Time Value of Money

Overview

• Future Value

• Present Value

• Finding I and N

• Annuities

• Rates of Return

• Amortization

Time Lines

• Show the timing of cash flows.

• Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.

Drawing Time Lines

• Cash flow timing can be represented visually using timelines.

• Examples:

  • $100 lump sum due in 2 years</p></li><li><p>3-year$100 ordinary annuity

  • Uneven cash flow stream

Future Value (FV)

• The future value (FV) of an initial $100 after 3 years, if I/YR = 4$FV_1 = PV(1 + I) = $100(1.04) = $104.00$</p></li><li><p>After 2 years:$FV_2 = PV(1 + I)^2 = $100(1.04)^2 = $108.16$</p></li><li><p>After 3 years:$FV_3 = PV(1 + I)^3 = $100(1.04)^3 = $112.49$</p></li><li><p>After N years (general case):$FV_N = PV(1 + I)^N$</p></li></ul><h4 id="7f66cdad-c1f9-44f5-a24b-cfa9578257e7" data-toc-id="7f66cdad-c1f9-44f5-a24b-cfa9578257e7" collapsed="false" seolevelmigrated="true">Solving for FV: Calculator and Excel Methods</h4><ul><li><p>Solves the general FV equation.</p></li><li><p>Requires 4 inputs into the calculator and will solve for the fifth. (Set to P/YR = 1 and END mode.)</p></li><li><p>Excel: =FV(rate,nper,pmt,pv,type)</p></li></ul><h3 id="d9dff930-2205-4c05-9390-a4c78137e0fb" data-toc-id="d9dff930-2205-4c05-9390-a4c78137e0fb" collapsed="false" seolevelmigrated="true">Present Value (PV)</h3><ul><li><p>What is the present value (PV) of$100 due in 3 years, if I/YR = 4%?

• Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding).

• The PV shows the value of cash flows in terms of today’s purchasing power.

Solving for PV: The Formula Method

• Solve the general FV equation for PV: $PV = \frac{FV_N}{(1 + I)^N}$

• PV = \frac{FV_3}{(1 + I)^3} = $\frac{100}{(1.04)^3} = $88.90

Solving for PV: Calculator and Excel Methods

• Solves the general FV equation for PV.

• Exactly like solving for FV, except we have different input information and are solving for a different variable.

• Excel: =PV(rate,nper,pmt,fv,type)

Solving for I

• What annual interest rate would cause $100 to grow to$119.10 in 3 years?

• Solves the general FV equation for I/YR.

• Hard to solve without a financial calculator or spreadsheet.

• Excel: =RATE(nper,pmt,pv,fv,type,guess)

Solving for N

• If sales grow at 10% per year, how long before sales double?

• Solves the general FV equation for N.

• Hard to solve without a financial calculator or spreadsheet.

• Excel: =NPER(rate,pmt,pv,fv,type)

Annuities

• What is the difference between an ordinary annuity and an annuity due?

Solving for FV

• Given a 3-Year Ordinary Annuity of $100 at 4$100 payments occur at the end of each period, but there is no PV.

• Excel: =FV(rate,nper,pmt,pv,type)

• Here type = 0.

Solving for PV

• Given a 3-year Ordinary Annuity of $100 at 4$100 payments still occur at the end of each period, but now there is no FV.

• Excel: =PV(rate,nper,pmt,fv,type)

• Here type = 0.

Solving for FV: 3-Year Annuity Due of $100 at 4%

• Now, $100 payments occur at the beginning of each period.</p></li><li><p>$FVAdue = FVAord(1 + I) = $312.16(1.04) = $324.65

• Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity due:

• Excel: = FV(rate,nper,pmt,pv,type)

• Here type = 1.

Solving for PV: 3-Year Annuity Due of $100 at 4%

• Again, 100 payments occur at the beginning of each period.

• PVAdue = PVAord(1 + I) = $277.51(1.04) = $288.61

• Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity due:

• Excel: = PV(rate,nper,pmt,fv,type)

• Here type = 1.

PV Calculation

• What is the present value of a 5-year $100 ordinary annuity at 4$445.18.

Annuities Over Time

The Power of Compound Interest

• A 20-year-old student wants to save $5 a day for her retirement. Every day she places$ 5 in a drawer. At the end of the year, she invests the accumulated savings ($1,825) in a brokerage account with an expected annual return of 8$705,373 when she is 65.

• Excel: = FV(.08,45,-1825,0,0)

Solving for FV

• If you don’t start saving until you are 40 years old, how much will you have at 65?

• If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $133,418 at age 65. This is$571,954 less than if starting at age 20.

• Lesson: It pays to start saving early.

• Excel: = FV(.08,45,-1825,0,0)

Solving for PMT

• How much must the 40-year old deposit annually to catch the 20-year old?

• To find the required annual contribution, enter the number of years until retirement and the final goal of $705,372.75, and solve for PMT.</p></li><li><p>Excel: = PMT(rate,nper,pv,fv,type) =PMT(.08,25,0,705373,0)</p></li></ul><h3 id="be46f9c4-928e-486d-b79a-995967140135" data-toc-id="be46f9c4-928e-486d-b79a-995967140135" collapsed="false" seolevelmigrated="true">Uneven Cash Flow Stream</h3><ul><li><p>What is the PV of this uneven cash flow stream?</p></li></ul><h4 id="c211ddfd-adc4-4390-b0ef-d06eea52bf6a" data-toc-id="c211ddfd-adc4-4390-b0ef-d06eea52bf6a" collapsed="false" seolevelmigrated="true">Solving for PV: Uneven Cash Flow Stream</h4><ul><li><p>Input cash flows in the calculator’s “CFLO” register:</p><ul><li><p>CF0 = 0</p></li><li><p>CF1 = 100</p></li><li><p>CF2 = 300</p></li><li><p>CF3 = 300</p></li><li><p>CF4 = -50</p></li></ul></li><li><p>Enter I/YR = 4, press NPV button to get NPV =$597.48. (Here NPV = PV.)

Compounding Frequency

• Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated I% constant?

• LARGER, as the more frequently compounding occurs, interest is earned on interest more often.

  • Annually: FV_3 = $100(1.04)^3 = $112.49

  • Semiannually: FV_6 = $100(1.02)^6 = $112.62

Classification of Interest Rates

• Nominal rate ($I_{NOM}$):

  • Also called the quoted or stated rate.

  • An annual rate that ignores compounding effects.

  • $I_{NOM}$ is stated in contracts.

  • Periods must also be given, e.g. 4% quarterly or 4% daily interest.

• Periodic rate ($I_{PER}$):

  • Amount of interest charged each period, e.g. monthly or quarterly.

  • $I<em>{PER} = \frac{I</em>{NOM}}{M}$, where M is the number of compounding periods per year.

  • M = 4 for quarterly and M = 12 for monthly compounding.

• Effective (or equivalent) annual rate (EAR = EFF%):

  • The annual rate of interest actually being earned, considering compounding.

  • EFF% for 4% semiannual interest $EFF$

  • Excel: =EFFECT(nominal_rate,npery) =EFFECT(.04,2)

  • Should be indifferent between receiving 4.04% annual interest and receiving 4% interest, compounded semiannually.

The Importance of Effective Rates of Return

• Investments with different compounding intervals provide different effective returns.

• To compare investments with different compounding intervals, you must look at their effective returns (EFF% or EAR).

• See how the effective return varies between investments with the same nominal rate, but different compounding intervals.

  • EARANNUAL

  • EARSEMIANNUALLY

  • EARQUARTERLY

  • EARMONTHLY

  • EARDAILY (365)

  • 4. 00%

  • 4. 04%

  • 4. 06%

  • 4. 07%

  • 4. 08%

When is each rate used?

• $I_{NOM}$: Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

• $I<em>{PER}$: Used in calculations and shown on time lines. If M = 1, $I</em>{NOM} = I_{PER} = EAR$.

• EAR: Used to compare returns on investments with different payments per year. Used in calculations when annuity payments don’t match compounding periods.

Effect of Compounding on FV

• What is the FV of $100 after 3 years under 4$100 annuity, if the quoted interest rate is 4%, compounded semiannually?

• Payments occur annually, but compounding occurs every 6 months.

• Cannot use normal annuity valuation techniques.

Method 1: Compound Each Cash Flow

• FV_3 = $100(1.02)^4 + $100(1.02)^2 + $100

• FV_3 = $312.28

Method 2: Financial Calculator or Excel

• Find the EAR and treat as an annuity.

• EAR = (1 + 0.04/2)2 – 1 = 4.04%.

• Excel: =FV(.0404,3,-100,0,0)

Find the PV of This 3-Year Ordinary Annuity

• Could solve by discounting each cash flow, or…

• Use the EAR and treat as an annuity to solve for PV.

• Excel: = PV(.0404,3,100,0,0)

Loan Amortization

• Amortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, etc.

• Financial calculators and spreadsheets are great for setting up amortization tables.

• EXAMPLE: Construct an amortization schedule for a $1,000, 4$INTt = Beg balt(I)$</p></li><li><p>$INT_1 = $1,000(0.04) = $40$</p></li></ul><h4 id="3b6d26f9-3067-4d04-b6da-c40b086871e4" data-toc-id="3b6d26f9-3067-4d04-b6da-c40b086871e4" collapsed="false" seolevelmigrated="true">Step 3: Find the Principal Repaid in Year 1</h4><ul><li><p>If a payment of$360.35 was made at the end of the first year and $40 was paid toward interest, the remaining value must represent the amount of principal repaid.</p></li><li><p>$PRIN = PMT – INT = $360.35 – $40 = $320.35$</p></li></ul><h4 id="8d167957-f0ec-487f-8805-7e51508f5882" data-toc-id="8d167957-f0ec-487f-8805-7e51508f5882" collapsed="false" seolevelmigrated="true">Step 4: Find the Ending Balance after Year 1</h4><ul><li><p>To find the balance at the end of the period, subtract the amount paid toward principal from the beginning balance.</p></li><li><p>$END BAL = BEG BAL – PRIN = $1,000 – $320.35 = $679.65$</p></li></ul><h4 id="061f294b-76a8-4ed4-a672-3680d4a0e6f4" data-toc-id="061f294b-76a8-4ed4-a672-3680d4a0e6f4" collapsed="false" seolevelmigrated="true">Constructing an Amortization Table: Repeat Steps 1-4 Until End of Loan</h4><ul><li><p>Interest paid declines with each payment as the balance declines. What are the tax implications of this?</p></li></ul><table style="min-width: 150px"><colgroup><col style="min-width: 25px"><col style="min-width: 25px"><col style="min-width: 25px"><col style="min-width: 25px"><col style="min-width: 25px"><col style="min-width: 25px"></colgroup><tbody><tr><th colspan="1" rowspan="1"><p>YEAR</p></th><th colspan="1" rowspan="1"><p>BEG BAL</p></th><th colspan="1" rowspan="1"><p>PMT</p></th><th colspan="1" rowspan="1"><p>INT</p></th><th colspan="1" rowspan="1"><p>PRIN</p></th><th colspan="1" rowspan="1"><p>END BAL</p></th></tr><tr><td colspan="1" rowspan="1"><p>1</p></td><td colspan="1" rowspan="1"><p>$1,000

  $360</p></td><td colspan="1" rowspan="1"><p>$40

  $320</p></td><td colspan="1" rowspan="1"><p>$360

  2

  680

  360

  27

  333

  347

  3

  347

  360

  14

  347

  0

  TOTAL

  –

  $1,081</p></td><td colspan="1" rowspan="1"><p>$81

  $$1,000

  –

  Illustrating an Amortized Payment: Where does the money go?

  • Constant payments

  • Declining interest payments

  • Declining balance