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Chapter 5 – Important Stuff. Mechanics of compounding / discounting PV, FV, PMT – lump sums and annuities Relationships – time, interest rates, etc Calculations: PV’s, FV’s, loan payments, interest rates. Time Value of Money (TVM).

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chapter 5 important stuff
Chapter 5 – Important Stuff
  • Mechanics of compounding / discounting
  • PV, FV, PMT – lump sums and annuities
  • Relationships – time, interest rates, etc
  • Calculations: PV’s, FV’s, loan payments, interest rates
time value of money tvm
Time Value of Money (TVM)
  • Time Value of Money – relationship between value at two points in time
    • Today versus tomorrow; today versus yesterday
    • Because an invested dollar can earn interest, its future value is greater than today’s value
  • Problem types: monthly loan payments, growth of savings account; time to goal
financial calculator keys
Financial Calculator Keys
  • PV - Present value
  • FV - Future value
  • PMT - Amount of the payment
  • N - Number of periods (years?)
  • I/Y - Interest rate per period
ti calculator manual strongly suggested readings
TI Calculator ManualStrongly Suggested Readings
  • Getting Started – page 6 and 7
  • Overview – page 1-4, 1-10 and 1-20
  • Worksheets – pages 2-14 and 2-15
  • TVM – 3-1 to 3-9
  • Cash Flow - All
calculator tips
Calculator Tips

Decimals and Compounding Periods

  • 2nd (gray), Format (bottom row), 4, enter, CE/C (lower left) - hit twice
  • Compounding: 2nd , I/Y, 1, enter, CE/C – extremely important !!
  • Right arrow key fixes “misteaks”
  • One cash flow must be negative or error
compound interest @ 6
Compound Interest @ 6%

YearBeginInterestFV

1 $100.00 $6.00 $106.00

2 106.00 6.36 112.36

3 112.36 6.74 119.10

future value fv
Future Value (FV)

Algebraically FVn = PV (1 + i)n

Underlies all TVM calculations

Keystrokes: 100 +/- PV; 3 N; 0 PMT;

6 I/Y; CPT FV = 119.10

One cash flow must be negative

Error 5 means you forgot a negative sign

future value interest factor
Future Value Interest Factor

[email protected]%@6%@10%

  • 1.020 1.060 1.100
  • 1.040 1.124 1.210

3 1.104 1.191 1.611

10 1.219 1.791 2.594

reading the formulas and tables fv n pv 1 i n
Reading the Formulas and TablesFVn = PV* (1 + i)n
  • Plain English = The future value in period n is the present value (PV) times the quantity (i plus the interest rate) raised to the nth power where n equals the number of compounding periods.
  • Future value of $500 invested 3 yrs @ 6%
    • From table: FV6%, 3 yr. = 500 *1.191 = 595.50
fv can be increased by
FV Can Be Increased By

1. Increasing the length of time it is compounded

2. Compounding at a higher rate

And/or

3. Compounding more frequently

fv other keystrokes
FV – Other Keystrokes
  • How long for an investment to grow from $15,444 to $20,000 if earn 9% when compounded annually? Must solve for N.

15444 +/- PV; 20000 FV; 0 PMT; I/Y 9;

CPT N = 3 years

  • What rate earned if start at $15,444 and reach $20,000 in 3 years? Solve for I/Y.

15444 +/- PV; 20000 FV; 0 PMT; 3 N;

CPT I/Y = 9%

time to double your money rule of 72
Time to Double Your Money“Rule of 72”
  • Enter 100 PV; 200 FV, 10 I/Y, solve for N or
  • Use Rule of 72 – says number of years to double is approximately equal to 72 divided by the interest rate.
  • Doubling time ≈ 72

Interest Rate

present value pv
Present Value (PV)

If I earn 10%, how much must I deposit

today to have $100 in three years? $75.10

This is “inverse compounding”

Discount rate – interest rate used to bring (discount) future money back to present

For lump sums (only) PV and FV are reciprocals

present value formula
Present Value Formula

[ 1 ]

PV = FVn[ (1 + i) n ]

PVIF and FVIF for lump sums only are reciprocals. For 5% over ten years

FVIF = 1.629 = 1 / .614

PVIF = .614 = 1 / 1.629

present value interest factor
Present Value Interest Factor

@2%@5%@10%

Year 1 .980 .952 .909

Year 2 .961 .907 .826

Year 3 .942 .864 .751

Year 10 .820 .614 .386

keystrokes 100 @5 for ten years
Keystrokes$100 @5% for ten years
  • For PV +/-100 FV; 0 PMT; 5 I/Y; 10 N;

CPT PV = 61.39

  • For I/Y 100 FV; 0 PMT; +/-61.39 PV;

10 N; CPT I/Y = 5

  • For N 100 FV; +/-61.39 PV; 0 PMT;

5 I/Y; CPT N = 10 years

pv decreases if
PV Decreases If
  • Number of compounding periods (time) increases,
  • The discount rate increases,

And/or

3. Compounding frequency increases

annuities
Annuities
  • Series of equal dollar payments
    • Usually at the end of the year/period
  • If I deposit $100 in the bank each year starting a year from now, how much will I have at the end of three years if I earn 6%? $318.36
  • We are solving for the FV of the series by summing FV of each payment.
fv of 100 annuity @ 6
FV of $100 Annuity @ 6%

End of

PMTFVIF $

Year 3 $100 1.0000 * $100.00

Year 2 100 1.0600 106.00

Year 1 100 1.1236 112.36

$318.36

* The payment at end Year 3 earns nothing

annuity keystrokes
Annuity Keystrokes

What will I have if deposit $100 per year starting at the end of the year for three years and earn 6%?

0 PV; 100+/- PMT; 3 N; 6 I/Y;

CPT FV = 318.36

PV is zero - nothing in the bank today

present value of an annuity
Present Value of an Annuity

Amount we must put in bank today to

withdraw $500 at end of next three years, earn 6% and have nothing left at the end?

Present valuing each of three payments

Keystrokes: 500+/- PMT; 0 FV; 3 N;

6 I/Y; CPT PV = 1,336.51

nonannual compounding
Nonannual Compounding
  • Invest for ten years at 12% compounded quarterly. What are we really doing?
    • Investing for 40 periods (10 * 4) at 3% (12%/4)
  • Make sure 2nd I/Y is set to 1.
  • Need to adjust rate per period downward which is offset by increase in N
nonannual compounding1
Nonannual Compounding
  • FVn = PV ( 1 + i/m) m * n
  • m = number of compounding periods per year so per period rate is i/m
  • And m * n is the number of years times the compounding frequency which adjusts to the rate per period
compounding 100 @10
Compounding $100 @10%

CompoundingOne Year10 Years

Annually $110.00 $259.37

Semiannually 110.25 265.33

Quarterly 110.38 268.51

Monthly 110.47 270.70

amortizing loans
Amortizing Loans
  • Paid off in equal installments
    • Makes it an annuity
  • Payment pays interest first, remainder goes to principal (which declines)
  • $600 loan at 15% over four years with equal annual payments of $210.16
600 loan amortization
$600 Loan Amortization

TotalTo IntTo PrinEnd Bal

Year 1 210.16 90.00 120.16 479.84

Year 2 210.16 71.98 138.18 341.66

Year 3 210.16 51.25 158.91 182.75

Year 4 210.16 27.41 182.75 0

calculate a loan payment
Calculate a Loan Payment
  • $8,000 car loan payable monthly over three years at 12%. What is your payment?

How many monthly periods in 3 yrs? 36 N

Monthly rate? 12%/12 = 1%/mo = I/Y

What is FV? Zero because loan paid out

8000+/- PV; 0 FV; 1.0 I/Y; 36 N;

CPT PMT=265.71

perpetuities
Perpetuities
  • Equal payments that continue forever
    • Like Energizer Bunny and preferred stock
  • Present Value = Payment Amount

Interest Rate

Preferred stock pays $8/yr, int rate- 10%

Payment fixed at $8/ .10 = $80 market price

npv irr uneven cash flows
NPV & IRR Uneven Cash Flows
  • Occur frequently in business problems
  • All we are doing is present valuing each cash flow, positive or negative
  • Need to switch to CF mode in calculator
    • Keystrokes in handout and on web page
    • Question on final but not FinCoach
  • Be sure to read 4-12 to 4-14 in manual and/or Table X in Appendix A of text
cash flow time line
Cash Flow Time Line

Understanding time – big problem

Remember number line from algebra

Visualize in picture form when each cash flow occurs by time period, amount and sign.

Time 0____1____2____3____4____5___→

Flows +200 +300 +50 →

-100 0 0

keystrokes you should know
Keystrokes You Should Know
  • Future value of a single payment
  • Present value for the same
  • Future value of an annuity
  • Annuity’s present value
  • Loans including monthly payments, effective rates and time to repay
  • Present value of a perpetuity
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