Pension Reform: Second Generation and
Implementation Issues in ECA
January 22-29, 2001 – Budapest, Hungary
Bond Analysis and Valuation:
Introductory Elements
Interest Rates and Term Structure
Dr. Martin Janssen
January 23
Bond Analysis and Valuation, Introductory
Elements: Overview
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Notion of a straight bond
Characteristics of other bonds
The discounted cash-flow approach
The yield to maturity
The term structure of interest rates
The risks of holding a bond
Credit risk and credit ratings
The concept of duration
Relative price changes and modified duration
Notion of A Straight Bond
• A straight bond is a tradable (negotiable) debt
instrument,
–
–
–
–
–
–
Issued by the borrower
For a fixed time period
During which a fixed interest rate is paid
To the owner of the bond
At a fixed interval until the bond is redeemed
And the principal amount is paid back
Characteristics of Other Bonds
• Under other bond definitions, there are many
exceptions to these rules, e.g.
– Bonds may be callable or perpetual
– There are zero-bonds, not paying interest rates during
the lifetime of the bond
– The right to get the interest rate payment (coupon) can
be separated from the bond („stripped“) and traded
separately
– The interest rate may be dependent upon another
interest rate (e.g. LIBOR)
– The bond may have other options tied to it than just
the right to call (e.g. on equity of another company)
Bond Valuation: the Discounted Cash-Flow
Approach (I)
• The time value of money
– Money, which can be spent today, is worth
more than money that can only be spent
tomorrow; this shows the existence of an
interest rate (irrespective of risk)
value of money today
= int erest rate per day
value of money tomorrow
– Using the time value of money concept and
the rules of compounded interest, the
following relation can be derived:
Bond Valuation: the Discounted Cash-Flow
Approach (II)
Price of a bond P0:
T
P0 =
∑ (1 + R
t =1
CFt
0, t )
t
=
CF1
(1 + R 0,1 )
1
+
CF2
(1 + R 0, 2 )
2
+ ... +
CFT
(1 + R 0,T )T
R0,T: spot rate between time zero and time T
Raising at least one of these interest rates, other
things equal, lowers the price of the bond (strong
indirect relationship)
Bond Valuation: the Yield to Maturity
The yield to maturity (YTM) is defined as
T
P0 =
∑ (1 + YTM)
t =1
CFt
t
=
CF1
(1 + YTM )
1
+
CF2
(1 + YTM )
2
+ ... +
CFT
(1 + YTM )T
YTM is the constant discount rate that equates
the present value of the bond’s future cash flows
with the current market price
YTM is a complex average of the spot rates over
the respective time period
Bond Valuation: the Term Structure of
Interest Rates
Based on zero-bonds, the term structure of interest
rates can be plotted (spot rates against the respective
time to maturity)
Spot Rates
Term Structure of Interest Rates
7
6
5
4
3
2
1
0
Government Bonds
AAA
AA
A
1
2
3
4
5
6
7
Time to Maturity
8
9
10
Bond Valuation: the Risks of Holding A
Bond (I)
• The investor holding a bond is exposed to
different forms of risk
– Interest rate risk (price risk)
°
°
Most important because of the strong negative
relationship between price and interest rate
Measured primarily by duration
– Reinvestment risk
°
Cash-flows from coupon payments will be
reinvested at a lower rate if interest rates fall
– Credit and default risk
°
Measured primarily by credit ratings
Bond Valuation, the Risks of Holding A
Bond: Credit Risk and Credit Rating (I)
• Credit risks, i.e. the likelihood that the issuer of the
bond will default on payments of interest and/or
principal payment, is measured by a „rating“
• The two main internationally active rating agencies,
Standard & Poor‘s and Moody‘s, differentiate between
investment grade bonds and speculative bonds
– Denotations for investment grade bonds
°
°
Standard & Poor‘s denotations: AAA, AA+, AA, AA-,
A+, A, A-, BBB+, BBB, BBBMoody‘s denotations: Aaa, Aa1, Aa2, Aa3, Baa1, Baa2,
Baa3
– Similar denotations for speculative bonds: BB+ (Ba1),
down to C (C) and D (-)
Bond Valuation, the Risks of Holding A
Bond: Credit Risk and Credit Rating (II)
• The table shows cumulative mortality losses in % in the years
after issuance
• Yet, it is open whether ratings have a true information content
(Do ratings change before the facts change?)
Original
Rating
AAA
AA
A
BBB
BB
B
CCC
1
2
3
4
0.00
0.00
0.00
0.02
0.00
0.43
1.07
0.00
0.00
0.06
0.42
0.58
1.53
1.83
0.00
0.19
0.12
0.60
0.93
3.88
3.76
0.00
0.27
0.12
1.04
3.56
6.46
12.78
Basis: S&P Ratings, 1971 – 1988
Years after issuance
5
6
0.00
0.29
0.12
1.25
3.85
8.54
14.74
0.06
0.29
0.17
1.54
5.03
11.67
N/A
7
8
9
10
0.10
0.47
0.24
2.07
9.80
15.87
N/A
0.10
0.47
0.26
2.07
9.80
18.48
N/A
0.10
0.54
0.30
2.17
9.80
26.16
N/A
0.10
0.62
0.30
2.51
12.08
28.05
N/A
Bond Valuation: the Risks of Holding A Bond
(II)
• Inflation risk
– Mainly important in connection with
unexpected changes in the inflation rate
• Exchange rate risk
– Describes the risk of unexpected changes in
the exchange rate
• Liquidity and marketability risk
– Smaller issues might tend to „dry out“ in the
secondary market and become illiquid
• Issue-specific risk
– Especially call features and reinvestment risks
Bond Valuation: the Concept of Duration
Duration is the weighted average length of time to the receipt
of a bond‘s cash-flows (coupon and redemption value); the
weights are the present values of the involved cash-flows
Duration
1200
1000
800
Present Value 600
400
200
0
Coupon and
Redemption Value
1
2
3
4
5
6
7
Time to Maturity
Duration
The Macaulay-Duration (F. Macaulay, 1938)
T
T
∑ t ⋅ PV(CF ) ∑ t ⋅ PV(CF )
t
Duration D =
t =1
P
t
=
t =1
T
∑ PV(CF )
t
t =1
The term structure of interest rates is assumed to be flat
Bond Valuation: Relative Price Changes and
Modified Duration
It can be shown that (YMarket being the market yield, p price):
D Macaulay
∆P
∆ Y Market
≈−
⋅ ∆ YMarket = −
⋅ D Macaulay
P
1 + Y Market
1 + YMarket
R elative Price Changes and Modified Duration D mod
D mod ≡
D Macaulay
1 + YMarket
∆P
≈ − D mod ⋅ ∆ Y Market
P
or ∆ P ≈ − D mod ⋅ ∆ YMarket ⋅ P