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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

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.

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

What is the present value (PV) of $100 received in 3 years, if i = 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

Now let’s study some interesting patterns of cash flows…

- Perpetuity
- Annuity

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

What is the present value of a 5-year $100 ordinary annuity at 10%?

- 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

What if it were a 10-year annuity? A 25-year annuity? A perpetuity?

- 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

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

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

What is the FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?

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.

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 CalculatorTexas 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).

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.

2

3

0

1

2

3

4

5

6

5%

100

100

100

110.25

121.55

331.80

Method 1:Compound each cash flowFV3 = $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?

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