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Lectures 2 & 3: Interest rate risk management
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Outline
• Different types of bonds
• Bond characteristics
• Risks of investing in bonds
• Shapes of the yield curve
• Theories behind the shapes of the yield curve
• Bond pricing, yield to maturity & spot rate
• Forward rate
• Measuring interest rate risk (i.e., duration)
• Some bond trading strategies using duration
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Different types of bonds
• In Australia, we have the following fixed income securities:
Note: Don’t be too concerned with the ‘short-term’ vs. ‘long-term’ terminologies.
• In U.S., a 2-year Treasury bond is called a Treasury note
• In Aust., CD could be between 1 day and 2 yrs
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Bank Accepted Bills (BAB)
• Similar to a ‘cheque’ but typically with a fixed maturity of 90, 120
or 180 days
• Example: A 90-day BAB
 A zero-coupon bond in which the ‘drawer’ promises to pay $100,000 in
90 days
 ‘Discounter’ (investor) longs BAB to earn interest
 ‘Drawer’ (company/issuer) draws or shorts BAB to raise money (i.e.,
borrow $) to fund its projects etc
 BAB is guaranteed by the ‘acceptor’ (bank aka middle person)
 Real-life example: CommBank Bank Bill.pdf (from Blackboard)
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Drawer Discounter
Acceptor
• Sell BAB i.e. borrow cash
• Promise to pay face value of BAB (e.g. $100,000) on
maturity
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Drawer
Acceptor
• Buy BAB i.e. lend cash to drawer
• Receives face value at maturity of bill
Discounter
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Drawer
Acceptor
Discounter
• The facilitator (i.e. bank)
• Guarantee (‘accept’) the BAB i.e. guarantee the drawer will pay
the face value on maturity by taking the cash from the drawer
and pass it to the discounter
Note: See CommBank Bank Bill.pdf (from Blackboard) for a real-life description of BAB issued by Comm Bank
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Certificates of Deposit (CD)
• Similar to BAB but with longer terms, between 1 day and 2 yrs
• Real-life example: CommBank CD.pdf (from Blackboard)
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Treasury notes
• Issued by the Aust gov to assist in ‘within-year’ funding purposes
• ‘Within-year’ funding needs arise because the timing of gov revenues does
not match expenditure profile
• In Aust, treasury notes are zero-coupon and mature in less than 6 mths
• Example: Treasury notes issued by Aust gov on 31 July 2015
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Commercial Papers (CP)
• Unsecured (not backed by
collateral) short-term debts
issued by companies to borrow
$$ at rates lower than bank
rates
• Companies (usually with high
credit rating) issue CP to raise
funds and finance their activities
• Typically mature in less than
270 days
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Interbank deposit
• Deposit between banks. Most popular is London Interbank Offered
Rate (LIBOR)
• LIBOR https://www.theice.com/iba/libor:
 Average of borrowing rates among a number of large banks that have London
offices
 Most important benchmark or reference rate for floating-rate debt securities
and short-term lending
 Published daily by Intercontinental Exchange (ICE) Benchmark Administration
for five currencies (i.e., USD, GBP, EUR, CHF and JPY)
 Seven maturities: overnight/spot next, one week, one month, two months,
three months, six months and 12 months
 Example: USD LIBOR - 1 week, Euro LIBOR - 6 months
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Importance of LIBOR (example):
• An investor borrows from a bank using a floating-rate security
• The floating/variable rate is LIBOR + a margin (e.g., 50 bps)
• The margin depends on the borrower’s credit risk
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• Euribor (Euro Interbank Offered Rate)
 Analogous to LIBOR except that Euribor is published by the European
Central Bank (ECB). See www.euribor-rates.eu
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Treasury bonds
• Coupon-paying bonds issued by the Aust gov; coupons are paid semi-
annually
• www.aofm.gov.au/ags/treasury-bonds/
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Treasury bonds
• On-the-run Treasury bonds
 Treasury bonds that are recently auctioned (newly issued bonds)
• Off-the-run Treasury bonds
 Older issued bonds but are still traded in the market
• On-the-run bonds are more actively traded (and hence more liquid) than off-the-
run bonds
• Thus, market prices of on-the-run bonds provide a better reflection and
information about the current market yield
• All else being equal, on-the-run bonds have lower yields (and thus higher prices)
than off-the-run bonds because on-the-run bonds are more actively traded
 On-the-run bonds have a lower liquidity premium, and hence a lower yield
 Also, on-the-run bonds are issued with limited size; thus interested buyers tend to push up
the price and depress the yield
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State bonds
• Semi-gov bonds issued by State Treasury (e.g., QTC) to meet funding
needs of state and local gov to build hospitals, infrastructures etc
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Source: Australian Financial Review, 31 July 2015
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Corporate bonds
• Issued by companies
• Yields are higher than Treasury yields
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Source: Australian Financial Review, 31 July 2015
Heritage bank ltd’s bond
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Eurobonds
• A Eurobond is an international bond denominated in a currency not local to
the country in which it is issued
• Eurobond is named after the currency it is denominated in.
 Euroyen = Eurobond denominated in Yen
 Eurodollar = Eurobond denominated in USD
• Typically issued by an oversea company outside the issuer’s home nation
 An Australian company issues a bond denominated in USD dollar in Japan; this bond
is called Eurodollar.
• Attractive because the issuer (e.g., the Australian company) can choose
the country/market (e.g., Japan) in which to issue the bond in its preferred
currency (e.g., USD)
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Bond characteristics
• Zero coupon bonds
 Bonds with no coupon rate; only has face/par value (principal amt, payable on maturity)
 These bonds are issued at a discount
 Example: BAB, CD, CP, Australian Treasury Notes
• Coupon paying bonds
 Bonds paying coupon rate + face/par value at maturity
 Example: Treasury bonds, corporate bonds
• Step-up coupon bonds
 Coupon rate increases over time
 Could be one-time step-up or multiple times over the bond life
• Step-down coupon bonds
 Coupon rate decreases over time
 Could be one-time step-down or multiple times over the bond life
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Bond characteristics
• Floating rate notes/bonds (FRN)
 Coupon rate varies over the life of the asset, based on a reference rate (usually the
LIBOR rate)
• Inverse floater
 A floating rate note/bond whose coupon rate increases when the reference LIBOR rate
decreases and vice-versa
 Example: The World Bank issued a 5-yr, USD-denominated FRN in 1992
 Its semi-annual coupon rate varies inversely to the 6-mth USD-LIBOR reference rate
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Bond characteristics
• Both bond buyers and sellers can impose upper and/or lower values to limit their
exposure to extreme interest rate movements
• Floor
 In the previous inverse floater example, the coupon rate has a floor i.e., a lower limit at 0%
 The lower limit could be of any figures e.g., 1%, 2% etc
• Cap
 For a normal FRN, we can put a cap e.g., an upper limit of 10% etc
• Collar
 A collar is a combination of both floor and cap
 Example: An FRN is issued with its coupon rate = LIBOR, subject to a floor of 2% and a
cap of 6%
 Draw the coupon diagram for the FRN’s holder (the person who longs the FRN)
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Risks of investing in bonds
• Yield curve risk
• Interest rate risk
 Risk due to the effect of changes in market interest rate
 When interest rate (yield) increases, bond price falls a loss to the bondholder
 Interest rate risk can be approximated by a measure called duration
 Components that make up the nominal interest rate:
 Real rate
 Inflation premium (due to inflation risk)
 Default premium(due to default risk)
 Liquidity premium (due to liquidity risk)
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The price of an asset is the present
value of its expected future payoffs
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• Yield curve risk
 Risk due to the effect of changes in the shape of the yield curve
 Won’t occur during a parallel shift in the yield curve
 Will occur during a non-parallel shift in the yield curve
• Scenario 1: No yield curve risk
 A parallel shift in yield curve
 Here, bonds with different maturities have the same change in their yields and prices
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yield
maturity
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• Scenario 2: Has yield curve risk
 A non-parallel shift in yield curve (see extreme example below)
 Here, bonds with different maturities have different changes in their yields and prices
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yield
maturity
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Components of nominal interest rate
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real rate + inflation premium
+ default premium + liquidity premium
Nominal interest rate =
Nominal risk-free rate
Risk premium
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Nominal risk-free rate = real rate + inflation premium
• Real rate of interest
- The basic rate of return (does not include other risks such as the risk of not getting paid
back)
- Simply to compensate the lender for the opportunity foregone for lending her money to you
• Inflation premium
- Inflation diminishes the purchasing power of money (i.e., in the presence of inflation, a $1
that you lend today will be valued at more than $1 on the maturity of the loan)
- Therefore, lenders must be compensated for inflation risk
• Real rate + inflation premium = nominal risk-free rate
- Proxied by Treasury yield (it is highly likely, but not 100% guaranteed, that you will get back
the money that you lent to the govn)
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Risk premium = Liquidity premium + default premium
• Liquidity premium
- To compensate for the risk that you may not be able to resell the bond that you previously
bought to another buyer in quick time
- If you are unable to resell the bond, you are then stuck with it (which means opportunity
foregone). Hence you must be compensated for this illiquidity risk
- Treasury bonds virtually have no illiquidity risk (i.e., they are the most liquid fixed-income
securities).
• Default premium
- To compensate for the risk that the bond’s issuer might fail to repay the interest and/or the
principal of the bond
- To gauge the creditworthiness of the bond’s issuer, we can use rating agencies such as
S&P, Fitch and Moody’s who use some sophisticated models to rate the bonds.
- The bond’s rating indicates of the likelihood of the bond’s issuer to default its repayments
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- Credit/default spread increases when the bond’s likelihood to default increases
- Very speculative grade bonds are also known as junk bonds; they command a very
high default premium
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Least likely to default
Likely to default
Very likely to default
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 https://fred.stlouisfed.org/series/DAAA#0
 The graph plots monthly Moody’s Aaa and Baa Industrial Corporate Bond Yields
 Moody’s Aaa is an index that ‘averages’ the yields of investment grade (Aaa) industrial
corporate bonds with maturities of at least 20 years
 Moody’s Baa is an index that ‘averages’ the yields of speculative grade (Baa)
industrial corporate bonds with maturities of at least 20 years
 Conclusion: Baa yield > Aaa yield
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- NBER does not define a recession in terms of 2 consecutive quarters of decline in real GDP
- Rather, a recession is deemed by NBER to occur if there is a significant decline in economic activities
(indicated by high unemployment rate, low industrial production etc), lasting for several months
 https://research.stlouisfed.org/fred2/graph/?g=f09
 By taking the differences of Baa and Aaa yields (i.e., Baa - Aaa), we have the default spread
 The shaded regions are recession periods indicated by NBER (white regions are expansion
periods)
 Conclusion: Default spread increases during recessions
 Default spread is a common variable used to measure the state of the economy, because it is a
good indicator of recessions and expansions (note: it is not a definite indicator but merely
suggesting the likelihood of recession/expansion)
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 https://fred.stlouisfed.org/series/DGS1#0
 The graph plots the monthly 10-yr and 1-yr Treasury yields; NBER recession indicator
is shaded
 By taking the differences in yields (i.e., long-term Treasury yield minus short-term
Treasury yield), we have the term spread
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 The graph plots monthly term spread of 10-yr Treasury yield minus 1-yr Treasury yield ; NBER
recession indicator is shaded
 Term spread is usually (but not always) positive during expansions and negative during recessions
 There are a few exceptions/false signals; for example, the term spread is negative in early 1980s
but there was no recession
 Conclusion: Term spread is another common variable used to measure the state of the economy,
because it is a good indicator of recessions and expansions (note: it is not a definite indicator but
merely suggesting the likelihood of recession/expansion)
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 If you only consider the stock market index movements and/or the NBER recession
indicator, you won’t know if we are in recession or expansion in real-time.
 We will be informed by the NBER Dating Committee that quarter t is in recession/expansion at t+2 quarters
later.
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S&P 500
Today Dec
2013
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- http://www.kamakuraco.com/Blog/tabid/231/EntryId/554/
Time-Warner-Cable-Inc-Bonds-Deja-vu-All-Over-
Again.aspx
- Dark blue line = US Treasury yield (UST) = nominal risk-
free rate
- Light blue line = UST + default spread of Time Warner
(almost zero default spread) + 0 liquidity spread
- Orange line = UST + default spread of Time Warner +
liquidity spread for Time Warner’s bonds with low yield
(investment grade bonds)
- Red line = UST + default spread of Time Warner +
liquidity spread for Time Warner’s bonds with high yield
(riskier than investment grade bonds e.g., unsecured
bonds)
- Finding: A liquidity premium is built on the yields of Time
Warner above the risk-free rate + default premium of the
firm
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Components of interest rate: A real-life example
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Shapes of the yield curve
• Yield curve (i.e., term structure of interest rate) defines the relationship
between yield and maturity of the bond
• Four general shapes of yield curve
 Normal upward sloping
 Downward sloping (inverted)
 Flat
 Humped
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 In reality, the shape is not-so-clear-cut. For example, a real-life upward sloping U.S.
Treasury yield curve on 4 Jan 2010
 http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/Historic-Yield-
Data-Visualization.aspx or
 https://www.treasury.gov/resource-center/data-chart-center/interest-
rates/Pages/TextView.aspx?data=yieldYear&year=2010
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 The curve looks about right  long term bonds (which are riskier than short term
bonds) have higher yields
 But why doesn’t the curve continue rising from 20 to 30 yrs?
 There is a typical high demand for very long-term (e.g., 30 yr) bonds from institutional
investors (e.g., pension fund managers etc)
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• Important to note that:
 Yield curve shows the yields of bonds with different maturities on a particular day t
 It is not a historical graph (it does not show the yields over time)
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 Going back to the steep upward sloping curve in Jan 2010 again (1-yr T-bill is 0.5%,
30-year bond is 4.5%, giving a term spread of +4% or 400 basis points (bps).
 Upward sloping yield curve implies 2 possibilities:
- It indicates a sustained economy recovery period (Federal Reserve maintained a
low short-term yield to encourage economy recovery)
- Long-term yield increases because investors sell long-term bonds in favor of rising
stocks. Also, the govn will price on-the-run bonds at a low price to encourage
investors to buy the long-term bonds. Overall, these indicate that the market is
expecting a period of inflation.
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 Curiously, a flat yield curve is typically followed by an inverted yield curve
 An inverted yield curve usually means economic recession is looming
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Flat
Inverted
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An inverted yield curve usually means economic recession is looming
 Inverted curves are rare, but could happen.
 Why would long-term investors (who are faced with a higher maturity risk) contend with a lower yield than
short-term investors (who are faced with a lower maturity risk)?
 Long-term investors will do that if they think the rates will drop even further in the future. For example,
suppose that they forecast that an economy recession is imminent over the next few years.
 Thus, the market rushes to buy long-term bonds (and lock-in the long-term yield before it drops even
further in the future)  demand for long-term bonds increases, increasing the prices of long-term bonds
and decreasing their yields
 In short, inverted yield curve signals that long-term investors (in particular) are expecting a recessionary
period
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• Note:
 An inverted yield curve means the term spread (long term yield minus short term yield)
is negative
 We argued previously that negative term spread is related to recession
 Hence, our current argument that an inverted yield curve, which implies a recession is
looming, is consistent with the negative term spread argument
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• Yield curve just before economic recessions begin
• Note that the yield curve is downward sloping in most cases
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Croushore, D. (2007) “Money & Banking: A policy-oriented approach”
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• Yield curve one year after economic expansions begin
• Note that the yield curve is much steeper and upward sloping after an economic expansion has begun.
• The steep upward sloping curve suggests that investors are expecting short-term rates to rise, which
typically occurs early in an expansion period
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Croushore, D. (2007) “Money & Banking: A policy-oriented approach”
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• Yield curve three years after economic expansions begin
• Note that the yield curve is not as steep as at the start of economic expansions
• Still, the curve is upward sloping
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Croushore, D. (2007) “Money & Banking: A policy-oriented approach”
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• Cool animation of the yield curve over time:
• http://www.businessinsider.com.au/us-treasury-yield-curve-evolution-1982-
2014-2014-12 or
• https://static-ssl.businessinsider.com/us-treasury-yield-curve-evolution-
1982-2014-2014-12 or
• https://www.businessinsider.in/videos/finance/This-Animation-Of-32-Years-
Worth-Of-Treasury-Yield-Curves-Is-Mesmerizing/videoshow/45528997.cms
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• Some common strategies for portfolio managers :
 If you predict the curve will steepen (see figure below), you can
 Buy short-term bonds now
 Sell long-term bonds now
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yield
maturity
Current curve
Predicted curve
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• Some common strategies for portfolio managers:
 If you predict the curve will flatten (see figure below), you can
 Buy long-term bonds now
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yield
maturity
Current curve
Predicted curve
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• Some common strategies for portfolio manager:
 If you predict the curve will flatten (see figure below), you can
 Sell short-term bonds now
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yield
maturity
Current curve
Predicted curve
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Theories behind the shapes of the yield curve
• Three general theories to explain the different shapes of the yield curve:
 Pure expectations theory
 Liquidity preference theory
 Market segmentation theory
• A yield curve shape may be explained by 1 or multiple theories
 Example: A flat yield curve may be explained by pure expectations theory only, or a
combination of pure expectations and liquidity preference theories.
 Or pure expectations theory may explain a flat yield curve, and adding the liquidity
preference theory may result the flat yield curve to become an upward sloping curve.
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Pure expectations theory
• In pure expectation theory, investors assume forward rates are unbiased
estimators of future short-term rates (i.e., spot rates)
 If the market expects future spot rates (i.e., future short-term rates) to rise, long term
rates will be higher than short-term rates, providing an upward sloping curve
 Will come back to this in the next few slides
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Expectations of future short-term rates Shape of yield curve
Short-term rates expected to rise in the future  Upward sloping
Short-term rates expected to fall in the future  Downward sloping
Short-term rates expected to remain constant in the future  Flat
Short-term rates expected to rise, then fall in the future  Humped
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Liquidity preference theory
• In addition to expectations about future short-term rates, investors require
a risk premium to hold especially long-term bonds
 Short-term bonds are more liquid because they can be easily converted to cash;
conversely, long-term bonds are less liquid
 Hence, investors must be compensated with liquidity premium for holding especially
long-term bonds
 The size of the liquidity premium depends on how much investors need to be
compensated to induce them to hold long term bonds which have higher risk, or
alternatively, how strong their preference for the greater liquidity of short-term bonds
 Thus, under liquidity preference theory, the yield curve can take on any shapes
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 Example:
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yield
maturity
Future short-term rates are
expected to remain constant
(pure expectations theory)
Liquidity premium
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Market segmentation theory
• Different investors and borrowers have different preferences for different
bond maturities
 Commercial banks prefer to buy short term bonds to match their short term liabilities
 Life insurers and pension fund managers prefer to buy long term bonds to match their
long term liabilities
 Legal or institutional policies may require some investors to hold some bonds with
specific maturities, thus preventing them from buying/selling bonds with different
maturities
 As such, the demand-supply of short term bonds (based on commercial banks) and
the demand-supply of long-term bonds (based on life insurers etc) determine the
shape of the yield curve
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yield
maturity
S D
S D
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• A different version of the market segmentation theory is the ‘preferred
habitat theory’
 Similar to the market segmentation theory, preferred habitat theory states that the
shape of the yield curve is determined by supply and demand for the bonds
 But under preferred habitat theory, investors can be induced to cross-over to another
‘habitat’ if there are sufficient incentives
 Example: Commercial banks are induced to hold long term bonds (instead of short
term bonds) because long term bond yield is sufficiently high and attractive
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Bond pricing: Pricing zero-coupon bonds
• A zero coupon bond pays no coupon; it only pays its face value at maturity
 BAB, CD and even Australian Treasury Notes (but not US Treasury notes) are zero
coupon bonds
 All zero coupon bonds are sold/issued at a discount (i.e., lower than their face values)
 For example, the price of a BAB is defined as:
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= 1 + 365
where face value = usually is $100,000 for BAB
= remaining days to maturity of the asset
y = annual yield
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 Example: Company A (drawer), through its bank (acceptor),
issued a $100,000 90-day BAB with a quoted price of $90.50 to
you (discounter). What is the price of the bill?
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= 1 + 365 = 100,0001 + 0.095 90365 = 97,711
Note:
Quoted price ≠ price of BAB
Quoted price = price quoted to get the implied annual yield
Annual yield (in %) = 100 – quoted price
 By holding the BAB until maturity, you will receive $100,000;
thus your investment earnings is $2289
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• In reality, investors prefer to talk in terms of the ‘quoted price’ of the bill
instead of annual yield
• The higher the quoted price, the higher is the actual traded price
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Source: Australian Financial Review, 9 July 2010
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 Prior to maturity of the bill, you may request the bank to repurchase the
bill from you.
 In CommBank Bank Bill.pdf (from Blackboard), it says that “request for a repurchase
depends at the discretion of the (Comm) Bank”
 Example: 60 days have passed. You request the bank to repurchase the
BAB. The (new) quoted price is $92
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= 1 + 365 = 100,0001 + 0.08 30365 = 99,347
 Investment earnings = $99,347-$97,711 = $1636
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• Pricing a longer maturity zero-coupon bond
 The price of a zero-coupon bond with longer term to maturity is defined as:
63
= 1 + ×
where face value = 100
m = compounding frequency
y = (discrete) annual yield
t = years to maturity
Note:
In reality, some bonds could have face values of $100, $1000, $10000 etc.
By assuming the face value in the above formula is equal to 100, we can
compare the prices of different bonds
FINM7405
24/02/2022 CRICOS code 00025B 64
• Example: Price a 4-year zero coupon bond with a yield to maturity of 8%
p.a. compounded semi-annually.
64
= 1001 + 0.082 2×4 = 73.07
FINM7405
24/02/2022 CRICOS code 00025B 65
Let’s test your knowledge
• Can a ZC bond be priced at premium? Why or why not?
 Answer: No. Recall that the price of an asset is the PV of its expected future payoff.
For ZC, its expected future payoff is its face value of $100. Hence, its PV (after
discounting) must be less than its face value.
65
FINM7405
24/02/2022 CRICOS code 00025B 66
Pricing coupon paying bonds
 When a bond is issued for the first time, its coupon rate is usually set equal to (or very close
to) the market yield of another bond with similar nature (similar maturity, similar credit risk
etc). Thus, the bond is sold at par (or very near to par) when issued
 The bond’s yield will subsequently change (e.g., the bond’s credit risk changes, the bond’s
liquidity changes etc)
 If the yield rises above the coupon rate, the bond price decreases and the bond is traded at
a discount (below its par value)
 If the yield remains equal to the coupon rate, the bond is (still) traded at par
 If the yield drops below the coupon rate, the bond price increases and the bond is traded at
a premium (above its par value)
66
Yield Bond price Bond sold at
Increase Decrease Discount
Constant Unchanged Par
Decrease Increase Premium
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24/02/2022 CRICOS code 00025B 67
67
 Example: A 8% coupon bond
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24/02/2022 CRICOS code 00025B 68
 Example: A 4-year coupon bond with a face value of $100 is currently on issue. The
bond, which is rated Aaa, pays coupons twice a year. Determine its coupon rate and
its current price.
 Solution 1: Use Moody’s Aaa index. But it has some weaknesses:
• Moody’s Aaa index is the aggregate of industrial corporate bonds with Aaa rating
and with maturities of at least 20 years
68
FINM7405
24/02/2022 CRICOS code 00025B 69
 Solution 2: Calculate the followings
 Default free yield (Treasury yield)
 Credit spread of the bond
 Liquidity premium
 Etc
yield on a risky bond = default-free yield + credit spread + liquidity spread
• Potential problems:
• Modelling credit spread is hard
• Modelling liquidity premium is subjective
69
FINM7405
24/02/2022 CRICOS code 00025B 70
 Solution 3: Set the bond’s coupon rate equal to the market yield of
another equivalent bond
 The yield to maturity of a 4-year Aaa bond issued by another company is 8% p.a. comp
semi-annually
• Thus, our bond’s coupon rate is 8% p.a. paid twice a year
70
FINM7405
24/02/2022 CRICOS code 00025B 71
 Proof that the bond must be priced at par/face value on issue date
• Recall that it is a 4-year bond
• Face value = $100
• Coupon = 8% paid twice a year
• Market yield = 8% p.a. comp semi-annually
• The bond’s time line:
71
Note: Two periods = 1 year
0 1 2 3 4 5 6 7 8
44 4 4 4 4 4
100
+ 4
remaining
periods
FINM7405
24/02/2022 CRICOS code 00025B 72
• Pricing a coupon-paying bond on a coupon date
 The price of coupon-paying bond on a coupon payment date is defined as:
72
where face value = 100
m = compounding frequency
y = (discrete) annual yield
c = annual coupon (in $)
t = years to maturity
Note: As we shall see later, the above formula will not work in some situations
m
y
m
y
m
C
m
y
valueFaceP
tm
tm










 +−
+




 +
=
×−
×
11
1
FINM7405
24/02/2022 CRICOS code 00025B 73
73
( )
( )
100
2
08.0
2
08.011
2
8
2
08.01
100
42
42 =



 +−
+
+
=
×−
×P
remaining
periods
FINM7405
24/02/2022 CRICOS code 00025B 74
 1 yr has passed. The bond’s riskiness has decreased. It is still rated as Aaa but the
Treasury yield has decreased significantly. Thus, the bond’s annual yield to maturity is
(now) 7% p.a. comp semi-annually. Calculate its current price
• We know that the bond must be trading at premium because its yield < coupon
The bond’s (current) time line:
74
44 4 4 4
100
+ 4
( )
( )
66.102
2
07.0
2
07.011
2
8
2
07.01
100
32
32 =



 +−
+
+
=
×−
×P
If you intend to buy this bond now, does it mean
you are buying at a loss (since its price > $100)?
remaining
periods
FINM7405
24/02/2022 CRICOS code 00025B 75
 New scenario: 1 yr has passed since it was first issued (the bond still has 3 years to
maturity). The bond’s riskiness has increased and is downgraded to Aa. Its annual
yield to maturity is (now) 9.5% p.a. comp semi-annually. Calculate its current price
• We know that the bond must be trading at discount because its yield > coupon
• The bond’s (current) time line:
75
44 4 4 4
100
+ 4
( )
( )
16.96
2
095.0
2
095.011
2
8
2
095.01
100
32
32 =



 +−
+
+
=
×−
×P
If you intend to buy this bond now, does it mean
you are buying at a gain (since its price <
$100)?
remaining
periods
FINM7405
24/02/2022 CRICOS code 00025B 76
76
Note: As the bond approaches maturity, its price will be worth
exactly its face value (see next slide)
FINM7405
24/02/2022 CRICOS code 00025B 77
 Another 0.5 yr has passed (the bond now has 2.5 years to maturity). The bond’s
riskiness remains with a current annual yield of 9.5% p.a. comp semi-annually.
Calculate its current price
• The bond’s (current) time line:
77
4 4 4 4
100
+ 4
now
x
Note: We now only have 5 periods remaining.
remaining
periods
FINM7405
24/02/2022 CRICOS code 00025B 78
78
4 4 4 4
100
+ 4
now
x
Method 1:
Note: We now only have 5 periods remaining.
( )
( )
73.96
2
095.0
2
095.011
2
8
2
095.01
100
5.22
5.22 =



 +−
+
+
=
×−
×P
remaining
periods
FINM7405
24/02/2022 CRICOS code 00025B 79
Accrued interest (AI), clean price and dirty price
o Accrued interest
- An interest that has been earned, but not yet been paid by the bond issuer, since the last coupon
payment.
- Interest accrued equally on every day during that period. As such, AI does not compound.
o Clean price
 Also known as “quoted price”, since this is the price quoted by the traders
 Clean price = Dirty price minus AI
o Dirty price
 Also known as “full price” (see the equation above and it is obvious why it is called the “full price”)
 This is the selling/buying price (e.g., if you want to buy or sell a bond, you pay/receive the dirty
price, not the clean price)
79
FINM7405
24/02/2022 CRICOS code 00025B 80
80
4 4 4 4
100
+ 4
now
x
Method 1 (again):
• As of today, we have 5 remaining periods, and the price of the
bond is $96.73
• We presume that the bond issuer has just paid $4 coupon to
us just before we calculate the price above.
• Therefore, from the time the bond issuer paid us the $4
coupon to the time when we price the bond, there is no AI.
• As such, $96.73 is the dirty price and clean price in this case.
remaining
periods
FINM7405
24/02/2022 CRICOS code 00025B 81
81
 Method 2:
The bond’s time line 0.5 yr ago:
We have calculated the bond price = $96.16 (valued at t=0 or 6 mths ago)
Today, the FV =
44 4 4 4
100
+ 4
now
( ) 73.1002095.0116.96
5.02
=+×=
×
P
- 100.73 is the dirty price.
- Over the last coupon paid at t=0 (6 mths ago) to t=1 (today), we have earned the
full $4 coupon.
- After the bond issuer paid this $4 to us, the bond price drops to 100.73 – 4 =
$96.73 (same as method 1)
remaining
periods
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24/02/2022 CRICOS code 00025B 82
82
 In most cases, bonds are traded on non-coupon payment dates.
 Example: Suppose that the bond has 2 years and 7 mths to maturity:
 Coupon rate is 8% p.a. paid twice a year
 Current yield is 9.5% p.a.
 In the figure below, one period = 6 mths 44 4 4 4
100
+ 4
now
remaining
periods
We have calculated the bond price = $96.16 (valued at t=0 or 5 mths ago)
Today, the FV = ( ) 95.992095.0116.96
12/52
=+×=
×
P (dirty price)
• $99.95 is the dirty price.
• Over the last coupon paid at t=0 to today, we have earned some interest for 5 mths.
• This interest earned is the AI, calculated as
AI = total interest x # of mths since last coupon/# of mths between 2 coupons
• Thus, AI = 4 x 5/6 = 3.33 (important: no compounding aka no FV calculation)
• Clean price = dirty price minus AI = 99.95 – 3.33 = $96.62
FINM7405
24/02/2022 CRICOS code 00025B 83
83
 Your turn: Suppose that the bond has 2 years and 10 mths to maturity:
 Coupon rate is 8% p.a. paid twice a year
 Current yield is 9.5% p.a.
 In the figure below, one period = 6 mths
44 4 4 4
100
+ 4
now
remaining
periods
• Calculate the followings:
- The fair price of the bond if you want to buy/sell this bond today
- Accrued interest
- The price quoted by traders
FINM7405
24/02/2022 CRICOS code 00025B 84
A real-life example
• Look at the following real-life example (Australian Financial Review, 31 July 2015)
84
bid asklast sale
(before the
mkt closed
yesterday)
Heritage bank ltd
bond, coupon
7.25% paid semi-
annually
next
coupon
date
YTM if bond
is converted
• In reality, we don’t observe the yield or YTM. Instead, we observe the sales price.
• Thus, we reverse-engineer to get the implied yield or YTM.
• In the above example, YTM must be less than the coupon rate. Why?
FINM7405
24/02/2022 CRICOS code 00025B 85
YTM and spot rates
 Going back to the previous example :
 1 yr has passed since the bond was first issued (the bond still has 3 years to maturity). The bond’s
coupon rate is 8% p.a. paid twice a year.
 Its current sales price is $96.16
 What is its yield?
• Note: Since the bond is trading at discount, its yield must be more than its coupon rate
85
44 4 4 4
100
+ 4
( )
( )
16.96
2
?
2
?11
2
8
2
?1
100
32
32 =



 +−
+
+
=
×−
×P
FINM7405
24/02/2022 CRICOS code 00025B 86
86
 Solution 1: Use trial-and-error
 Solution 2: Use optimization program (e.g., Excel’s Solver)
 Solution 3: Use Excel’s “RATE” function
Answer: y = 9.5% p.a. compounded semi-annually
44 4 4 4
100
+ 4
( )
( )
16.96
2
?
2
?11
2
8
2
?1
100
32
32 =



 +−
+
+
=
×−
×P
Yield to maturity (internal rate of return): A constant rate that makes the PV
of future cash flows equals to its current market value (i.e., current sales
price)
FINM7405
24/02/2022 CRICOS code 00025B 87
Spot rate and stripping the coupon-paying bond
o YTM is a constant rate that makes the PV of future cash flows equals to its current market
value
o That is, YTM is the ‘average’ (but not equally weighted) annual rate of investment if you hold the
bond until maturity.
o But, in reality, the appropriate discount rates for different cash flows that come at different times
are not typically the same (unless the yield curve is completely flat).
o Consider this:
 Do you think a 1-yr zero coupon bond will have the same yield as a 10-yr zero-coupon bond, all else
being equal?
87
FINM7405
24/02/2022 CRICOS code 00025B 88
88
o The discount rates for individual future cash flows are called spot rates
o Assume we want to value a 2-yr 7% p.a. coupon Treasury bond:
 The spot rate for the coupon received in 0.5 yr is 0.2% p.a.
 The spot rate for the coupon received in 1.0 yr is 0.5% p.a.
 The spot rate for the coupon received in 1.5 yr is 0.75% p.a.
 The spot rate for the coupon received in 2 yrs is 1% p.a.
FINM7405
24/02/2022 CRICOS code 00025B 89
• The bond is treated as a package of zero coupon bonds with different
maturities, face values and discount/spot rates (i.e., the bond is stripped
into different zero coupon bonds):
• What is the bond’s YTM implied by the fair price calculated above?
89
FINM7405
( ) ( ) ( ) ( )
90.111
2
01.01
50.103
2
0075.01
50.3
2
005.01
50.3
2
002.01
50.3
225.12125.02
=
+
+
+
+
+
+
+
= ××××P
( )
( )
annuallysemicompap
P
−=
=



 +−
+
+
=
×−
×
..%98.0?
90.111
2
?
2
?11
2
7
2
?1
100
22
22
24/02/2022 CRICOS code 00025B 90
o The fair price of the bond is around $112; YTM = 0.98% pa comp semi annual
o What if the bond is traded at $108 (corresponding YTM = 2.86%)?
o Too cheap, thus long the 2-yr coupon bond at $108 and short-sell a package of
ZC bonds:
 A 0.5 yr ZC bond with face value of $3.5
 A 1.0 yr ZC bond with face value of $3.5
 A 1.5 yr ZC bond with face value of $3.5
 A 2.0 yr ZC bond with face value of $103.5
90
FINM7405
24/02/2022 CRICOS code 00025B 91
Forward rate & pure expectations theory
• In pure expectations theory, investors assume forward rates are unbiased
estimators of future short-term rates (i.e., spot rates)
 If the market expects future spot rates (i.e., future short-term rates) to rise, long term
rates will be higher than short-term rates, providing an upward sloping curve
 And so on…
91
FINM7405
Expectations of future short-term rates Shape of yield curve
Short-term rates expected to rise in the future  Upward sloping
Short-term rates expected to fall in the future  Downward sloping
Short-term rates expected to remain constant in the future  Flat
Short-term rates expected to rise, then fall in the future  Humped
24/02/2022 CRICOS code 00025B 92
 If the market expects future spot rates (i.e., future short-term rates) to rise, long term
rates will be higher than short-term rates, providing an upward sloping curve. Proof is
based on no-arbitrage strategy. Example:
 1-year short term rate is 5% p.a. comp annually
 Market expects future spot rate (i.e., forward rate) is 7% p.a. comp annually
 What is the 2-year long term rate?
92
t=0 t=1 t=2
S0,1 = 5% f0 = 7%
$1 $1.05 ?
FINM7405
24/02/2022 CRICOS code 00025B 93
 What if we have the following situation?
 1-year short term rate is 5% p.a. comp annually
 Market expects future spot rate (i.e., forward rate) is 7% p.a. comp annually
 2-year long term rate is 5.5% p.a. comp annually
 Arbitrage by …
93
t=0 t=1 t=2
S0,1 = 5% f0 = 7%
$1 $1.05
FINM7405
24/02/2022 CRICOS code 00025B 94
Measuring interest rate risk: Full valuation approach
• Basically, it is a sensitivity analysis analyzing what will happen to the bond price given a
set of scenarios by changing the yields. Consider the following bond:
 Years to maturity = 4
 Coupon = 7% pa paid semi-annually
 Face value = $100
 Current yield = 7% p.a. comp semi-annually
94
Yield changes to New bond price Change in bond price
from its current value
5% 107.17 7.17
6% 103.51 3.51
7% 100.00 0
8% 96.63 -3.37
FINM7405
24/02/2022 CRICOS code 00025B 95
Measuring interest rate risk: Full valuation approach
• Advantages
 Useful in complicated situations e.g., analyzing the effect of a non-parallel shift in
yield curve or a steeping in yield curve etc to the bond price
 Can be used to test extreme yield changes e.g., how will the bond price react to a
change in yield from 7% to 20%?
 ‘Exact’ testing on the bond price sensitivity
• Disadvantages
 Not useful if the bond has exotic features (e.g., call provisions i.e., issuer has the right
to buy back the bond from the bondholder at a strike price sometime in the future or
put provisions i.e., bondholder has the right to sell back the bond to the issuer at a
strike price sometime in the future )
 Involve complicated calculations if it is a bond portfolio involving multiple bonds, and
more so if those bonds have exotic callable features etc
95
FINM7405
24/02/2022 CRICOS code 00025B 96
Measuring interest rate risk: Duration
96
• Interest rate risk is measured using duration; it gives a good
approximation of a bond price change to a change in the yield
Maturity of bond increases Int rate risk increases Duration increases
Coupon rate increases Int rate risk decreases Duration decreases
Yield increases Int rate risk decreases Duration decreases
≈ −
%
( %)% ≈ − × ( %)
FINM7405
24/02/2022 CRICOS code 00025B 97
 Example:
 A bond has a duration of 5.2
 The yield is expected to increase from 7% to 8%
 What is the approximated change in the bond price?
97
% ≈ − × ( %)
 Bond price will change by approximately 5.2 x 1 = 5.2%.
 -ve sign in the formula: Because yield is expected to increase by 1%,
bond price is expected to decrease by 5.2%
FINM7405
24/02/2022 CRICOS code 00025B 98
98
Maturity of bond increases Int rate risk increases Duration increases
 Long term bonds have higher interest rate risk, hence they are more
sensitive to a change in the yield i.e. they have higher durations
 Macaulay duration for a zero coupon bond is equal to the bond’s term to
maturity
 Modified duration for a zero coupon bond is slightly less than the bond’s
term to maturity
Coupon rate increases Int rate risk decreases Duration decreases
 The prices of bonds with higher coupon rates will change less for a
given change in the yield compared to the prices of bonds with lower
coupon rates
FINM7405
24/02/2022 CRICOS code 00025B 99
 Instead of focusing on the bond price change, some traders look at price value of a
basis point (PVBP or PV01) or duration value of a basis point (DV01)
 What is the change in the bond price if the yield changes by 1 bp?
 Example:
 A bond has a duration of 5.2
 The current market value is $90
 What is the bond’s PV01?
 Bond price will change by approximately 5.2 x 0.01 = 0.052% if yield changes by 1bp
 PV01 = 5.2 x 0.0001 x 90 = $0.0468
 Bond price will change approximately from $90 to $89.9532 (if yield increases by 1 bp
i.e., bond price changes by around 0.052%) or from $90 to $90.0468 (if yield
decreases by 1 bp i.e., bond price changes by around 0.052%)
99
% ≈ − × ( %)
≈ × 0.0001 × current mkt value
FINM7405
24/02/2022 CRICOS code 00025B 100
 Portfolio duration is defined as:
where wi = mkt value of bond i divided by mkt value of the portfolio
Di = duration of bond i
N = number of bonds in the portfolio
100
= 11 +2 2 + ⋯+
FINM7405
24/02/2022 CRICOS code 00025B 101
 Example: A portfolio is constructed based on 2 bonds, each has a face value of $100
per bond
 Buy 6 bonds A, each bond A is a 2-year zero coupon bond with YTM = 5% p.a. comp semi-annually
 Buy 4 bonds B, each bond B is a 8-yr 6% p.a. semi-annual bond with YTM = 9% p.a. comp semi-
annually
101
Bond
(1)
# of bonds
(2)
Current
market value
per bond
(3)
(2) x (3) Modified
duration per
bond
A 6 90.60 543.57 1.95
B 4 83.15 332.60 6.02
Sum 876.17
= 543.57876.17 × 1.95 + 332.60876.17 × 6.02 = 3.50
FINM7405
24/02/2022 CRICOS code 00025B 102
 Bond portfolio will change by approximately 3.5 x 0.01% = 0.035% if yield changes by
1bp
 PV01 for portfolio = 3.5 x 0.0001 x 876.17 = $0.31
 Bond portfolio will decrease approximately from $876.17 to $875.86 if yield increases
by 1 bp or
 Bond portfolio will increase approximately from $876.17 to $876.48 if yield decreases
by 1 bp
102
% ≈ − × ( %)
FINM7405
24/02/2022 CRICOS code 00025B 103
 How good is the approximation given by the duration measure?
103
Bond
(1)
# of bonds
(2)
YTM changes
(3)
New market
value per
bond
(4)
(2) x (4)
A 6 5%  5.01% 90.58 543.46
B 4 9%  9.01% 83.10 332.40
Total 875.86
All yields increase by 1 bp
Bond
(1)
# of bonds
(2)
YTM changes
(3)
New market
value per
bond
(4)
(2) x (4)
A 6 5%  4.99% 90.61 543.68
B 4 9%  8.99% 83.20 332.80
Total 876.47
All yields decrease by 1 bp
FINM7405
24/02/2022 CRICOS code 00025B 104
 What if we want to test the effect of a 30 bps change in yield?
 PV01 for portfolio = 3.5 x 0.0030 x 876.17 = $9.20
 Bond portfolio will decrease approximately from $876.17 to $867 if yield increases by
30 bps or
 Bond portfolio will increase approximately from $876.17 to $886 if yield decreases by
30 bps
104
FINM7405
24/02/2022 CRICOS code 00025B 105
 How good is the approximation given by the duration measure?
105
Bond
(1)
# of bonds
(2)
YTM changes
(3)
New market
value per
bond
(4)
(2) x (4)
A 6 5%  5.30% 90.07 540.40
B 4 9%  9.30% 81.66 326.65
Total 867.06
All yields increase by 30 bps
Bond
(1)
# of bonds
(2)
YTM changes
(3)
New market
value per
bond
(4)
(2) x (4)
A 6 5%  4.70% 91.13 546.76
B 4 9%  8.70% 84.67 338.67
Total 885.44
All yields decrease by 30 bps
FINM7405
24/02/2022 CRICOS code 00025B 106
 What if we want to test the effect of a 200 bps change in yield?
 PV01 for portfolio = 3.5 x 0.0200 x 876.17 = $61.33
 Bond portfolio will decrease approximately from $876.17 to $815 if yield increases by
200 bps or
 Bond portfolio will increase approximately from $876.17 to $938 if yield decreases by
200 bps
106
FINM7405
24/02/2022 CRICOS code 00025B 107
 How good is the approximation given by the duration measure?
107
Bond
(1)
# of bonds
(2)
YTM changes
(3)
New market
value per
bond
(4)
(2) x (4)
A 6 5%  7% 87.14 522.87
B 4 9%  11% 73.84 295.38
Total 818
All yields increase by 200 bps
Bond
(1)
# of bonds
(2)
YTM changes
(3)
New market
value per
bond
(4)
(2) x (4)
A 6 5%  3% 94.22 565.31
B 4 9%  7% 93.95 375.81
Total 941
All yields decrease by 200 bps
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 Approximation is still ok, but not as good as before
 The more extreme is the change in yield, the worse is the
approximation given by the duration measure convexity effect
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Measuring interest rate risk: Duration
• Advantages
 Simple to use (as compared to full valuation approach)
 Useful in analyzing how yield changes will affect a bond portfolio, especially if the
portfolio’s components have callable feature etc
• Disadvantages
 Not very useful in complicated situation e.g., analyzing the effect of a non-parallel shift
in yield curve or a steeping in yield curve etc to the bond price
 Not an ‘exact’ testing on the bond price sensitivity i.e., it only approximate how a
change in yield will affect the bond price change. The approximation will break down
when testing for extreme changes e.g., how the bond price will react when the yield
changes from 7% to 20%?
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Calculating Macaulay duration and modified duration
• Example:
 A 5.3% pa semi-annual coupon bond with 2 years to maturity
 YTM is 6% p.a. comp semi-annually, face value is $100
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Time to maturity
(1)
Cash flow (2) Discount factor
(3)
PV of cash
flow (4)
Weight (5) =
(4) / PV of bond
Time x Weight
(1) x (5)
0.5 2.65 0.9709 2.57 0.02607 0.0130
1.0 2.65 0.9426 2.50 0.02531 0.0253
1.5 2.65 0.9151 2.43 0.02457 0.0369
2.0 102.65 0.8885 91.20 0.92405 1.8481
Sum 98.70 1 1.9233
Mac duration = 1.92 yrs
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• Macaulay duration
 Earliest measure of duration
 How long, on average, do you have to wait (in years) to receive all promised cash flow
 Not the same as the bond’s term to maturity, which measures the waiting time for the
bond’s face value
 Macaulay duration = fulcrum
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• Macaulay duration
 Macaulay duration = fulcrum (see figure below)
 Consider a 5.3% pa semi-annual 2 year bond with $100 face value and YTM = 6% pa
(see previous calculation)
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$91.20
2yr
PV
$2.43
Time to
maturity
0.5yr 1.0yr 1.5yr 2.0yr
$2.50$2.57
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• Macaulay duration
 Macaulay duration = fulcrum (see figure below)
 Consider a 5.3% pa semi-annual 2 year bond with $100 face value and YTM = 6% pa
(see previous calculation)
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$91.20
1.92yr
PV
$2.43
Time to
maturity
0.5yr 1.0yr 1.5yr 2.0yr
$2.50$2.57
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• Modified duration
 A better and enhanced version of Macaulay duration
 Modified duration accounts for YTM
 Investors tend to use modified duration i.e., when we mention duration, we mean
“modified duration” and not “Macaulay duration”
 Modified duration is defined as:
 Calculation based on previous example:
114





 +
=
m
y
MacDModD
1
867.1
2
06.01
9223.1
=





 +
=ModD
Recall that duration
(i.e., modified duration)
decreases when yield
increases
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• Bond price vs YTM (inverse relationship)  convexity effect
 But the inverse relationship is NOT 1-to-1 (not a straight line)
 Example: Consider a 8% p.a. semi-annual coupon, 20-yr bond with $100 face value
and YTM = 8%
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90.8
0
100.0
0
110.68
When y drops by 1%,
price rises by 10.68%
($10.68)
When y rises by 1%,
price drops by 9.20%
($9.20)
Main point: Price rises
at an increasing rate
as YTM decreases
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• Mathematical interpretation of duration
 Duration is the first derivative (i.e., slope) of the price-yield curve
 It can be shown that the duration (i.e., modified duration) of the bond (8% p.a. semi-
annual coupon, 20-yr and YTM = 8%) is 9.41
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90.80
100.00
110.68
Slope is -9.41 = duration
Convexity effect
(i.e., actual bond-yield relationship)
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• Because of the convexity effect, duration is not a good interest rate sensitivity measure in
extreme cases
 According to the duration measure, bond price will increase by D x % change in yield = 9.41 x 1% = 9.41% if
yield decreases by 1%. Thus, bond price will increase to $109.41 (1.0941 x 100)
 But according to the actual (full) valuation, a 8% p.a. semi-annual coupon bond with $100 face value and YTM
= 7% is priced at $110.68
 The difference $1.27 (110.68-109.41) is due to convexity effect
 The difference is magnified if y decreases to 6%, 5% etc
 The same applies when y increases
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100.00
90.80
110.68
109.41
90.59
Price approximation
based on duration
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• It is straightforward to observe why portfolio duration approximation will
breakdown when we predict a non-parallel shift in yield curve
118
%
≈ − × ( %)
yield
maturity
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• How to interpret duration?
1. A weighted average of time (in years) until all future cash flows are received
(Macaulay definition): not very useful
2. The slope of the bond’s current YTM: not very useful
3. Bond price sensitivity as a result of a change in yield
• Approximate percentage change in bond price for a 1% change in yield, or in terms of
PV01, approximate change in bond price for a 1 bp change in yield
• Most useful interpretation
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Some trading strategies using duration
• A portfolio with long duration (duration = interest rate sensitivity) will have high price volatility
• Thus, if you were an asset manager/investor and you predict the yield will decrease (bond
price will increase  the value of your asset will increase),
 You should increase your bond portfolio’s duration to maximize bond price increase (i.e., maximize bond
price volatility) maximize your gain
 For example, sell/dispose short-term bonds (eg commercial papers) and use the proceeds to buy long-term
bonds or long-term futures
• A portfolio with short duration (duration = interest rate sensitivity) will have low price volatility
• If you were an asset manager/investor and you predict the yield will increase (bond price will
decrease the value of your asset will decrease),
 You should decrease your bond portfolio’s duration to minimize bond price decrease (i.e., minimize bond
price volatility) minimize your loss
 For example, sell/dispose long-term bonds or long term futures and use the proceeds to buy short-term
bonds
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Some trading strategies using duration
• Thus, if you were a liability manager/issuer and you predict the yield will
decrease (bond price will increase your liability value will increase),
 You should decrease your bond portfolio’s duration  reduce an increase in the value of
your liability (bonds that you have issued)
 For example, buy-back long-term bonds or long-term futures that you have issued and
instead, sell or issue short-term bonds
• Thus, if you were a liability manager/issuer and you predict the yield will
increase (bond price will decrease your liability value will decrease),
 You should increase your bond portfolio’s duration  increase the reduction in the value
of your liability (bonds that you have issued)
 For example, buy-back short-term bonds that you have issued and instead, sell or issue
long-term bonds or long-term futures
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Thank you