Valuation
(Chapter 8)
Growing Perpetuity

Year
Cash
Is a perpetual stream of Cash Flows. The first
Cash Flow will start one period from now and
will occur with the same periodicity. The Cash
Flows will grow at a constant rate of g%
0
1
2
3
…
T
T+1
…
C
C(1+g)1
C(1+g)2
…
C(1+g)T-1
C(1+g)T
…
What is the Present Value of this stream of
Cash Flows if the interest rate is r%?

The Sum of the Present Value of the Cash
Flows in this Growing Perpetuity is just
the Sum of a Geometric Series of rate
(1+g)/(1+r). The Sum of such series is
C/(r-g)

For this Sum to work you need r>g.

Why?
Growing Annuity
Is a stream of Cash Flows. The first Cash Flows occurs
one period from now and the last will occur T periods
from now and occur with constant periodicity. The Cash
Flows grow at constant rate of g%.

Year
Cash
0
1
2
3
…
T
T+1
…
C
C(1+g)1
C(1+g)2
…
C(1+g)T-1
0
…
What is the Present Value of this stream of Cash Flows,
if the interest rate is r%?
Growing Annuity
C * GArT%, g %
T

1
1+ g  
=C*
 
1 − 
r − g   1 + r  
Delayed Annuity
It’s a stream of Cash Flows that is just like
an Annuity but starts t-periods later than a
regular Annuity with T Cash Flows.
 Things to look out for:

◦ It starts at time t+1
◦ It ends at time T+t-1

What is the Present Value of this stream of
Cash Flows?

This is how the Cash Flows look like
Year
1
…
t
t+1
…
T+t-1
T+t
Cash Flow
0
…
0
C
…
C
0
PV(Delayed Annuity in t-periods)=
PV(Annuity)/(1+r)t
The Present Value is the Present Value of a
regular Annuity, but Discounted t times
more!!!
This rule works also for a Perpetuity.
Annuity Due

This is the opposite of a Delayed Annuity. It is an
Annuity that has its first Cash Flow today. So, it
starts one period before than a regular Annuity,
which means t=0, and ends at T-1.

PV(Annuity Due)=PV(Annuity)*(1+r)

The Present Value of an Annuity Due is the
Present Value of a regular Annuity, only it is
Compounded once.

This rule also applies to a Perpetuity
Intrinsic value of an asset

Is the PRESENT VALUE of all the FUTURE INCOME
that the owner will RECEIVE
∞
P=∑
t =1
Income in Period t
(1+discount rate in period t )
t
FINANCIAL ASSETS: Stocks, Bonds, Bank Loans,
Lease Obligations
REAL ASSETS: Tangible (estate, machinery,
factories), Intangible (technical expertise, trademarks,
patents)
Bond

A fixed obligation debt security, issued by
governments, governmental agencies,
municipalities and corporations.

A promissory note that entitles its holder
to a series of COUPON (interest)
payments and the repayment of the FACE
(or par) VALUE at MATURITY.
Elements of a Bond

FACE/PAR VALUE: The amount of money
borrowed. Usually $1,000 per bond.

COUPON: Specified dollar amount to be
paid periodically (usually every six months)

COUPON RATE: Stated Annual Rate of the
Interest of the Bond

MATURITY: The date of the last payment
(the last coupon and the face value)
Types of Bond

PURE DISCOUNT BONDS: It promises only the
re-payment of the FACE VALUE at MATURITY.
AKA zero-coupon bonds, zeroes, deep discount
bonds, or original issue discount bonds (OIDs)
FV
P=
T
(1 + r )
Treasury Bills and principal-only Treasury
strips are good examples of zeroes.

LEVEL COUPON BONDS: It promises COUPON
payments in regular time intervals and the re-payment of
the FACE VALUE at MATURITY.
P
=
T
C
t =1
(1 + rt )
∑
t
+
FV
(1 + rT )
T
When the interest rate is constant , the
expression simplifies to
FV
P =C*A +
T
(1 + r )
T
r

CONSOLS: they represent perpetual COUPON
paying Bonds.
∞
P=∑
t =1
C
(1 + rt )
t
If the interest rate is constant
C
P=
r
Yield To Maturity





The yield to maturity is the required market
interest rate on the bond, the interest rate at
which the coupon payment is reinvested
Bond prices and market interest rates move in
opposite directions
When coupon rate = YTM, price = par value
When coupon rate > YTM, price > par value
(premium bond)
When coupon rate < YTM, price < par value
(discount bond)
Example 1

60
Example 1: 30-year maturity Bond, with 8%
semi-annual coupons and $1,000 Face Value,
has a YTM of 10%. What is the price of the
Bond?
$40
$1,000
+
= $757.18 + $53.54 = $810.71
∑
60
t
(1 + 0.05)
t =1 (1 + 0.05)
Example 2

Example 2: A Bond, currently selling for $950,
has 10 years before it matures and a 7% semiannual coupon. What is its YTM?
20
$35
t =1
(1 + x )
∑
t
+
$1, 000
(1 + x )
YTM
x=
2
20
$950
=

Solving by trial and error, x=3.86%.

This is a 6-month interest rate

YTM: 3.86%*2=7.72%

Effective Annual Yield: (1+0.0386)2 -1=7.869%
Return from a bond

The return to a Bond, and generally to any
financial asset, will have two components:
◦ Current Yield: The return coming from the
Coupon payments generated by the Bond
◦ Capital Gain/Loss: The return coming from
the difference in prices paid to buy and
received when selling the Bond
Return from a bond

If you hold the Bond just for the time it pays a
single coupon, the return to the Bond will be
C P1 − P0
R=
+
P0
P0
Where C is the Coupon, P0 is the Price you
paid to buy the Bond and P1 is the Price you
received when you sold the Bond

If you hold a Bond for a period of time that
allows for receiving several Coupons, the total
Holding Period Return depends on the rate of
return you can earn by re-investing the Coupons.

You have Re-Investment Risk!!

The Yield to Maturity assumes you can reinvest the Coupons at the Yield to
Maturity and you hold the Bond to
Maturity.

If this does not happen the return to the
Bond is usually not the YTM
Example

You bought a Par Bond with a Coupon
Rate of 8% and maturity of 10 years.

Six months later, the Yield to Maturity of
the Bond changed to 7% and you sold the
Bond.

What return did you get?

Step 1: Calculate new price
19
$40
$1,000
$1,068.55 = ∑
+
19
t
(1 + 0.035)
t =1 (1 + 0.035)
Calculate the Holding Period Return
$40 + ($1,068.55 − $1,000 )
= 10,85%
R=
$1,000
The return was 10.85% (semi-annual rate)
Practice Exercises (Chapter 8)
Easy: 2,3
 Intermediate: 13, 16,17
 Challenge: 23,25
