CHAPTER 5

Time Value of Money

Overview

Future value

Present value

Annuities

Perpetuity

Uneven cash flow

Interest rates

Amortization

시간가치의 위력

Manhattan

면적: 22.96 mile2 (59.47km2=180만평)

가격: 2007.7.2에 450 Park Avenue의 9,135평이 $510 million에 매각

($1,589/feet²= $17,224/m²=$56,000/평)

=> $102 billion

역사: 1626.5.24 네덜란드 총독 Peter Minuite이 인디언들에게 60길드(=$1,000)에 매입

네덜란드 총독과 인디언, 둘 중 누가 투자 의사결정을 더 잘했나?

Time lines

Show the timing of cash flows.

Tick marks 각 기간 말을 의미. 즉 Time 0 는 오늘; Time 1 은 첫 기간 말 (년, 월 등등) or 두번째 기간의 초를 의미.

CF0 CF1 CF3 CF2

0 1 2 3

I%

Terminology

PV: Present value

FV: Future value

CF: Cash flow

r(I/YR or I): Interest rate

Int: Dollar of interest

PMT: Payment

N: Number of period

Future Value

Finding the FV of a cash flow or series of cash flows is called compounding.

FV can be solved by using the step-by-step (Formula), financial calculator, and table methods.

What is the future value (FV) of an initial $100 after 3 years, if r = 10%?

FV = ?

0 1 2 3

10%

100

Solving for FV: The Formula

After 1 year: FV1 = PV (1 + r) = $100 (1.10)

= $110.00 After 2 years:

FV2 = PV (1 + r)2 = $100 (1.10)2

=$121.00 After 3 years:

FV3 = PV (1 + r)3 = $100 (1.10)3

=$133.10 After N years (general case):

FVN = PV (1 + r)N = PV (FVIFr,n)

이자율수준과 미래가치

PV = ? 100

Present Value

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.

What is the present value (PV) of $100 due in 3 years, if r = 10%?

0 1 2 3

10%

Solving for PV: The formula

Solve the general FV equation for PV:

PV = FVN / (1 + r)N = FVN (PVIFr,n)

PV = FV3 / (1 + r)3

= $100 / (1.10)3

= $75.13

이자율(할인율)과 현재가치

Interest Rate

Solves the general FV equation for r.

What interest rate would cause $100 to grow to $125.97 in 3 years?

125.97 = 100(1 + r)3

TrCFV )1(0

Number of years

Solves the general FV equation for T.

If sales grow at 20% per year, how long before sales double?

2 = 1(1 +0.2)x

TrCFV )1(0

Solving by Spreadsheet

Use the following formulas for TVM calculations FV(rate,nper,pmt,pv)

PV(rate,nper,pmt,fv)

RATE(nper,pmt,pv,fv)

NPER(rate,pmt,pv,fv)

The formula icon is very useful when you can’t remember the exact formula

Click on the Excel icon to open a spreadsheet containing four different examples.

Annuity

Ordinary Annuity

PMT PMT PMT

0 1 2 3 i%

PMT PMT

0 1 2 3 i%

PMT

Annuity Due

A series of equal payments at fixed intervals for a specified number of periods

Solving for FVA: The formula

3-year ordinary annuity of $100 at 10%

FVA = 100(1.1)2 +100(1.1)1 +100(1.1)0 = 331

FVA = PMT(1 + r)N-1 +PMT(1 + r)N-2 + PMT(1 + r)N-3

+ PMT(1 + r)N-4 ……..+ PMT(1 + r)0

=

= PMT(FVIFAr,n)

I

1)(1 PMT

N

I

Solving for PVA: The formula

3-year ordinary annuity of $100 at 10%

PVA = 100/(1.1)1 +100/(1.1)2 +100/(1.1)3 = 248.69

PVA = PMT/(1 + r)1 +PMT/(1 + r)2 + PMT/(1 + r)3

+ PMT/(1 + r)4 ……..+ PMT/(1 + r)n

=

= PMT(PVIFAr,n)

)I(1

1-)(1 PMT

n

n

r

r

Annuity Due: Solving for FVA and PVA

Solving for FVA Now, $100 payments occur at the

beginning of each period.

FVAdue= FVAord(1+r) = $331(1.10) = $364.10

Solving for PVA PVAdue= PVAord(1+r) = $248.69(1.10) =

$273.55

Perpetuity

A stream of equal payments at fixed intervals expected to continue forever

What is the present value of a perpetuity $100 at 10%?

PV = PMT / r = $100/0.1 = $1,000.

Uneven cash flow

0

100

1

300

2

300

3 10%

-50

4

90.91

247.93

225.39

-34.15 530.08 = PV

Compounding

Definition

The arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added

Semiannually, quarterly, monthly, daily

FVn

N: number of years, M: periods per year

NMNOM ) M

r 1 ( PV

Solving for Compounding FVA

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.

0 1 2 3 4 5

100

5%

100 100

6

재미나는 예

하루 한 잔씩 마시는 4천원짜리 카페베네 아메리카노 커피 값을 30년간 복리로 저축한다면 ?

올림픽에서 금메달을 획득하면 월100만원씩 연금을 받는다. 이 연금의 현재가치는?

Loan amortization (할부)

Amortization: A loan that is repaid in equal payments over its life

Amortization use: home mortgages, auto loans, business loans, retirement plans, etc.

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

Amortization cont.

EXAMPLE: Construct an amortization schedule (table) for a $1,000, 10% annual rate loan with 3 equal payments

PV

PMT

10%

PMT PMT 1000

3

1t

t

3

3

2

2

1

1

0.1)(1

PMT

)1.0(1

PMT

)1.0(1

PMT

)1.0(1

PMT

t

Step 1: the annual payment

All input information is already given, just remember that the FV = 0 because the reason for amortizing the loan and making payments is to retire the loan.

INPUTS

OUTPUT

N I/YR PMT PV FV

3 10

402.11

0 -1000

Step 2: the interest and the principal paid in Year 1

The borrower will owe interest upon the initial balance at the end of the first year. Interest to be paid in the first year can be found by multiplying the beginning balance by the interest rate.

INTt = Beg balt (r) INT1 = $1,000 (0.10) = $100 If a payment of $402.11 was made at the end of the

first year and $100 was paid toward interest, the remaining value must represent the amount of principal repaid.

PRIN = PMT – INT = $402.11 - $100 = $302.11

Step 3: the ending balance after Year 1

To find the balance at the end of the period, subtract the amount paid toward principal from the beginning balance.

END BAL = BEG BAL – PRIN

= $1,000 - $302.11

= $697.89

Constructing an amortization table: Repeat steps 1 – 3 until end of loan

Interest paid declines with each payment as the balance declines.

Year BEG BAL PMT INT PRIN END BAL

1 $1,000 $402 $100 $302 $698

2 698 402 70 332 366

3 366 402 37 366 0

TOTAL 1,206.34 206.34 1,000 -

Illustrating an amortized payment: Where does the money go?

Constant payments.

Declining interest payments.

Declining balance.

$

0 1 2 3

402.11 Interest

302.11

Principal Payments

실제 생활에서 적용되는 예제

3,000만원짜리 차 구입시 1,700만원을 선수금으로 지급하고 나머지를 할부로 구입하려고 한다. 36개월, 연 9.5% 금리의 할부조건이라면 월 지급해야 하는 할부액은?

당신은 현재 35세이고 연간소득이 5,000만원이다. 앞으로 25년 동안 일을 하고 60세에 은퇴할 예정이며 90세까지 건강하게 살 것으로 예상한다. 은퇴 후에 소비하는 수준이 현재 소비하는 수준과 동일하게 되려면 연간 얼마를 저축하고 연간 얼마를 소비해야 하는가?