2.
Principles of Valuation
First
Value of financial securities = PV of expected
future cash flows
To
Principles:
value financial securities we need to:
Estimate future cash flows:
• Size (how much) and
• Timing (when)
Discount future cash flows at an appropriate
rate:
• The rate should be appropriate to the risk presented
by the financial security.
3.
Definition of a Bond
A bond is a legally binding agreement between a
borrower and a lender:
Specifies the principal amount of the loan (face
value or par value)
Specifies the size and timing of the cash flows
(coupon payments):
• In dollar terms (fixed-rate borrowing)
• As a formula (adjustable-rate or floating-rate borrowing)
4.
Example of a Bond
Consider a U.S. government bond listed as 6 3/8 of
December 2011.
The Par Value of the bond is $1,000.
Coupon payments are made semi-annually (June 30 and December
31 for this particular bond).
Since the coupon rate is 6 3/8 the payment is $31.875.
Principal repayment is on December 31, 2011.
On January 1, 2006 the size and timing of future cash flows are:
$31.875
$31.875
$31.875
$1,031.875
30 / 6 / 11
31 / 12 / 11
1 / 1 / 06
30 / 6 / 06
31 / 12 / 06
5.
How to Value Bonds
Identify
the size and timing of cash flows.
Discount at the correct discount rate.
If you know the price of a bond and the size and
timing of cash flows, then you can calculate the
discount rate
This discount rate that discounts future cash
flows to the current price is the Yield to Maturity
or YTM of the bond
6.
Pure Discount Bonds
Also called zero coupon bonds
Information needed for valuing pure discount bonds:
Time to maturity (T) = Maturity date - today’s date
Face value (F)
Discount rate (r)
$0
$0
$0
$F
T −1
T
0
1
2
Present value of a pure discount bond at time 0:
F
PV =
T
(1 + r )
7.
Pure Discount Bonds: Example
Find the value of a 30-year zero-coupon bond
with a $1,000 par value and a YTM of 6%.
$0
$0
$0
$1,000
0$ 0,1$ 0
0 1 30
229
0
1
2
29
F
$1,000
PV =
=
= $174.11
T
30
(1 + r )
(1.06)
30
8.
Level-Coupon Bonds
Information needed to value level-coupon bonds:
Coupon payment dates and time to maturity (T)
Coupon payment (C) per period and Face value (F)
Discount rate (r)
$C
$C
$C
$C + $ F
0
1
2
T −1
T
Value of a Level-coupon bond
= PV of coupon payment annuity + PV of face value
C
1
F
PV = 1 −
+
T
r (1 + r ) (1 + r )T
9.
Level-Coupon Bonds: Example
Find the present value (as of January 1, 2006), of a 6-3/8
coupon T-bond with semi-annual payments, and a maturity
date of December 2011 if the YTM is 5-percent.
On January 1, 2006 the size and timing of cash flows
are:
$31.875
$31.875
$31.875
$1,031.875
30 / 6 / 11
31 / 12 / 11
1 / 1 / 06
30 / 6 / 06
31 / 12 / 06
$1,000
$31.875
1
PV =
1 − (1.025)12 + (1.025)12 = $1,070.52
.05 2
10.
Duration
Duration of a Zero Coupon Bond
0
Maturity date
Duration
11.
Duration (contd.)
Duration of a Coupon Paying Bond
0
Duration
Maturity date
12.
Duration (contd.)
Measure
of bond price sensitivity to
interest rate risk
A bond with longer duration has higher
relative price change than one with shorter
duration when interest rate (YTM)
changes
13.
Bond Concepts
1.
Bond prices and market interest rates move in opposite
directions.
2.
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)
3.
A bond with longer maturity has higher relative price
change than one with shorter maturity when interest rate
(YTM) changes, all other features being identical.
4.
A lower coupon bond has a higher relative price change
than a higher coupon bond when YTM changes, all other
features being identical.
14.
YTM and Bond Value
When YTM < coupon, the bond
trades at a premium.
Bond Value
$1400
1300
1200
When YTM = coupon, the bond
trades at par.
1100
1000
800
0
0.01
0.02
0.03
0.04
0.05
0.06 0.07
6 3/8
0.08
0.09
0.1
YTM
When YTM > coupon, the bond trades at a discount.
15.
Bond Value
Maturity and Bond Price
Volatility
Consider two otherwise identical
bonds.
The long-maturity bond will have
much more volatility with respect to
changes in the discount rate
Par
Short Maturity Bond
C
YTM
Long Maturity Bond
16.
Bond Value
Coupon Rate and Bond Price
Volatility
Consider two otherwise identical bonds.
The low-coupon bond will have much more
volatility with respect to changes in the
discount rate (YTM)
High Coupon Bond
YTM
Low Coupon Bond
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