1+r)^{-T}}{r}

Annuity Cash Flows

Definition

  • An annuity consists of a level cash flow stream at regular intervals with a finite duration.

  • Two Types:

    • Annuity Due: Payments made at the start of each period (e.g., rent, insurance). ( HAPPENS AT THE BEGINNING)

    • Ordinary Annuity: Payments made at the end of each period (e.g., water bill). (HAPPENS AT THE END)

Present Value of an Annuity (PVA)

  • Formula:

    • PV=PMTimesrac1−(1+r)−trPV = PMT imes rac{1 - (1+r)^{-t}}{r}

  • Variables:

    • PMT: periodic payment amount

    • r: periodic interest rate

    • t: number of periods

    • PVAF: Present Value Annuity Factor

Example 1: Car Purchase and Installments

  • Scenario:

    • Making 3 annual installments of $8,000 each, with a 10% interest rate

  • Formula Application:

    • PV=PMTimesrac1−(1+r)−TrPV = PMT imes rac{1 - (1+r)^{-T}}{r}

    • Calculation of price paid for the car:

    • PV=8,000imesrac1−(1+0.10)−30.10=19,894.82PV = 8,000 imes rac{1 - (1+0.10)^{-3}}{0.10} = 19,894.82

Future Value of an Annuity (FVA)

  • Formula:

    • FV=PMTimesPVAFimes(1+r)tFV = PMT imes PVAF imes (1 + r)^{t}

  • Variables:

    • PMT: periodic annuity payment

    • r: periodic interest rate

    • t: number of periods

    • PVAF: formula rac1−(1+r)−trrac{1 - (1 + r)^{-t}}{r}

Example 2: Future Savings

  • Scenario:

    • Saving $3,000 annually for the next 4 years at an 8% interest rate.

  • Future Value Calculation:

    • Application of FVA formula:

    • FV=3,000imesrac1−(1+0.08)−40.08imes(1+0.08)4=13,518FV = 3,000 imes rac{1 - (1 + 0.08)^{-4}}{0.08} imes (1 + 0.08)^{4} = 13,518

Example 3: Retirement Savings Goal

  • Scenario:

    • Saving for 40 years with a goal of $1,000,000 at retirement and a 10% interest rate.

  • Calculation of Required Savings:

    • Rewrite FVA formula to find PMT needed:

    • 1,000,000=PMTimesrac1−(1+0.10)−400.10imes(1+0.10)401,000,000 = PMT imes rac{1 - (1 + 0.10)^{-40}}{0.10} imes (1 + 0.10)^{40}

    • Result:

    • PMT=1,129.71PMT = 1,129.71

Annuity Due

  • Difference:

    • Annuity Due cash flow starts immediately rather than at the end of the period.

  • Adjustment Formulas:

    • PV<em>AnnuityextDue=PV</em>OrdinaryextAnnuityimes(1+r)PV<em>{Annuity ext{ Due}} = PV</em>{Ordinary ext{ Annuity}} imes (1 + r)

    • FV<em>AnnuityextDue=FV</em>OrdinaryextAnnuityimes(1+r)FV<em>{Annuity ext{ Due}} = FV</em>{Ordinary ext{ Annuity}} imes (1 + r)

Example 4: Retirement Savings with Annuity Due

  • Scenario:

    • Depositing $3,000 annually for 20 years at an 8% interest rate.

  • Difference between making deposits at the beginning vs. end:

  • Calculation:

    1. Ordinary future value:

    • Calculate using ordinary annuity formula.

    1. Adjust for annuity due:

    • FV<em>AnnuityextDue=FV</em>OrdinaryextAnnuityimes(1+0.08)FV<em>{Annuity ext{ Due}} = FV</em>{Ordinary ext{ Annuity}} imes (1 + 0.08)

      • Difference in values:

      • Result: 148,268.76−137,285.89=11,000.87148,268.76 - 137,285.89 = 11,000.87

Perpetuities

Definition

  • A perpetuity is an annuity cash flow that continues indefinitely.(never ending)

  • Formula for Present Value (PV):

    • PV=racPMTrPV = rac{PMT}{r}

  • Re-expressed to show the perpetuity payment calculated from present value and interest rate.

Example 5: Endowed Scholarship Contributions

  • Scenario:

    • Creating a scholarship offering $10,000 annually with a 5% rate of return.

  • Contribution Calculation:

    • Use formula for present value of perpetuity:

    • PV=rac10,0000.05=200,000PV = rac{10,000}{0.05} = 200,000

Commercial Lending and Amortization

Definition

Commercial lending is to business not individuals

Paying back loan in equal payments over time

  • Description of loan repayment via an annuity cash flow.

  • Characteristics:

    • Consistent periodic payments that cover interest and part of the principal.

    • The last payment addresses any remaining interest and principal.

    • Reliable cash flow for lenders; regular scheduled payments for consumers.

Example 6: Home Purchase Mortgage

  • Scenario:

    • Buying a home for $300,000, with $50,000 down, and a 30-year mortgage at a 5% annual fixed rate.

  • Monthly Payment Calculation:

    • Use the formula for Present Value (PV):

    • PV=PMTimesrac1−(1+r)−TrPV = PMT imes rac{1 - (1+r)^{-T}}{r}

    • Find monthly payment PMTPMT:

    • PMT=rac250,000imes(0.05/12)1−(1+0.05/12)−360=894.33PMT = rac{250,000 imes (0.05/12)}{1 - (1 + 0.05/12)^{-360}} = 894.33

Annual and Effective Rates

Annual Percentage Rate (APR)

  • Definition:

    • The commonly quoted annual interest rate treating the charge as if incurred once per year.

  • Relation between APR and periodic rates:

    • r=racAPRmr = rac{APR}{m}

    • APR=rimesmAPR = r imes m

  • Variables:

    • APR: Annual percentage rate

    • R: periodic rate

    • M: number of compounding periods per year

Effective Annual Rate (EAR)

  • Definition:

    • The true rate of return for lenders and the actual cost for borrowers.

  • Formula to calculate EAR:

    • EAR=(1+r)m−1EAR = (1 + r)^{m} - 1

  • Alternatively:

    • EAR=racAPR+1m−1EAR = rac{APR + 1}{m} - 1

Example 7: Interest Calculation and EAR

  • Scenario:

    • Borrowing $100,000 at an 8.5% APR, compounded monthly for 1 year.

  • Calculation to derive EAR:

    • EAR=(1+rac0.08512)12−1=0.0884EAR = (1 + rac{0.085}{12})^{12} - 1 = 0.0884

  • Total interest paid after one year:

    • 100,000imes0.0884=8,840100,000 imes 0.0884 = 8,840

Example 8: Lottery Prize Options

  • Scenario:

    • Winning a lottery prize of $50,000,000 with options for lump-sum or annuity. Options:

    • Lump-sum of $13,000,000 today or 50-year annuity of $1,000,000 annually.

  • Decision-making based on the greater future value when invested at 5% APR.

Example 9: Business Loan Alternatives

  • Scenario:

    • Jim needs a $50,000 loan at a 9% APR with options for annual, quarterly, or monthly payments.

  • Calculation of periodic payments for each option, comparing total payments made annually.

Example 10: Retirement Withdrawal Savings

  • Scenario:

    • Intention to withdraw $10,000 per month for 20 years after retiring at age 65.

  • Required monthly deposit for account earning 8% per year, with monthly compounding.

  • Decision-making on deposit amount needed based on calculations to ensure sufficient funds upon retirement withdrawal.