Sub-sovereign Credit Markets
Fixed Income Securities
Applications to Municipal
Markets
Samir El Daher
The World Bank
May 1999
Fixed Income Securities
Main Parameters
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Definitions
Market Risk
Liquidity Risk
Credit Risk
Risk/Return Analysis
Structured Products/Derivatives
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Definitions
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Bond is a financial asset represented by
a schedule of cash flows
Amounts and timing of payments are
“fixed” in advance or predictable
Difference with equities, real-estate or
industrial investments
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Bond Price Formula
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P = i Ci / (1+r)i
P is a function of the variable r as
cashflows Ci are constant numbers
Equation indicates that P is a
decreasing function of variable r
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Market Risk
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Unanticipated change in value of asset
Duration as a measure of risk of fixed
income security
dr
dP
Measuring Duration: P = - D 1 r
Duration D: Price elasticity of interest
rate
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Duration -- Some
Characteristics
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Duration: increasing function of maturity
Duration: decreasing function of yield
level
Duration: decreasing function of coupon
level (ex. zero-coupon)
Duration: increasing function of
frequency of coupon payments
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Duration Versus Maturity
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Maturity relates to the timing of final
cash-flow only
Duration includes all cash-flows timeweighted
Duration carries more information, and
is more relevant, than maturity
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Case of Discount Note:
Zero Coupon Bond
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Discount note consists of:
» one initial outlay (I0) at time zero
» one final payment at time n
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Cn
(1 r )n
P=
dP = - n [Cn / (1+r)n] [dr / (1+r)]
dr
dP / P =  n 1  r
D = n ==> Duration = Maturity for zero
coupon bond
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Example of Zero Coupon
Bond
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Example: P0 = 100 / (1+ 0.065)30 = 15
Only in case of bond with single
cashflow payment would duration and
maturity be the same
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Comparison Between
Duration and Maturity
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Example of 8% coupon rate
Maturity 1y, 3y, 5y, 7y, 10y, 20y, 30y
Duration 1y, 2.5y, 4.2y, 5.6y, 6.8y,10y,12y
30-Y, zero coupon bond is three times
riskier than a 10-Y zero coupon
30-Y, 8% coupon bond is only 12/6.8 = 1.75
time riskier than 10-Y coupon
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Duration of a Portfolio
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Portfolio is a group of securities
Portfolio concept essential for investors
for whom “munis” are part of diversified
basket of investment instruments
Duration of a portfolio of securities is
equal to the sum of the market-valueweighted durations of its component
securities
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“Additivity” of Duration
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Securities
S1, S2, S3,....Si,....Sn
Weights in portfolio a1, a2, a3,.... ai,....an
with i ai = 1
Durations D1, D2, D3,........Di,.....Dn
Maturities M1, M2, M3,.......Mi,....Mn
Average Portfolio Duration = i ai Di
Average Portfolio Maturity = i ai Mi
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Comparing Portfolios (A) & (B):
Portfolio (A)
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Portfolio (A) includes 50% 10-Year note
and 50% 30-Year note
10-Year note ===> Duration 7 years
30-Year note ===> Duration 12 years
Average Duration, Portfolio (A)
(50% x 7) + (50% x 12) = 9.5 years
Average Maturity, Portfolio (A)
(50% x 10) + (50% x 30) = 20 years
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Comparing Portfolios (A) & (B):
Portfolio (B)
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Portfolio (B) includes 100% 20-Year zero
coupon note
20-Year zero coupon note ===> Duration
20 years
Average Duration, Portfolio (B)
= 100% x 20 = 20 years
Average Maturity, Portfolio (B)
= 100% x 20 = 20 years
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Comparing Duration & Maturity
of Portfolios (A) & (B)
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Portfolio (A) and (B) have same average
maturity and different durations
Average Maturity, Portfolio (A) = Average
Maturity, Portfolio (B) = 20 years
Average Duration, Portfolio (B) = 20 years
Average Duration, Portfolio (A) = 9.5 years
Portfolio (B) twice as risky as Portfolio (A)
for same average maturity
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Implication for Investor -Portfolio Approach
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Question: What is meaning of: “a 20-year
municipal bond is “too risky” for an
investor”
Answer: Meaning not clear if the 20-year
bond is part of a balanced portfolio
Importance of assessing contribution of a
security, or asset, within a portfolio
approach [ex: fixed income and real
estate (inflation hedge)]
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Example of “Balanced”
Fixed Income Portfolio
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One third Cash
===> Duration zero
One third 1-year bill ===> Duration 1 year
One third 20-year municipal bond ===>
Duration 10 years
Average duration of portfolio:
(1/3 x 0) + (1/3 x 1) + (1/3 x 10) = 3.6 years
Result might well be within risk tolerance of
investor
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Liquidity Risk
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Liquidity Spectrum
More liquid ---------to---------> Less Liquid
Cash, Gov Securities, ........... Fixed Assets,..
Liquidity risk is associated with existence of
“ready market” where assets may be
exchanged at a small difference between
sale and purchase price
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Liquidity Risk
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For fixed income securities, liquidity is
measured by bid-ask spread in secondary
markets
Bid-ask spread constitutes margin of
market-makers (small 1/32nd in US)
For cash ==> bid-ask spread = 0
For real estate ==> bid-ask spread > 6%
(agent’s fee)
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Credit Risk -- Definition
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Loss of value of an asset as a result of a
party to a contract (seller, issuer,...) not
fulfilling a contractual obligation Loss of value due to default: spot loss
might overestimate real loss
Loss might affect principal and/or interest
Securities trading: c.o.d - Opportunity loss
due to change in market value between
trade date and delivery
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Credit Risk -- Estimation
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Example: Fixed income Security
Years 0----(i) ----(10)------(j)----------(20)
Cashflows -----------(C10)-----(Cj)-------(C20)
Income -------------<---Forgone Income--->
Cj
Actual Loss at Year-10 = 
j
j (1 r )
(In year-10 value)
Potential Loss at time Zero = Actual Loss at


C
Year 10 / (1+r)10 =


(1  r )
j

j
j
(1  r )10

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Implication for Investor -An Example
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Investor must chose 10-Y versus 20-Y
Question: If investor’s risk tolerance for a
given credit is 10 years
Would investor not buy a 20-Y instrument
from same credit?
Answer: Not necessarily ==> Several
scenarios
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Implication for Investor -Scenarios
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1st Scenario ===> Yield Premium
(compensating for risk)
2nd Scenario ===> Guarantee or
insurance beyond 10-Y
3rd Scenario ===> Derivatives, such as
put option
4th Scenario ===> Collateral, such as
mortgage-backed security
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Guarantee or Insurance
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Insurance may be full or partial (say
beyond 10 years)
Case 1: Insurance may be necessary for
debt acceptance
Case 2: Insurance would reduce price of
debt issue and enhance liquidity (USA)
Feasibility: Interest without Insurance >
Interest with Insurance + Insurance Fee
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Other Enhancement
Mechanisms
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Structured finance and derivatives (e.g.
put option allowing maturity “reduction”)
Other non-maturity related
enhancements
» Collateral (revenue pledge, MBS,...)
» Bank letter of credit
» Other features such as convertible debt
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Example of Credit Enhancement
“Zero Coupon Collateral”
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Several recent cases for bullet repayment
Zero coupon “deposited” in segregated
account in “highest credit quality (say US
treasuries)
Zero coupon to accrue interests so as to
become equivalent to face value of
principal upon maturity
Definition of real cost of zero-coupon
collateralized principal (ex. of calculation)
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Risk/Return Analysis -Definitions
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Return = Enhancement in market value
of an asset
Risk is measure of uncertainty of
outcome
Risk = volatility of returns expressed by
Standard Deviation
(Example daily price changes during
one year)
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Simple Risk/Return
Measures
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A number of ways to define and measure
risk
Information Ratio =
Return
Standard Deviation
Sharpe Ratio = (Return - Risk-free Return)
Standard Deviation
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Benchmark for Municipal
Debt Security
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“Risk -free” Government Securities in
relevant maturity range
Ymuni = Ygvn + dYn
dYn = premium that covers, inter-alia,
two categories of risks:
» credit risk
» liquidity risk
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Risk/Return: Portfolio
Approach
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Long-term investors and hedgers need
to know the relative risk of securities so
that they may construct portfolios that
match their preferences for risk and
expected return
Optimize risk/return function:
» For one unit of risk ===> Highest return
(e.g. Foundations,...)
» For one unit of return ===> Lowest risk
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Defining a Portfolio for
Institutional Investors
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Definition of Classes of Assets: Fixed
income, equities, commodities, realestate, currencies,...
Analyses of historical returns (over
representative period, say 20 years)
Analyses of volatilities
Analyses of correlations
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Defining a Portfolio for
Institutional Investors
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Optimization function (e.g. “Efficient
Frontier”) with iterations providing
percentages for each “class” of assets
==> Asset allocation process
Example of pension fund: government
securities, high yield, MBS, marketable
equities, private equities, real-estate,
currencies,...
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Structured Finance -Derivatives
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Main categories: Futures, Options, Swaps
Derivatives as investment vehicles: leverage
Derivatives as hedge and credit enhancement
vehicles
Derivatives transfer investment risks to those
(speculators) willing to assume risks
Futures, options and swaps traded over the
counter (OTC) or on regulated exchanges
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Futures
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Futures contract is a commitment to buy or
sell a security at a future specified date
and specified price
Future neutralizes price uncertainties
Example of farmer hedging crop with
futures
Same result may be achieved by put
option as both futures and options provide
hedge
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Swaps
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Swaps are arrangements between two
parties
Swaps entail exchange of mutual liabilities as
these come due
Swaps may involve currencies (US$/DM)
Swaps may involve interest rates
(floating/fixed)
A
11%
9.95%
B
LIBOR+2%
LIBOR
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Options
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Option is a right -- not an obligation -- to buy
or sell an asset at a pre-established price
within a specified time period (US), or at a
specified time (Europe)
Privilege to exercise such a right entails a
fee, or option “premium”
Calculation of “premium” is crucial element
Option pricing models for equities and
interest rate instruments
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Options
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Call Option - Market position of an
investor/hedger speculating that asset
price would increase
Put Option - Market position of an
investor/hedger speculating that asset
price would decline
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Option Value -- Bond Example
25
20
Profit ($)
15
10
5
0
70
80
90
100
110
120
130
-5
Term inal Bond Price ($)
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Option Parameters
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Definition of “underlying” asset
Exercise or strike price
Time to expiration
“In-the-money”, “Out-of-the-money” and
“At-the-money” options
Premium = Intrinsic Value + Time Value
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Option Parameters
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Delta  = dP/dF or Hedge Ratio
(change in Price P of option to change
in price F of underlying security)
Gamma = d /dF = d2P/ (dF)2
Zeta  = dP / dV (ratio of change of
price to change in volatility V)
Theta  = dP/dt (time dimension, time
“decay”)
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Option Pricing
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Call price: Increasing function of price
of “underlying” relative to “strike” price
Call price: Increasing function of “time
to expiration”
Call option price: Increasing function of
riskless rate of return (on treasury)
Call price: Increasing function of
volatility
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