Financial Management:

Financial Arithmetic

My T. Nguyen
Fulbright University Vietnam

January 2025

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Shareholder value

Shareholder value or shareholder wealth is a number today that we

want to maximise

How can we obtain that number

Forecast the cash flows in the future generated by a particular project

Turn these future cash flow forecasts into a single number

This process is called discounting

But in order to understand discounting, we should start with a

process called compounding

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Compounding
You invest $1 today at the interest rate of 10% per year, compounded
annually

Today, you put $1 in your saving account

After 1 year, you get $1 × (1+0.10) =$1.10

1050
After 2 years, you get $1 × (1+0.10) × (1+0.10)=
$1 × (1+0.05)2 = $1.21

After 3 years, you get $1 × (1+0.10)

× (1+0.10) × (1+0.10)= $1 ×
(1+0.05)3 = $1.33

After 4 years, you get $1× (1+0.10) ×

(1+0.10) × (1+0.10) × (1+0.10) = $1 ×
(1+0.10)4 = $1.4641

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Compounding in general formula

In general, $1 invested today for a period of N years at interest rate

R, compounded annually, will grow to:

$1 × (1 + R)N

If compounded M times per year at regular intervals

R M ×N


$1 × 1 + M

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Quiz

How much will you get back in a 6 years if you invest 50,000,000 VND
today at the interest rate of 4% per annum, compounded annually?
A. 52,000,000VND
B. 63,265,950VND
C. 63,486,732VND
D. 64,109,256VND

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Quiz

How much will you get back in a 6 years if you invest 50,000,000 VND
today at the interest rate of 4% per annum, compounded quarterly?
A. 63,412,090VND
B. 63,265,951VND
C. 63,486,732VND
D. 64,109,256VND

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Compounding and discounting

What we just did is compounding: Present value ⇒ Future value

Discounting is flipping the coin: Future value ⇒ Present value

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Discount Factors and Present Values

Conversely, in order to end up with $1 at the end of N years, today

we need to invest only:
1
$1 × (1+R)N

R is the discount rate or the cost of capital

1
(1+R)N is the discount factor

This is the present value of $1 that we expect to receive N years from

now.

The discount factor that multiplies $1 in the above formula reflects

the time value of money:

$1 expected N years from now is not as valuable as $1 today

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Discounted Cash Flows

Company invests $1,000 today in machinery that is expected to

generate incremental (i.e. additional) cash flows of $300 at the end of
each Years 1-5. Assume that the discount rate is 10% per year:

Present value of expected future cash flows:

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Net Present Value

Net Present Value of project equals:

NPV = 1,137.24 - 1,000 = 137.24

If capital is not rationed, accept all projects with

NPV > 0

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Quiz

Company invests $1,200 today in a project that is expected to

generate incremental (i.e. additional) cash flows of $300 at the end of
Year 1, $400 at the end of Year 2, $500 at the end of Year 3 .
Assume that the cost of capital is 5% per year. What is the present
value of expected future cashflows?
A. 1,000
B. 1,060
C. 1,080
D. -119

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Annuities
An annuity is a stream of N equal cash flows C:

The annuity factor AN,R is the sum of the corresponding N

present-value factors, calculated using R as the discount rate:

The present value of an annuity is the product of annual cash flow C

and the annuity factor AR,T :

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Annuities

Company invests $1,000 today in machinery that is expected to

generate incremental (i.e. additional) cash flows of $300 at the end of
each Years 1-5. Assume that the discount rate is 10% per year:

Present value of expected future cash flows:

 
1 1
P V = 300 ∗ · 1− = 1, 137.24 (1)
0.1 (1 + 0.1)5

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Annuities

Examples of annuities in real life:

Bonds
Pension payment

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Perpetuities

A perpetuity is an infinite stream of equal future cash flows C:

The present value of a perpetuity is obtained by letting N → ∞ in

the annuity formula:
C
PV = R

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Quiz

How much more is a perpetuity of $1,000 worth than an annuity of

the same amount for 20 years? Assume an interest rate of 10% and
cash flows at the end of each period.

A. $297.29
B. $1,486.44
C. $1,635.08
D. $2,000.00

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Growing Perpetuities

Suppose expected future cash flows grow indefinitely at constant rate

G:

Present value of stream of expected future cash flows is given by

Gordon growth model:
C
PV = , R>G (2)
R−G

where C is cash flow exactly one period from now

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Growing Annuities
Now the expected future cash flows grow at constant rate G for N
periods only:

Present value of the growing annuity equals the difference between

two growing perpetuities, staggered in time by N periods:
"  #
C · (1 + G)N 1+G N

C 1 C
PV = − × = · 1−
R − G (1 + R)N R−G R−G 1+R
(3)
N.B. We recover the annuity formula when G=0

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Important notes about annuities and
perpetuities formula

Formulas you need to memorise:

Present value of perpetuities
Present value of growing perpetuities

Formulas I will give you on the test and exam front sheet:
Annuity factor
Present value of growing annuities

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Important notes about annuities and
perpetuities formula

Annuities and perpetuities formulas are applicable for finding PV with

cashflows stream starting 1 period from now

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Tricky cases

Delayed annuity: cashflows starting later than 1 period from now

Annuity due: cashflows starting now

Infrequent annuities

Equating the present value of two annuities

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Example: Delayed Annuity

Dealayed annuity happens when cashflow starts later than 1 period

out into the future

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Delayed Annuity

Grace will receive a four-year annuity of €500 per year, beginning at

date 6. If the interest rate is 10 percent, what is the present value of
your annuity? How do you do it?

1.
2.

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Delayed Annuity

Step 1: Discount annuity to year 5

1
" #
1− (1.10)4
500 = 1, 584.95
0.10 (4)

Step 2: Discount year 5 value back to year 0

1, 584.95
= 984.13 (5)
(1.10)5

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Annuity Due

Tom receives $50,000 a year for 20 years from a competition. Assume

that the first payment occurs immediately and that the discount rate
is 8 percent. What is the value of the prize?

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Infrequent Annuities

You receive an annuity of $450, payable once every two years. The
annuity stretches out over 20 years. The first payment occurs at date
2 that is, two years from today. The annual interest rate is 6 per cent.

Step 1: Find correct interest rate (2 - year interest rate)

(1.06 ×1.06) − 1 = 12.36%

Step 2: Discount using the correct number of periods and interest

rate (10 periods and 12.36%)

P V = 450 × A10
0.126 = 2, 505.57

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Equating the Present Value of Two Annuities

William and Kate are saving for the university education of their
newborn daughter, Susan. They estimate that university expenses will
be $30,000 per year when their daughter reaches university in 18
years. The annual interest rate over the next few decades will be 14
percent. How much money must they deposit in the bank each year
so that their daughter will be completely supported through four years
of university?

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Equating the Present Value of Two Annuities

Three steps:
Step 1: Calculate the Year 17 Value of the University payments
Step 2: Calculate the Year 0 value of the university payments
Step 3: Calculate the cash flow that equates the year 1 and 17
payments to the year 0 value of the university payments

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Financial applications

Some applications of annuities and perpetuities in Finance:

Bond valuation

Firm’s share price

Firm’s valuation

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Bonds

A bond is a loan instrument that typically pays a fixed coupon payment C (a

fixed percentage of the face value of the loan) at regular intervals until the
loan expires, at which point it also repays the face value (FV):
+ FV

Present value of the loan is:

C C C C FV
PV = 1+R + (1+R)2 + (1+R)3 + ... + (1+R)N
+ (1+R)N

FV
PV = C × AN
R + (1+R)N

We can then compare the PV with its market value

If PV is greater than the market price: bond is traded cheaply
If PV is less than the market price: bond is traded expensive

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Quiz

What is the present value of a 5-year bond with face value of $1000,
coupon rate of 8%. The discount rate is 5%.

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Quiz

A corporation has promised to pay $1,000 20 years from today for

each bond sold now. No interest will be paid on the bonds during the
20 years, and the bonds are discounted at an interest rate of 7%,
compounded semiannually. Approximately how much should an
investor pay for each bond?

A. $70
B. $252.57
C. $629.56
D. $857.43

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Share Price Valuation

We can use the growing perpetuity formula to derive the share price
of a firm, because in principle firms can exist forever.

Example: A firm pays dividends per share that start at $10 next year
and grow at g=5% perpetually. The discount rate is 10%

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Firm Valuation

We can use the growing perpetuity formula to value a firm.

Example: A firm has profits of $1M next year, which grow perpetually
at 2% per year thereafter. The discount rate is 8%

Using the growing perpetuity formula, the value of the firm is:

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