Time Value of Money (CH 4)

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Time Value of Money (CH 4). TIP If you do not understand something, ask me!. Future value Present value Annuities Interest rates. Last week. Objective of the firm Business forms Agency conflicts Capital budgeting decision and capital structure decision.

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### Time Value of Money (CH 4)

TIP

If you do not understand

something,

Future value

Present value

Annuities

Interest rates

Last week
• Objective of the firm
• Agency conflicts
• Capital budgeting decision and capital structure decision
The plan of the lecture
• Time value of money concepts
• present value (PV)
• discount rate/interest rate (r)
• Formulae for calculating PV of
• perpetuity
• annuity
• Interest compounding
• How to use a financial calculator
Financial choices
• Which would you rather receive today?
• TRL 1,000,000,000 ( one billion Turkish lira )
• USD 652.72 ( U.S. dollars )
• Both payments are absolutely guaranteed.
• What do we do?
Financial choices
• We need to compare “apples to apples” - this means we need to get the TRL:USD exchange rate
• From www.bloomberg.com we can see:
• USD 1 = TRL 1,637,600
• Therefore TRL 1bn = USD 610.64
Financial choices with time
• Which would you rather receive?
• \$1000 today
• \$1200 in one year
• Both payments have no risk, that is,
• there is 100% probability that you will be paid
Financial choices with time
• Why is it hard to compare ?
• \$1000 today
• \$1200 in one year
• This is not an “apples to apples” comparison. They have different units
• \$1000 today is different from \$1000 in one year
• Why?
• A cash flow is time-dated money
• It has a money unit such as USD or TRL
• It has a date indicating when to receive money
Present value
• To have an “apple to apple” comparison, we
• convert future payments to the present values
• or convert present payments to the future values
• This is like converting money in TRL to money in USD
Some terms
• Finding the present value of some future cash flows is called discounting.
• Finding the future value of some current cash flows is called compounding.

0

1

2

3

10%

100

FV = ?

What is the future value (FV) of an initial \$100 after 3 years, if i = 10%?
• Finding the FV of a cash flow or series of cash flows is called compounding.
• FV can be solved by using the arithmetic, financial calculator, and spreadsheet methods.
Solving for FV:The arithmetic method
• After 1 year:
• FV1 = c ( 1 + i ) = \$100 (1.10) = \$110.00
• After 2 years:
• FV2 = c (1+i)(1+i)= \$100 (1.10)2 =\$121.00
• After 3 years:
• FV3 = c ( 1 + i )3 = \$100 (1.10)3 =\$133.10
• After n years (general case):
• FVn = C ( 1 + i )n
Set up the Texas instrument
• 2nd, “FORMAT”, set “DEC=9”, ENTER
• 2nd, “FORMAT”, move “↓” several times, make sure you see “AOS”, not “Chn”.
• 2nd, “P/Y”, set to “P/Y=1”
• 2nd, “BGN”, set to “END”
• P/Y=periods per year,
• END=cashflow happens end of periods
Solving for FV:The calculator method
• Solves the general FV equation.
• Requires 4 inputs into calculator, and it will solve for the fifth.

3

10

-100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

133.10

• 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 worth.

0

1

2

3

10%

PV = ?

100

Solving for PV:The arithmetic method
• i: interest rate, or discount rate
• PV = C / ( 1 + i )n
• PV = C / ( 1 + i )3

= \$100 / ( 1.10 )3

= \$75.13

Solving for PV:The calculator method
• Exactly like solving for FV, except we have different input information and are solving for a different variable.

3

10

0

100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-75.13

Solving for N:If your investment earns interest of 20% per year, how long before your investments double?

20

-1

0

2

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

3.8

Solving for i:What interest rate would cause \$100 to grow to \$125.97 in 3 years?

3

-100

0

125.97

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

8

Ordinary Annuity

0

1

2

3

i%

PMT

PMT

PMT

Annuity Due

0

1

2

3

i%

PMT

PMT

PMT

ordinary annuity and annuity due
Value an ordinary annuity
• Here C is each cash payment
• n is number of payments
• If you’d like to know how to get the formula below, see me after class.
Example
• you win the \$1million dollar lottery! but wait, you will actually get paid \$50,000 per year for the next 20 years if the discount rate is a constant 7% and the first payment will be in one year, how much have you actually won?
Solving for FV:3-year ordinary annuity of \$100 at 10%
• \$100 payments occur at the end of each period. Note that PV is set to 0 when you try to get FV.

3

10

0

-100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

331

Solving for PV:3-year ordinary annuity of \$100 at 10%
• \$100 payments still occur at the end of each period. FV is now set to 0.

3

10

100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-248.69

Solving for FV:3-year annuity due of \$100 at 10%
• \$100 payments occur at the beginning of each period.
• FVAdue= FVAord(1+i) = \$331(1.10) = \$364.10.
• Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity:

BEGIN

3

10

0

-100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

364.10

Solving for PV:3-year annuity due of \$100 at 10%
• \$100 payments occur at the beginning of each period.
• PVAdue= PVAord(1+I) = \$248.69(1.10) = \$273.55.
• Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity:

BEGIN

3

10

100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-273.55

• Be sure your financial calculator is set back to END mode and solve for PV:
• N = 5, I/YR = 10, PMT = 100, FV = 0.
• PV = \$379.08
• 10-year annuity
• N = 10, I/YR = 10, PMT = 100, FV = 0; solve for PV = \$614.46.
• 25-year annuity
• N = 25, I/YR = 10, PMT = 100, FV = 0; solve for PV = \$907.70.
• Perpetuity (N=infinite)
• PV = PMT / i = \$100/0.1 = \$1,000.

What is the present value of a four-year annuity of \$100 per year that makes its first payment two years from today if the discount rate is 9%?

\$297.22

\$323.97

\$100 \$100 \$100 \$100

0 1 2 3 4 5

4

0

1

2

3

10%

100

300

300

-50

90.91

247.93

225.39

-34.15

530.08 = PV

What is the PV of this uneven cash flow stream?
Solving for PV:Uneven cash flow stream
• Input cash flows in the calculator’s “CF” register:
• CF0 = 0
• CF1 = 100
• CF2 = 300
• CF3 = 300
• CF4 = -50
• Enter I/YR = 10, press NPV button to get NPV = \$530.09. (Here NPV = PV.)
Detailed steps (Texas Instrument calculator)
• To clear historical data:
• CF, 2nd ,CE/C
• To get PV:
• CF ,↓,100 , Enter , ↓,↓ ,300 , Enter, ↓,2,

Enter, ↓, 50, +/-,Enter, ↓,NPV,10,Enter, ↓,CPT

• “NPV=530.0867427”
The Power of Compound Interest

A 20-year-old student wants to start saving for retirement. She plans to save \$3 a day. Every day, she puts \$3 in her drawer. At the end of the year, she invests the accumulated savings (\$1,095=\$3*365) in an online stock account. The stock account has an expected annual return of 12%.

How much money will she have when she is 65 years old?

Solving for FV:Savings problem
• If she begins saving today, and sticks to her plan, she will have \$1,487,261.89 when she is 65.

45

12

0

-1095

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

1,487,262

Solving for FV:Savings problem, if you wait until you are 40 years old to start
• If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have \$146,000.59 at age 65. This is \$1.3 million less than if starting at age 20.
• Lesson: It pays to start saving early.

25

12

0

-1095

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

146,001

0

1

2

3

10%

100

133.10

0

1

2

3

4

5

6

0

1

2

3

5%

100

134.01

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: FV3 = \$100(1.10)3 = \$133.10

Semiannually: FV6 = \$100(1.05)6 = \$134.01

Classifications of interest rates
• 1. Nominal rate (iNOM) – also called the APR,quoted rate, or stated rate. An annual rate that ignores compounding effects. Periods must also be given, e.g. 8% Quarterly.
• 2. Periodic rate (iPER) – amount of interest charged each period, e.g. monthly or quarterly.
• iPER = iNOM / m, where m is the number of compounding periods per year. e.g., m = 12 for monthly compounding.
Classifications of interest rates
• 3. Effective (or equivalent) annual rate (EAR, also called EFF, APY) : the annual rate of interest actually being earned, taking into account compounding.
• If the interest rate is compounded m times in a year, the effective annual interest rate is
Example, EAR for 10% semiannual investment
• EAR= ( 1 + 0.10 / 2 )2 – 1 = 10.25%
• An investor would be indifferent between an investment offering a 10.25% annual return, and one offering a 10% return compounded semiannually.

keys:

description:

Sets 2 payments per year

[↑] [C/Y=] 2 [ENTER]

[2nd] [ICONV]

Opens interest rate conversion menu

[↓][NOM=] 10 [ENTER]

Sets 10 APR.

[↓] [EFF=] [CPT]

10.25

EAR on a Financial Calculator

Texas Instruments BAII Plus

Why is it important to consider effective rates of return?
• An investment with monthly payments is different from one with quarterly payments.
• Must use EAR for comparisons.
• If iNOM=10%, then EAR for different compounding frequency:

Annual 10.00%

Quarterly 10.38%

Monthly 10.47%

Daily 10.52%

If interest is compounded more than once a year
• EAR (EFF, APY) will be greater than the nominal rate (APR).

1

2

3

0

1

2

3

4

5

6

5%

100

100

100

What’s the FV of a 3-year \$100 annuity, if the quoted interest rate is 10%, compounded semiannually?
• Payments occur annually, but compounding occurs every 6 months.
• Cannot use normal annuity valuation techniques.

1

2

3

0

1

2

3

4

5

6

5%

100

100

100

110.25

121.55

331.80

Method 1:Compound each cash flow

FV3 = \$100(1.05)4 + \$100(1.05)2 + \$100

FV3 = \$331.80

Method 2:Financial calculator
• Find the EAR and treat as an annuity.
• EAR = ( 1 + 0.10 / 2 )2 – 1 = 10.25%.

3

10.25

0

-100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

331.80

When is periodic rate used?
• iPER is often useful if cash flows occur several times in a year.
Exercise

You agree to lease a car for 4 years at \$300 per month. You are not required to pay any money up front or at the end of your agreement. If your discount rate is 0.5% per month, what is the cost of the lease?