2 Time Value of Money

ĐẠI HỌC HOA SENKhoa Kinh tế Thương mại

KHOA KINH TẾ THƯƠNG MẠI

CORPORATE

FINANCEThS. Nguyễn Tường Minh

Email: minh.nguyentuong@yahoo.com.vn

CORPORATE FINANCE

CHAPTER 2

THE TIME VALUE OF MONEY

References

• Fundamentals of Corporate Finance, Brealey et al., McGraw Hill, 5th edition, USA, 2007.

• Foundation of Financial Management, Block & Hirt, McGraw Hill, 12th edition,USA, 2008.

• Other relevant materials.

Chapter 2: THE TIME VALUE OF MONEY

• Main Contents:

1. Future values and Compound interest

2. Present values

3. Multiple cash flow

4. Level cash flow: Perpetuities and Annuities

5. Inflation and the time value of money

6. Effective annual interest rate

I. Future values and Compound interest

Interest = interest rate x initial investment

Capital after the 1st year = initial investment x (1 + interest rate)

Capital after the 2nd year = capital after the 1st year x (1 + interest rate)= initial investment x (1 + interest rate)2

Capital after the t year = initial investment x (1 + interest rate)t

Future value

Present value

I. Future values and Compound interest (cont’d)

Future value after the t year = Present value x (1 + interest rate)t

0 1 2 3

Saving

Present value

r = 6% $106 $112.36 $119.10

+ $6 + $6.36 + $6.74

Future valueWhy do the interest after each year higher

than the privous ones ?

…because interests are calculated based on original investment and its interest of previous years

I. Future values and Compound interest (cont’d)

I. Future values and Compound interest (cont’d)

I. Future values and Compound interest (cont’d)

�Compound interest

Interest= + x

Original investment

Accumulated interest

over periods

…earning interest on interest

�Simple interest

Interest= + x

Original investment

Accumulated interest

over periods

…interest only from the original investment

I. Future values and Compound interest (cont’d)

�Do you know ???

MANHATTAN Island

Peter Minuit

1626, bought with 24$

??? How much equivalent in 2006 value ?

The average standard of interest rate is 3.5%

24(1+3.5%)380 = 11,427,000 USD

outstanding successful deal!!!

I. Future values and Compound interest (cont’d)

II. Present Values

Now!!!!offered$100,000

At the year-end!!

offered$100,000

0 1 2 3 4 5 tTime

Time value of money

•A dollar today is worth more than 1 dollar tomorrow

II. Present Values (cont’d)

0 1 2 3 t

Time

Receiving value

(Future Value)

Original investment(Present

Value) Int 1 Int 2 Int 3

+ ++

( )

( )t

t

rFVPV

rPVFV

+=

+=

1

1

II. Present Values (cont’d)

•How much do we need to invest now in order to produce $106 at the end of the year with interest rate of 6% ?

( )100$%)61(

1061 1 =

+=

+= tr

FVPV

II. Present Values (cont’d)

•Which strategy should he select ?

Strategy 1:Save money in 1 year, interest rate 8%

Strategy 2:Save money in 2 year, interest rate 8%

$3,000

$2,600

suggestion•Calculate PV of each strategy, and compare with his available fund If PV < available fund: select the strategy

II. Present Values (cont’d)

The longer the time before you must make a payment, the less you need to invest today

II. Present Values (cont’d)

�Discount factor

trFVPV )1(

1+

=

Discount factor To measure the PV of $1 received in year n

II. Present Values (cont’d)

�Finding the value of free credit

•Down payment: $8,000

•The 2nd pay out: $12,000Free credit provider

No free credit prividedDiscount $1,000

•Which company should you select ?

•PV = 20,000 – 1,000 = $19,000

Choose Toyota for cheaper purchasing

$20,000

II. Present Values (cont’d)

�Finding the interest rate

issue

•Repay $1,000

•…paid at the end of 25 years

•Price of IOU: $129.20

•How much is the interest rate ?

...)1(

1

=⇒

+=

rr

FVPV t

III. Multiple Cash Flow

A single cash flow

( )

( )t

t

rFVPV

rPVFV

+=

+=

1

1

Now, we calculate the FV, PV of a Multiple Cash Flow…

Single CF1

Single CF4

Single CF3

Single CF2

Multiple CF

III. Multiple Cash Flow (cont’d)

�Future Value of multiple cash flow

2 years later

•Year 1: deposit $1,200

•r = 8%

•Year 2: deposit $1,400

•How much will he spend on a laptop after 2 years ?

III. Multiple Cash Flow (cont’d)

III. Multiple Cash Flow (cont’d)

�Present Value of multiple cash flow

drawing 2 strategies

Pay $15,500 at once(deducted $500)

Installment plan

•Down payment: $8,000

•Year 1: $4,000

•Year 2: $4,000

$16,000

•Which strategy should be chosen ?

1

2

r = 8%

III. Multiple Cash Flow (cont’d)

�Present Value of multiple cash flow (cont’d)

< $15,500

The strategy 2 of installment plan should be chosen

III. Multiple Cash Flow (cont’d)

�Present Value of multiple cash flow (cont’d)

Characteristics of PV of a stream of future cash flows

…is the amount that needs to be invested today to generate the stream of future cash flows.

Total future cash flow: - $16,000

Available cash: $15,133.06

Total of PV of future cash flow = available cash = $15,133.06

Don’t worry

III. Multiple Cash Flow (cont’d)

�Present Value of multiple cash flow (cont’d)

…to prove this:

IV. Level Cash flows: Perpetuity and Annuity

Iphone5

0 1 2 3 4

$x $x $x $x

Annuity

0 1 2 3 4

$x $x $x $x

Perpetuity

….

….

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

Perpetuity

…sequence of equal cash flows that never end

�What is an annuity and a perpetuity ?

Annuity

…sequence of equal cash flow with a determined last period

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�How to value perpetuity

Bank of England

issue

Consols

0 1 2 3 4

$10 $10 $10 $10 ….

….

Market interest rate: 10%

Cash flow of 1 Consol

Value of the consol = PV of the endless cash flow

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�How to value perpetuity (cont’d)

0 1 2 3 4

$10 $10 $10 $10 ….

….

Market interest rate: 10%

Cash flow of 1 Consol

Cash payment from perpetuity = interest rate x PV

C = r x PV

rCPV = PV of 1 consol = 10/10% = $100

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�How to value perpetuity (cont’d)

How much is the amount that the man must set aside today ?

000,000,1$%10000,100

===rCPV

Generous man

Endow in finance

$100,000 per year, forever

r= 10%

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�How to value perpetuity (cont’d)

80.314,751$%)101(1

%10000,100

)1(1

3 =+

=+

= xr

xrCPV t

Generous man

0 1 2 3 4

$10 ….

….

Market interest rate: 10%

5

$10

PV

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�How to value perpetuity (cont’d)

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�How to value annuities

+−=⇔

+−=

−

rrCPV

rrrCPV

t

t

)1(1)1(

11Annuity factor

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�How to value annuities (cont’d)

41.947,9$%10%)101(14000)1(1 3

=+−

=

+−=

−−

rrCPV

t

Kangaroo Autos offer a payment scheme of $4,000 a year at the end of each of the next 3 years, r = 10%

0 1 2 3

$4,000 $4,000 $4,000

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�How to value annuities (cont’d)

miorrCPV

t

6.152$%9.5%)9.51(1828.11)1(1 25

=+−

=

+−=

−−

0 1 2 25

$11.828

Lottery winner of $295.7 mio

•Receive equally installments each year: $11.828 mio.•Total year: 25.•Interest rate: 5.9%

$11.828 $11.828

…

…

It is a fair trade ? - No!!!

What is a solution ? -Lottery winner receives the down payment of $143.1 mio

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�How to value annuities (cont’d)

bioC

rrCPV

t48.4$

%9%)91(1

46

)1(1 25 =+−

=⇔

+−=

−

−

0 1 2 30

$?

Bill Gatesthe richest man of $46 bio

•If he could live more 30 years, how much could Bill Gates spend yearly as taking his $46bio ?•His money is invested to earn 9%.

$? $?

…

…PV = $46 bio

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�How to value annuities (cont’d)

•Lending 80% of the cottage price

Price: $125,000

Pay down 20%

•r= 1% per month

•t= 30 years

0 1 2 360

$? $? $?

…

…PV = $100,000

What is the monthly mortgage payment ?

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

61.028,1$%1%)11(1000,100

)1(1

360 =+−

=⇔

+−=

−

−

C

rrCPV

t�How to value annuities (cont’d)

$99,971.39

$99,942.49

$1,028.61

$1,028.61

$28.61

$28.9

$1,000

$999.71

$100,000

$99,971.39

1

2

…

360

End of month

balance

Month-end

payment

Amortization of loan

InterestBeginning of month balance

Months of repayment

DETAIL OF THE MONTHLY DEBT PAYMENT

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�How to value annuities (cont’d)

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�Annuities Due

)1()1(1 rrrC

t+

+−−

…value of a stream of cash payments starts immediately (at the beginning of a period).

PV of an annuities due =

−+

rrC

t 1)1(FV of an annuity =

�Future value of an annuity

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�Future value of an annuity (cont’d)

0 1 2 3

$3,000

4

$3,000 $3,000 $3,000

$13,000

Can you buy this red car at the end of year 4 ?

r= 8%

34.518,13$%81%)81(30001)1( 4

=−+

=

−+

rrC

tFV of an annuity =

You can buy the red car

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�Future value of an annuity (cont’d)

How much could she save each year from this year ?

59.429$%10

1%)101(000,500

1)1(

50 =−+

=⇒

−+

C

rrC

tFV of an annuity =

…will be retired

$500,000

…in 50 more years

r= 10%

IV. Level Cash flows: Perpetuity and Annuity (cont’d)

�Annuities due )1(1)1( rrrC

t+

−+FV of an annuities due =

If she save the money at the beginning of each year, how much should she deposit ?

…will be retired

$500,000

…in 50 more years

r= 10%

C = ???Compare outcome with the previous FV annuity,

any conclude about this?

V. INFLATION AND THE TIME VALUE OF MONEY

Investment return6%

Inflation10%

…value of money is eroded

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Real versus Nominal Cash flow

…CPI used for measuring the inflation rate.

What can be used for measuring the inflation rate ?

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Real versus Nominal Cash flow (cont’d)

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Real versus Nominal Cash flow (cont’d)

…refer to the actual number of dollars

What is the nominal dollar ?

…refer to the amount of purchasing power

What is the real dollar ?

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Real versus Nominal Cash flow (cont’d)

buy

provide loanPay monthly

$800 for 30 years

In 1990

190.32011

133.81990

CPIYear

??? What is the real monthly payment of 2011 compared with real 1990 dollar ?

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Real versus Nominal Cash flow (cont’d)

??? What is the real monthly payment of 2011 compared with real 1990 dollar ?

CPI increased (1990/2011): …………

Real payment of 2004 compared with 1990:……

What do you think the real amount paid in 2011 with that in 1990 ?

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Inflation and interest rate

rate inflation1

rate interest normal1 rate interest Real1

rate inflation1

rate) interest normal(1investment investment of FV Real

+

+=+

+

+=

…rate at which money invested growths

What is the nominal interest rate ?

…interest rate board of commercial banks

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Inflation and interest rate (cont’d)

…invest to earn interest rate: 6%

…simultaneously, reduce the income with inflation rate of 2%

…how much is the real interest rate ?

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Inflation and interest rate (cont’d)

In reality, if nominal interest rate and inflation rate are small, the real interest rate will be…

Attention!!!

Real interest rate = nominal interest rate – inflation rate

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Inflation and interest rate (cont’d)

…compare the nominal and real values of investment under the inflation rate of 7% and nominal interest rate of 10%

$90.91$90.91PV

$93.46$100FV (after 1 year)

2.8%10%Interest rate

RealNominal

Nominal PV and Real PV are equal to each other

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Inflation and interest rate (cont’d)

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Inflation and interest rate (cont’d)

His total assets: $46 bio.

Spend $4.5 bio per year, in 30 years

•Interest rate = 9%•Inflation rate = 5%

He would like to ensure the same

power of purchasing of

2042 as in 2012

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Inflation and interest rate (cont’d)

He would like to ensure the same

power of purchasing of

2042 as in 2012

Spend less in 2012 and then increase expenditure in line with inflation

Solution

•Real interest rate: …………….

•Annual spending in 2012:………………….

V. INFLATION AND THE TIME VALUE OF MONEY (cont’d)

�Inflation and interest rate (cont’d)

VI. EFFECTIVE ANNUAL INTEREST RATE

Borrow $100

Interest 1% per month

Putting off the payment up to 1 year Total payment after 1 year:

100(1+1%)12= $112.68

VI. EFFECTIVE ANNUAL INTEREST RATE (cont’d)

How much is the equivalent interest rate?

$100 $112.68

0 1

•12.68%

Effective annual interest rate

1 + effective annual interest rate = (1 + monthly rate)12

VI. EFFECTIVE ANNUAL INTEREST RATE (cont’d)

�Method to convert to effective annual interest rate from an annual percentage rates (APRs)

•APRs: annualized by multiplying the rate per period by the number of period in a year.

Steps to convert to effective annual interest rate

1 Take the quoted APR divided by the number of compounded period in a year

•Monthly interest: APR / 12

•Quarterly interest: APR / 4

•Semi-annually interest: APR / 2

VI. EFFECTIVE ANNUAL INTEREST RATE (cont’d)

Steps to convert to effective annual interest rate (cont’d)

2 Convert to effective annual interest rate

1 + effective annual interest rate = (1 + monthly rate)12

1 + effective annual interest rate = (1 + quarterly rate)4

1 + effective annual interest rate = (1 + semi-annually rate)2

VI. EFFECTIVE ANNUAL INTEREST RATE (cont’d)

Why do we use the effective annual interest rate ?

•To measure the actual income of the depositors or expense of the borrowers

LAST SELF TESTA car loan requiring quarterly payments carries an ARP of 8 percent.

What is the effective annual rate of interest ?

Thank you for your attention !