Global Fixed Income Portfolio

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Global Fixed Income Portfolio

Asset Management

Dr. Enzo Mondello, CFA, FRM, CAIA

August 2014

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Risks Associated with Investing in Bonds

Fixed-Income Valuation

Term Structure of Interest Rates

Yield Measures

Interest Rate Risk: Duration and Convexity

Credit Risk: Fundamentals of Credit Analysis

Managing Bond Portfolio

Content

Relative-Value Methodologies for Global

Corporate Bond Portfolio Management

Exchange Rate Risk: International Bond Investing

Managing Interest Rate Risk with Derivatives

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August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA

Overview of Risk Factors

Interest rate risk is the risk that when interest rates increase, bond prices decline. Interest rate risk is the greatest risk faced by bond market investors.

Call risk is the risk that a bond will be paid off before its maturity date. The risks are: – Higher uncertainty of the cash flows of the bond.

– Risk that principal proceeds will have to be reinvested at lower rates. – Reduced capital gain potential in a falling rate environment.

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Yield curve risk is the risk that the value of a bond portfolio might deteriorate because of a change in the shape of the yield curve

Reinvestment risk is defined as the risk that the received cash flows must be reinvested at a rate lower than the original investment. If coupon payments must be reinvested at lower rates, overall returns of the investment will be less than initially projected.

Credit (default) risk is the possibility that the issuer will be unable to repay the coupon payments and/or the principal amount to the bondholder as defined by the indenture

Liquidity risk is the risk that a security will not be able to be sold quickly without giving up a large price concession. This bid-ask-spread is the best measure of liquidity risk; the wider the spread, the greater the liquidity risk is.

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Exchange rate risk is the risk that the exchange rate between the currency in which a bond is denominated and the currency of the investor’s home country might change

Volatility risk describes the risk that changes in volatility of interest rates will affect the value of options embedded in a bond

Inflation risk is the risk that the purchasing power of the cash flows received from a bond (interest and principal) will decline over time because of inflation

Event risk is the risk that some unusual event could cause the price of bonds to decrease (e.g. natural disaster, corporate takeover, a regulatory change, or political factors)

Sovereign risk has two components:

1. A sovereign may be unable to service its bonds (no ability to pay)

2. A sovereign may be unwilling to service its bonds even though it has the resources to do so

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Bond Price and Risk Factors

Interest rate risk is the major risk faced by fixed-income investors

The bond price is the present value of the sum of future cash flows (coupon payments plus the principal amount)

Therefore, if the discount rate r, which is the yield required by the market (which is related to interest rate levels)

increases, the price of the bond decreases, and vice versa. This is true for almost all bonds.

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Coupon Rate > Required Market Yield:

Bond Price > Par Value (Premium Bond)

Coupon Rate < Required Market Yield:

Bond Price < Par Value (Discount Bond)

Coupon Rate = Required Market Yield:

Bond Price = Par Value (Par Bond)

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The volatility of a bond (i.e. by what percentage the price of a bond will change for a given basis point change in interest rates) depends upon:

– Maturity: Long-term bonds are more volatile than short-term bonds, all other factors being equal – Coupon: The lower a bond’s coupon is, the greater its volatility will be

– Yield: The higher the yield at which a bond trades, the lower its price sensitivity for a given basis point change in interest rates will be, all other factors being equal

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Call Risk

Some bonds are callable. Therefore, they have a call option embedded in their price structure. This means that the

owner of a callable bond can be viewed as owning a portfolio consisting of an option-free (straight) bond and a short position in a call option on the bond. The investor has a short position in the embedded call option because the issuer has the right to call the bond from the bondholder.

Therefore, the price structure of a callable bond can be modeled as follows:

Price Callable Bond = Price Noncallable Bond Price – Price of embedded Options

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Call and prepayment risk is the risk that a bond will be paid off before its maturity date. The reasons why this is disadvantageous for an investor are:

– The cash flows are unknown.

– Reinvestment risk because bonds are usually called when interest rates are low so that the investor is forced to reinvest the proceeds at lower interest rates.

– The appreciation potential of a callable bond is limited (price compression).

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Interest Rate Risk of a Floating-rate Security

The future cash flows of floating-rate notes are not fixed. They rise and fall directly with changes in interest rates. Consequently, the price of a typical floating-rate security should change very little when interest rates change because the dollar value of its coupon (future cash flows) will change in the same direction as the rate at which the future cash flows will be discounted.

There are three reasons why floating-rate security prices can be affected by changes in interest rates: – Reset dates: the longer the time between reset dates, the greater the interest risk will be.

– Reference rate: the quoted margin above the reference interest rate that the market requires can change. – Cap risk: this is the risk that the reference interest rate will rise enough that a floating-rate security’s coupon rate

will be capped out.

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Yield Curve Risk

The yield curve risk is the risk that the value of a bond portfolio will deteriorate because of the change in the shape of the yield curve

The yield curve is a graphical representation of interest rates across all maturities. When interest rates move, they do not change in an equal amount for all maturities.

Duration has a very restrictive interpretation: it is the percentage change in the value of a bond that will occur if the

entire yield curve shifts in a parallel manner (all maturities move by the same increment)

Since a parallel shift is an unlikely scenario, duration is considered at best to be an approximation of a bond’s sensitivity, and is only accurate for small changes in interest rates

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Reinvestment Risk

The factors that affect the reinvestment risk of a security are:

Changes in interest rates: The greater the change, the higher the risk

The size of the cash flows: The larger the cash flows to be reinvested, the greater the risk

The timing of the reinvestment cash flows: The faster the cash flows are received, the greater the reinvestment risk

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Amortizing securities, such as mortgage-backed securities, have more reinvestment risk than non-amortizing securities, such as conventional bonds. There are two reasons for this:

– The periodic cash flows paid by amortizing securities consist of both coupon interest and the repayment of a portion of principal

– Amortizing securities typically pay their cash flows monthly, rather than semi-annually. The more frequently the cash flows are paid, the more frequently the reinvestments occur, and the higher the reinvestment risk.

Note that with zero-coupon bonds, there are no coupon payments to be reinvested over their term to maturity, and thus have no reinvestment risk

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Types of Credit Risk

Default risk: is the risk that an issuer will fail to make interest or principal payments when they are due. If a bond defaults, the investors do not necessarily suffer a total loss. The recovery rate is the percentage of the investor’s investment that is not lost, but recovered.

Credit spread risk: is the risk that the market yield (due to the credit spread) will rise, causing the price of the bond to decline

Downgrade risk: is the risk that the price of a bond might fall because the credit rating agencies reduce its credit rating

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AAA (or Aaa) Prime Grade AA (or Aa) High Quality Grade A Upper Medium Grade BBB (or Baa) Medium Grade

Investment Grade

Non Investment

Grad

BB (or Ba) Low Grade B Speculative

CCC (or Caa) Poor Grade (substantial risk) CC (or Ca) Very Speculative

C Extremely Speculative

CI Noninterest Bearing Income Bonds DDD, DD, D Default

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Liquidity Risk

Liquidity risk is the risk that a security will not be able to be sold quickly without giving up a large price concession The bid/asked spread is the best measure of liquidity risk: the wider the spread, the greater the liquidity risk

The market bid/asked spread can be determined by simply taking the difference between the lowest dealer asked and the highest dealer bid for an issue at a particular point in time

Liquidity risk is mostly a concern for investors who do not expect to hold a security to maturity or who must periodically mark it to market

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Liquidity risk can change for a number of reasons:

– If the structure of a bond structure is popular, the bid-ask-spread will narrow. If the structure of a bond is unpopular, the bid-ask-spread will widen

– When interest rates become more volatile, the demand of bonds decline which will cause the bid-ask-spread to widen

– When important traders of a certain type of bond exit the market, the spreads will tend to widen because of a lack of liquidity

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Exchange Rate Risk

Exchange rate risk is the risk that the exchange rate between the currency in which a bond is denominated and the currency of the investor’s home country might change

Inflation Risk

Inflation risk is the risk that the purchasing power of the cash flows received from a bond (interest and principal) will decline over time because of inflation

If an investor buys a bond with a 7% return but inflation is 6% the real return is only 1%

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Volatility Risk

Volatility risk is the risk that changes in the expected volatility of interest rates can affect the value of any embedded options in a bond’s pricing structure, thereby affecting the value of the bond.

Price Callable Bond = Price Noncallable Bond – Price embedded Call Option

Price Putable Bond = Price Nonputable Bond + Price embedded Put Option

The value of both puts and calls increase if volatility increases. Because the embedded option values are affected by changes in volatility, the price of bonds with embedded options will also be affected.

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Event Risk

Event risk is the risk that some unusual event could cause the price of bonds to decline: – Natural disasters such as famine, war etc.

– Takeover, leveraged buy-out, or corporate debt restructuring that substantially increases an issuer’s debt-to-equity ratio and causes downgrading of its credit rating

– A regulatory change that requires an issuer to conduct its affairs in ways that result in a downgrading of its credit rating (i.e. lower regulatory capital for banks)

– Political factors or actions taken by government that impair an issuer’s ability or willingness to pay its debt service

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Risks Associated with Investing in Bonds

Fixed-Income Valuation

Term Structure of Interest Rates

Yield Measures

Interest Rate Risk: Duration and Convexity

Credit Risk: Fundamentals of Credit Analysis

Managing Bond Portfolio

Content

Relative-Value Methodologies for Global

Corporate Bond Portfolio Management

Exchange Rate Risk: International Bond Investing

Managing Interest Rate Risk with Derivatives

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Valuation Principles

Interest rate risk is the major risk faced by fixed-income investors

The bond price is the present value of the sum of future cash flows (coupon payments plus the principal amount)

Therefore, if the discount rate r, which is the yield required by the market (which is related to interest rate levels)

increases, the price of the bond decreases, and vice versa. This is true for almost all bonds

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Fixed-Income Valuation

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The appropriate discount rate is the sum of the risk-free rate and a risk premium (nominal spread).

The yield on a U.S. Treasury security with the same maturity as the bond being valued can be used as a proxy for the risk-free rate.

Discount Rate for the Bond = Yield-to-Maturity

= Yield-to-Maturity of Treasury Security + Nominal Spread

where:

C = Coupon payments

Par = Par value of the bond at maturity r = Yield for maturity (discount rate)

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The appropriate discount rate (i) is the sum of the risk-free rate and a risk premium (i.e. the nominal spread). Discount Rate = Yield to Maturity = Risk-free Rate + Nominal Spread

Example (valuing annual-pay bonds):

A 3-year corporate bond has an annual coupon rate of 5% and a face value of USD 1,000. The discount rate is 4%. The bond is paid back at par at maturity. Calculate the price of the bond!

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Example (valuing semiannualy-pay bonds):

A 3-year corporate bond has a coupon rate of 5%, coupons are paid semiannualy and the bond has a face value of USD 1,000. The discount rate is 4%. The bond is paid back at par at maturity. Calculate the price of the bond!

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Relationships among a Bond’s Price, Coupon Rate, Maturity, and Market Discount Rate (Yield-to-Maturity)

The farther into the future a cash flow is received, the lower its present value will be. The higher the discount rate (yield to maturity) is, the lower the value of the bond will be, all other factors being equal.

A bond’s price and YTM are inversely related. An increase in YTM decreases the price and vice versa.

Prices are more sensitive to changes in YTM for bonds with lower coupon rates and longer maturities, and less sensitive to changes in YTM for bonds with higher coupon rates and shorter maturities.

If the yield-to-maturity of a bond is higher (lower) than its coupon rate, the bond will sell below (above) its par value. If the yield-to-maturity equals its coupon rate, the bond will sell at its par value.

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Coupon Rate > Yield-to-Maturity:

Bond Price > Par Value (Premium Bond)

Coupon Rate < Yield-to-Maturity:

Bond Price < Par Value (Discount Bond)

Coupon Rate = Yield-to-Maturity:

Bond Price = Par Value (Par Bond)

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There is a convex relationship between a bond’s price and its yield-to-maturity: (Bond Price) (Yield-to-Maturity)

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At a bond’s maturity date, the bond’s value is equal to its par value. As a bond moves closer to maturity (constant discount rate assumed), a bond’s value:

– A bond selling at a premium decreases over time – A bond selling at a discount increases over time

Example:

A 7% coupon, 4-year semiannual paid bond is priced at 96.63 and has a yield-to-maturity of 8%. If the yield to maturity remains unchanged, what will the bond’s price be in one year?

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Spot Rates and the Price of a Bond

Spot rates are market discount rates for single payments to be made in the future. The discount rates for zero-coupon bonds are spot rates because these rates do have no reinvestment risk.

Example:

The spot rates over the next 6 months, 12, months, 18 months, and 24 months are 4.0%, 4.2%, 4.4%, and 4.5%. Therefore the no-arbitrage price of a 2-year, 6% coupon Treasury note is:

The YTM of this issue is 4.489% (2,244% x 2);

N = 4, PMT = 3, PV = -102.86, FV = 100  I/Y = 2.244

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Flat Price, Accrued Interest, and Full Price

Full Price = Flat Price + Accrued Interest

The full price (dirty price) of a bond includes interest accrued between coupon dates. The flat price (quoted or clean price) of a bond is the full price minus accrued interest.

Accrued interest for a bond transaction is calculated as the coupon payment times the portion of the coupon period from the previous payment date to the settlement date

Methods for determining the period accrued interest include actual (typically used for government bonds) or 30-day months and 360-day years (typically used for corporate bonds)

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The full price of a fixed-income bond between two coupon payments given the market discount rate resp. the YTM can be calculated as:

The next coupon payment (C) is discounted for the remainder of the coupon period, which is 1 – t / T. The second coupon payment is discounted for that fraction plus another full period, 2 – t / T. This equation is simplified by multiplying the numerator and denominator by the expression (1 + YTM) ^ t/T:

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Example

A 6% German corporate bond is priced for settlement on 18 June 2015. The bond makes semi-annual coupon payments on 19 March and 19 September of each year and matures on 19 September 2026. The corporate bond uses the 30/360 day-count convention for accrued interest. Calculate the full price, the accrued interest, and the flat price per EUR 100 par value for a yield-to-maturity of 6,2%?

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Matrix Pricing

Matrix pricing is a method used to estimate the YTM for bonds that are not traded or infrequently traded. The yield is estimated based on the yields of traded bonds with the same credit quality. If these bonds have different maturities than the bond being valued, linear interpolation is used to estimate the subject bond’s yield.

For example, suppose that an analyst needs to value a 3-year, 4% semi-annual coupon payment corporate bond, Bond X. This bond is not actively traded. However, there are quoted prices for four corporate bonds that have similar credit quality:

– Bond A: 2-year, 3% semi-annual coupon paying bond with a price of 98.500, – Bond B: 2-year, 5% semi-annual coupon paying bond with a price of 102.25, – Bond C: 5-year, 2% semi-annual coupon paying bond with a price of 90.250 – Bond D: 5-year, 4% semi-annual coupon paying bond with a price of 99.125

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The semi-annual YTM of the four bonds are:

– Bond A: YTM = 3.786% (N = 4, PMT = 1.5, PV = - 98.5, FV = 100  I/Y = 1.8929) – Bond B: YTM = 3.821% (N = 4, PMT 2.5, PV = -102.25, FV = 100  I/Y = 1.9104) – Bond C: YTM = 4.181% (N = 10, PMT = 1, PV = -90.250, FV = 100  I/Y = 2.0906) – Bond D: YTM = 4.196% (N = 10, PMT = 2, PV = -99.125, FV = 100  I/Y = 2.0979)

The average yields for the 2-year bonds of 3.8035% and for the 5-year bond of 4.1885% are calculated:

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The estimated 3-year YTM can be obtained with linear interpolation. The interpolated yield is 3,9318%:

Thus, the 3-year Bond X has an estimated price of 100.191 (N = 6, PMT = 2, I/Y = 1.9659, FV = 100)

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Example

An analyst needs to assign a value to an illiquid 4-year, 4.5% annual coupon payment corporate bond. The analyst

identifies two corporate bonds that have similar credit quality: One is a 3-year, 5.5% annual coupon payment bond priced at 107.5 per 100 of par value, and the other is a 5-year, 4.5% annual coupon payment bond priced at 104.75 per 100 of par value. Using matrix pricing, the estimated price of the illiquid bond per 100 of par value is closest to:

A. 103.895 B. 104.991 C. 106.125

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August 2014 © Dr. Enzo Mondello, CFA, FRM, CAIA 39

Risks Associated with Investing in Bonds

Fixed-Income Valuation

Term Structure of Interest Rates

Yield Measures

Interest Rate Risk: Duration and Convexity

Credit Risk: Fundamentals of Credit Analysis

Managing Bond Portfolio

Content

Relative-Value Methodologies for Global

Corporate Bond Portfolio Management

Exchange Rate Risk: International Bond Investing

Managing Interest Rate Risk with Derivatives

Managing Credit Risk with Derivatives

Currency Risk Management

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The three different shapes of the yield curve are: – Normal: upward sloping (short rates < long rates) – Flat: no slope (short rates = long rates)

– Inverted: (short rates > long rates)

Historically the yield curve has been upward sloping more often than the other shapes.

The slope of the yield curve captures what the market is willing to pay for bonds of different maturities. The yield curve expresses the relationship between yield and maturity.

Typically the yields being measured are U.S. Treasury yields as these are default risk-free yields.

Term Structure of Interest Rates

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(Yields)

(Maturity) Normal Yield Curve

Inverted Yield Curve Flat Yield Curve

Treasury Yield Curve

Term Structure of Interest Rates

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The problems with using on-the-run Treasury issues are:

– On-the-run Treasury curve consists only of 6 points. Interpolation is needed

– Due to the strong dealer demand Treasury yields tend to be abnormally low, reducing their usefulness as a good benchmark

– They tend to have abnormally low reinvestment rate risk, and abnormally high interest rate risk

Despite the drawbacks of the on-the-run Treasury yield curve as the benchmark for valuing other fixed-income securities, it is the most widely used benchmark

An alternative to the on-the-run Treasury yield curve that is sometimes used is the yield curve for zero-coupon Treasury securities

The yield on zero-coupon bond securities is called the spot rate, with the yield on Treasury strips called the Treasury spot rate. When Treasury spot rates are plotted versus their maturities, the resulting curve is called the term

structure of interest rates

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A shift in the yield curve occurs when yields change.

In a parallel shift all yields change across the term structure by the same amount. A nonparallel shift occurs when the changes in yield are different for different maturities.

Yield Maturity Initial Curve Yield Maturity Initial Curve

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A yield curve twist occurs when the curve (the slope of the curve) flattens or steepens due to a nonparallel shift. The yield curve can become flatter (less difference between long and short rates) or steeper (more difference between long and short rates).

A butterfly twist occurs when the curvature of the curve changes. – Positive butterfly: the curve becomes more straight (less humped).

– Negative butterfly: the curve becomes less of a straight line (more humped).

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Factors that Drive U.S. Treasury Security Returns

Researchers generally agree that three factors are responsible for changes in Treasury returns:

– Changes in the level of interest rates. This is by far the most important factor. It accounts for about 90% of historical returns and is measured by duration.

– Changes in the slope of the yield curve (distant second most influential factor). It accounts for about 8.5% of historical returns and is measured by key rate duration.

– Changes in the curvature of the yield curve (slight impact). It accounts for about 1.5% of historical bond returns.

Bond portfolio managers who want to hedge their interest rate risks, therefore, should be most concerned about protecting against the adverse effects of changes in the level of interest rates.

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Various Universes of Treasury Securities used to Construct the Theoretical Spot Rate Curve

Constructing a theoretical spot rate yield curve is not simple. Ideally we would use the yield on default risk-free zero coupon bonds (to abstract from the coupon effect) for each maturity in the maturity spectrum.

There are several different combinations of Treasury securities that can be used to construct a default-free theoretical spot rate curve:

– On-the-run Treasury issues are the most recently auctioned issues of a given maturity. The Treasury is currently issuing bills with maturities of 1, 3, and 6 months, notes and bonds with maturities of 2, 5, 10, and 30 years. The bills are issued at a discount while the notes and bonds carry coupons. The resulting on-the-run yield curve is a par coupon curve because the notes and bonds are issued at par. Securities issued at par eliminate the tax effect that exist for securities issued at a discount or premium. The bootstrapping methodology is used to generate the theoretical spot rate curve. A potential criticism is that large maturity gaps exist, particularly after 5 years.

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Selected off-the-run Treasury issues can be added to the on-the-run issues to bridge the caps in on-the-run maturities. The par coupon yield curve is estimated and remaining gaps are filled by interpolation. Like with all on-the-run issues, bootstrapping is used to generate the theoretical spot rate curve.

All Treasury issue so that all coupon securities and bills are used. As a practical matter issues that have special circumstances such as tax advantages, illiquid markets, futures contract delivery are usually omitted to avoid yield distortions. Adjustments are made for taxes and call features. The advantage is that all information available in prices can be used.

Treasury coupon strips are observable zero-coupon securities that can be used directly to create an actual spot rate curve. The relative illiquidity in the strips market implies that strip rates include a premium.

Term Structure of Interest Rates

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Swap Rate Curve (LIBOR Curve)

LIBOR is the rate at which high quality banks will borrow or lend U.S. dollars outside the U.S. amongst themselves, and 3 months LIBOR is the most common floating rate used in interest rate swap agreements. The LIBOR spot rate curve is calculated using the same bootstrapping procedure used to calculate Treasury spot rates.

The swap rate curve represents the swap rates available at various future time periods to convert fixed rates to floating rates and vice versa

The swap rate curve is used to hedge interest rates, to value bonds, and for performance evaluation

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The swap rate curve tends to be better benchmark than the government bond curve for the following reasons: – There is little or no government regulation of the swaps market.

– A large demand for government bonds in the repo market can unrealistically change the yield curve. The swaps market does not have these yield problems.

– The swap curve has the credit risk of the underlying banks. Credit risks are thus more similar in the swaps market (LIBOR) than when comparing various government bond market.

– The swaps market has more bond maturities to construct a yield curve than the government bond market. Swap rates quoted in the swap market have maturities of 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, and 30 years.

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Expectations Theory Biased Expectations Theory Pure Expectations Theory Broadest

Interpretation Expectations Local Liquidity Theory Habitat Theory Preferred Segmentation

Theory

Theories of the Term Structure of Interest Rates

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Pure (unbiased) expectations theory says the investor’s expectations of future interest rates alone creates the shape of the yield curve. Forward rates are the expected future spot rates. This implies that if the yield curve is upward (downward) sloping, term rates are expected to rise (fall), and if the yield curve is flat, the market expects short-term rates to be constant. The drawback is that it fails to consider price risk and reinvestment risk, but interest risk increases as the term to maturity increases.

– The broadest interpretation is that given any investment horizon, investors expect the same return, regardless of the maturity of the investment vehicle selected. This ignores the price risk associated with selling a bond prior to its maturity.

– The local expectations form of the pure expectation theory is an interpretation that suggests that the return on bonds with different maturities will be identical over a short-term investment period, commencing immediately. This is the only interpretation of the pure expectation theory that can be sustained in equilibrium.

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The two form of biased expectations theory are the liquidity theory and the preferred habitat theory.

– Liquidity theory: duration measures the price risk of holding a bond. Duration increases as the bond’s maturity lengthens. Liquidity theory says that investors will demand a risk premium for holding bonds with long maturities because the risk of this bonds is higher. - The yield curve will typically be upward sloping as investors demand higher yields on longer bonds. The yield curve could slope downwards, however, if expectations for lower rates in the future overwhelm the risk premium.

– Preferred habitat theory also proposes that forward rates represent expected future spot rates plus a premium, but it does no support the view that this premium is directly related to maturity. The existence of an imbalance between supply and demand for funds in a given maturity range will induce lenders and borrowers to shift from their preferred habitat (maturity range) to one that has the opposite imbalance. To do so, they must be offered a risk premium to compensate for the price and/or reinvestment risk.

Term Structure of Interest Rates

(53)

Market segmentation theory is similar to the preferred habitat theory in that it agrees that lenders and borrowers have preferred maturity ranges and there is no premium (or discount) large enough to induce investors out of their preferred maturity range.

Instead, the shape of the yield curve is proposed to be determined by the supply and demand for securities within a given maturity range. In the extreme, the segmentation theory implies that rates for a given maturity segment will be determined independently of all other maturities.

The shape of the yield curve depends exclusively on the supply and demand within maturity segments. Under this theory the yield curve can take any shape.

Term Structure of Interest Rates

3

(54)

Spot Rate Curve, Yield Curve on Coupon Bonds, Par Curve, and Forward Rate Curve

A yield curve shows the term structure of interest rates by displaying yields across different maturities (i.e., yields of U.S. Treasury coupon bonds). Yields are calculated for several maturities and yields for bonds with maturities between these are estimated by linear interpolation.

The spot rate curve is a yield curve for single payments in the future, such as 0%-bonds or stripped Treasury par bonds. Yields on zero-coupon government bonds are spot rates.

The par curve shows the coupon rates for bonds of various maturities that would result in bond prices equal to their par values. It is not calculated from yields on actual bonds but is constructed from the spot rate curve.

Term Structure of Interest Rates

(55)

With spot rates of 1%, 2%, and 3%, a 3-year annual par bond will have payments that are:

Thus, the payment is 2,96 and the par bond coupon rate is 2.96%

A forward curve is a yield curve composed of forward rates, such as 1-year rates available at each year over a future period

 

(

1 .

03 )

100 PMT

2 .

96

100 PMT

)

02 .

1 (

PMT

01 .

1 PMT

3 2









Term Structure of Interest Rates

3

(56)

Forward Rates

The general formula to calculate a semi-annual forward is:

where:

1fm forward rate that starts in m semi-annual periods for 6 months zm spot rate for a period of m semi-annual periods

zm+1 spot rate for a period of m semi-annual periods plus 6 months

Doubling the forward rate 1fm gives the bond equivalent yield for the forward rate that starts in m months for 6 months

1 )

z

1 (

)

z

1 (

f

_m m 1 m 1 m m 1

_







 

Term Structure of Interest Rates

(57)

Example

The spot rates for 6-month Treasury bills and 1-year Treasury bills are 2.50% and 2.80% respectively, expressed as bond equivalent yields. The 6-month forward rate expressed as bond equivalent yields is closest to:

A) 2.96% B) 3.00% C) 3.10%

Term Structure of Interest Rates

3

(58)

Example

Given the following spot rate curve, the implied forward rate in 12 months for 6 months is closest to:

Maturity Spot Rate

6 months 3.00% 12 months 4.00% 18 months 5.00% 24 months 6.00% A) 6.55% B) 7.02% C) 7.54%

Term Structure of Interest Rates

(59)

The relationship between a T-period spot rate (zT), the current 6-month spot rate (z1), and the 6-month forward rates is stated as:

where:

1f1 forward rate that starts in 6 months for 6 months 1f2 forward rate that starts in 12 months for 6 months

Just know that a spot rate is a package of forward rates and that discounting at either the forward rates or the spot rate will give the same present value.



(

1 z

)(

1 f

)(

1 f

)...(

1 f



1 z

_T





₁



₁ ₁



₁ ₂



₁ _T_₁ 1/T



Term Structure of Interest Rates

3

(60)

Example

The following forward rates are given:

Semi-annual Periods Notation Forward Rate

1 1f0 4.00%

2 1f1 4.60%

3 1f2 5.00%

4 1f3 5.20%

The 2-year spot rate is closest to: A) 4.40%

B) 4.70% C) 4.95%

Term Structure of Interest Rates

(61)

Example

The following forward rates are given:

Semi-annual Periods Notation Forward Rate

1 1f0 4.00%

2 1f1 4.60%

3 1f2 5.00%

4 1f3 5.20%

A 6% coupon bond pays the coupons semi-annually and has a remaining maturity of 1.5 years. The price of the bond is

closest to: A) 101.56% B) 102.00% C) 102.12%

Term Structure of Interest Rates

3

(62)

Risks Associated with Investing in Bonds

Fixed-Income Valuation

Term Structure of Interest Rates

Yield Measures

Interest Rate Risk: Duration and Convexity

Credit Risk: Fundamentals of Credit Analysis

Managing Bond Portfolio

Content

Relative-Value Methodologies for Global

Corporate Bond Portfolio Management

Exchange Rate Risk: International Bond Investing

Managing Interest Rate Risk with Derivatives

1

2

3

4

5

6

7

8

9

10

(63)

Yield Measures for Fixed-Rate bonds, Floating Rate Notes, and Money Market Instruments

The effective yield of a bond depends on its periodicity, or annual frequency of coupon payments. For an annual-pay bond the effective yield is equal to the yield-to-maturity. For bonds with greater periodicity, the effective yield is greater than the YTM. For example, a semi-annual coupon paying bond with a YTM of 8% has a yield of 4% every 6 months and an effective yield of 1.04 ^2 – 1 = 8.16%.

A YTM quoted on a semi-annual basis is two times the semi-annual discount rate: 2 x 4% = 8%.

Yield Measures

4

(64)

An important tool used in fixed-income analysis is to convert an annual yield from one periodicity to another. These are called periodicity, or compounding, conversions. A general formula to convert an annual percentage rate for m periods per year, denoted as APRm, to an annual percentage rate for n periods per year, APRn, is the following equation: n n m m

n

APR

1 m

APR

1 









 













 

Yield Measures

4

(65)

Example

A 5-year, 4.5% semi-annual coupon payment government bond is priced at 98 per 100 of par value. Calculate the annual yield-to-maturity stated on a semi-annual bond basis, rounded to the nearest basis point. Convert the annual yield to: A. An annual rate that can be used for direct comparison with otherwise comparable bonds that make quarterly coupon

payments and

B. An annual rate that can be used for direct comparison with otherwise comparable bonds that make annual coupon payments

Yield Measures

4

(66)

Investors of fixed-income securities obtain their total return from the following three sources: – Coupon interest

– Capital gains / losses resulting from buying at a different price than the one received when the security is sold or matures

– Reinvestment income from investing interim cash flows (interest on interest)

Yield Measures

(67)

Current Yield

The current yield on a security is simply its coupon rate, divided by its market price:

This yield calculation ignores potential capital appreciation / depreciation and reinvestment income, and does not incorporate the influence of the time value of money.

Price

Bond

Year

per

Payment

Coupon

Cash

Yield

Current



Yield Measures

4

67

(68)

Yield to Maturity (YTM)

This yield is the single discount rate that when applied to all of the cash flows generated by a fixed-income security over its term to maturity, will make the present value of those cash flows equal to the current price of the bond:

The yield to maturity takes into account the time value of money and the potential for capital appreciation

This measure does not consider the reinvestment rate. In reality, the reinvestment rate rarely ever equates to the YTM. The YTM also is based upon the assumption that the bond will be held to maturity.

n n n 2 2 1 1

)

YTM

1 (

Par

)

YTM

1 (

C

...

)

YTM

1 (

C

)

YTM

1 (

C

ice

Pr





Yield Measures

4

(69)

Example:

Suppose we have a 10-year, USD 1‘000 par value, 7% semi-annual coupon paying bond. The bond price today is USD 895.80. Compute the current yield and the yield to maturity!

Yield Measures

4

(70)

Yield to Call (YTC)

Callable bonds might not reach maturity because they can be called before their maturity date. Therefore, the yield-to-call was developed to measure the return on a bond if it were to be yield-to-called on a particular date. The yield to first yield-to-call is the same as yield to maturity, calculated through the first call date with the call price as the maturity value.

where: n = number of periods to first call date

The YTC should be used whenever a callable bond is trading at a price greater than or equal to its par value. Any additional premium above this price could be lost if the bond were called away, and thus the YTC will be a more conservative return measure.

n n n 2 2 1 1

)

YTC

1 (

Price

Call

)

YTC

1 (

C

...

)

YTC

1 (

C

)

YTC

1 (

C

ice

Pr





Yield Measures

4

(71)

There are several problems with these yield-to-call measures: – They assume the bond is held to the call date

– They assume the issuer calls the bond on the call date – They assume the coupons are reinvested at the yield-to-call

Example

Compute the yield-to-first-call and the yield-to-first-par-call for a 8% coupon (paid semi-annually), 7-year bond priced at 93 that is callable in 4 years at 106 and in 6 years at 100!

Yield to first par call date

This measure is the same as YTC, using expected cash flows to the first date at which the issuer can call the bond at par

Yield Measures

4

(72)

Yield to Worst (YTW)

Yield to worst involves the calculation of YTC for every possible call date as well as the YTM, and determining which of these results is the lowest expected return. This yield is supposed to be the worst possible yield that can be realized by the investor.

In reality, the YTW measure has little meaning because it does not identify a bond’s true return, except in the rare event that the worst possible conditions do happen to materialize

Furthermore, the YTW measure incorporates a conglomeration of different reinvestment risk exposures

Yield Measures

(73)

The yield to worst is a commonly cited yield measure for fixed-rate callable bonds used by bond dealers and investors. However, a more precise approach is to use an option pricing model and an assumption about future interest rate volatility to value the embedded call option.

Option-adjusted price = flat price of bond + value of embedded call

The investor bears the call risk, so the embedded call option reduces the value of the bond from the investor’s perspective. The investor pays a lower price for the callable bond than if it were option-free. If the bond were non-callable, its price would be higher. The option-adjusted price is used to calculate the option-adjusted yield. – The option-adjusted yield is the required market discount rate whereby the price is adjusted for the value of the embedded call option.

Value of Call = Price of option-free Bond – Price of Callable Bond

Yield Measures

4

(74)

Floating-rate Note Yields

Floating rate notes have a quoted margin relative to a reference rate, typically LIBOR

The quoted margin is positive for issuers with more credit risk than the banks that quote LIBOR and may be negative for issuers that have less credit risk than loans to these banks

The required margin on a floating rate note may be greater than the quoted margin if credit quality has decreased, or less than the quoted margin if credit quality has increased

Yield Measures

(75)

The valuation of a floating-rate note needs a pricing model:

where:

Index reference rate, QM quoted margin

FV future value paid at maturity or par value of bond

m periodicity of the floating-rate note (number of payment periods per year)

DM discount margin

N number of evenly spaced periods to maturity

N 1

m

DM

Index

1 FV

m

FV

)

QM

Index

(

...

m

DM

Index

1 m

FV

)

QM

Index

(

P













_



















_









Yield Measures

4

75

(76)

Example

A 4-year, French floating-rate note pays three-month Euribor plus 1.25%. The floater is priced at 98 per 100 of par value. Calculate the discount margin for the floater assuming that three-month Euribor is constant at 2%. Assume the 30/360 day-count convention and evenly spaced periods.

Yield Measures

(77)

Yields for Money Market Instruments

For money market instruments yields may be quoted on a discount basis or an add-on basis, and may use 360-day or 365-day years. A bond-equivalent yield is an add-on yield based on a 365-day year.

Commercial papers, T-bills, and bankers’ acceptances often are quoted on a discount basis

Bank certificates of deposits, repos, and such indices as LIBOR and Euribor are quoted on an add-on basis

Yield Measures

4

(78)

The pricing formula for money market instruments quoted on a discount rate basis is:

where:

PV present value, or price of the money market instrument

FV future value paid at maturity, or face value of the money market instrument

Days number of days between settlement and maturity Year number of days in the year

DR discount rate (stated as an annual percentage)













































_

_





FV

PV

FV

Days

Years

DR

Year

Days

1 FV

PV

Yield Measures

4

(79)

Example

91-day U.S. Treasury bill with a face value of USD 1 million is quoted at a discount rate of 2.25% for an assumed 360-day year. What is the price of the T-bill?

Example

91-day U.S. Treasury bill with a face value of USD 1 million is quoted at a price of USD 976,450 for an assumed 360-day year. What is the quoted discount rate of the T-bill?

Yield Measures

4

(80)

The pricing formula for money market instruments quoted on an add-onrate basis is:

where:

PV present value, or price of the money market instrument

FV future value paid at maturity, or face value of the money market instrument

Days number of days between settlement and maturity Year number of days in the year

AOR add-on rate (stated as an annual percentage)













































_

_



PV

FV

Days

Years

AOR

Year

Days

1 FV

PV

Yield Measures

4

(81)

Yield Spread Measures

Yield Spread = Bond Yield – Benchmark Yield

If the benchmark is a government bond yield, the spread is known as a government spread or G-spread. If the benchmark is a swap rate, the spread is known as an interpolated spread or I-spread.

A disadvantage of G-spreads and I-spreads is that they are theoretically correct only if the spot yield curve is flat and approximately the same across maturities. However, the spot yield curve is normally upward-sloping.

Yield Measures

4

(82)

A zero-volatility spread or Z-spread is the percent spread that must be added to each spot rate on the benchmark yield curve to make the present value of a bond equal to its price. – Thus, the Z-spread accounts for the shape of the yield curve.

where:

z benchmark spot rates Z Z-spread

In practice, the Z-spread is usually calculated in a spreadsheet using a goal seek function or similar solver function. n n n 2 2 2 1 1 1

)

Z

z

1 (

Par

C

...

)

Z

z

1 (

C

)

Z

z

1 (

C

P





Yield Measures

4

(83)

Example

A 6% annual coupon corporate bond with two years remaining to maturity is trading at a price of 100.125. The 2-year, 4% annual payment government benchmark bond is trading at a price of 100.750. The 1-year and 2-year government spot rates are 2.1% and 3.635%, respectively, stated as effective annual rates.

1. What is the G-spread?

2. Demonstrate that the Z-spread is 234.22 bps

Yield Measures

4

(84)

Investors will require a larger spread for an issue with an embedded option that is favourable to the issuer (call option). If interest rates fall, the issuer will call the bond forcing the investor to reinvest at lower rates and

reducing their return. The option-adjusted spread (OAS) is used to price bonds with embedded options. The OAS for a

callable bond is calculated as follows:

OAS (bps.) = Z-Spread (bps.) – Option Value (bps.)

Since embedded options will clearly impact the spread one way or another, a Z-spread calculation does not give nearly as accurate a picture as an option-adjusted spread calculation.

Yield Measures

(85)

Option-adjusted spread (OAS) removes the effect of the embedded options and shows the average spread the investor will actually earn over a comparable Treasury security.

7-Year Maturity Bonds

Credit First Call in Z-Spread OAS Quality

AA 1-year 55 bps. 30 bps. AA 3-years 89 bps. 28 bps. Baa 3-years 77 bps. 40 bps.

The Z-spreads cannot be used to compare the bonds because they are based only on spot rates and do not take into account the impact of call features. The OAS is lower then the Z-spread for all the callable bonds because it considers the adverse affect of the embedded call.

Yield Measures

4

(86)

Risks Associated with Investing in Bonds

Fixed-Income Valuation

Term Structure of Interest Rates

Yield Measures

Interest Rate Risk: Duration and Convexity

Credit Risk: Fundamentals of Credit Analysis

Managing Bond Portfolio

Content

Relative-Value Methodologies for Global

Corporate Bond Portfolio Management

Exchange Rate Risk: International Bond Investing

Managing Interest Rate Risk with Derivatives

1

2

3

4

5

6

7

8

9

10

(87)

Macaulay Duration

Macaulay duration is named after Frederick Macaulay, a Canadian economist who first wrote about the statistic in 1938.

Macaulay duration is the weighted average of the time to receipt of the bond’s promised payments, where the weights are the shares of the full price that correspond to each of the bond’s promised future payments.

Interest Rate Risk

Duration and Convexity

5

(88)

Example

A 4-year 5% annual coupon paying bond is trading at par. What is the Macaulay duration? 1

2 3 4



The Macaulay duration is 3.7232 years

Period Cash Flow PV Weight Period x Weight

5 5 5 105 4.76 4.54 4.32 86.38 100.00 0.0476 0.0454 0.0432 0.8638 1.0000 0.0476 0.0908 0.1296 3.4552 3.7232

Interest Rate Risk

Duration and Convexity

(89)

There is also a general closed-form solution which can be used to calculate the Macaulay duration (MD):

where:

r expected return (yield-to-maturity) c coupon rate

N maturity of the bond













(

t

/

T

)

r

1 r

1 c

)

r

c

(

N

r

1 r

r

1 MD

_N

_













































0

3 .

7232

05 .

0

1

05 .

1

05 .

0 )

05 .

0

05 .

0 (

4

05 .

1

05 .

0

05 .

1 MD

₄

_































Interest Rate Risk

Duration and Convexity

5

(90)

Modified Duration

The calculation of the modified duration statistic of a bond requires a simple adjustment to Macaulay duration.

where:

r expected return (yield-to-maturity)

For the example, the modified duration of the 4-year, 5% annual coupon paying bond is 3.546:

r

1 Duration

Macaulay

Duration

Modified





546 .

3

05 .

1 7232

.

3 Duration

Modified



Interest Rate Risk

Duration and Convexity

(91)

Interpretation of duration:

1st interpretation of duration:

– Effective duration is the first derivative of the price-yield relationship of a security, divided by the initial price of the security:

– While correct, this interpretation is mathematically.

P

dP/dy

D

_E





Interest Rate Risk

Duration and Convexity

5

(92)

2nd interpretation of duration:

– Unadjusted duration (Macaulay duration) is a weighted average of time. – The Macaulay duration may be

meaningful to an investment professional who understands that a bond with a duration of 6 years is more volatile than a bond with a duration of 2 years, but it does not have much meaning to most clients.

3rd interpretation of duration:

– Effective duration is a measure of how sensitive the return on a bond is to small changes in interest rates

– This interpretation is easily understood by almost anybody. It indicates that if a bond has a duration of 4.0, its price will rise or fall by 4% every time interest rates fall or rise by 100 basis points.

dy

dP/P

D

_E





Interest Rate Risk

Duration and Convexity

(93)

Approximate Modified Duration

An alternative approach is to approximate modified duration directly:

where:

P– Price of the bond after a decrease in yield P+ Price of the bond after an increase in yield.

Let’s assume a yield change of 50 bps. for your 4-year, 5% annual coupon paying bond. The approx. modified duration is: 0

P

)

Yield

(

2 P

P

Duration

Modified

.

Approx











 

547 .

3

100 )

005 .

0 (

2

247 .

98

794 .

101 Duration

Modified

.

Approx









Interest Rate Risk

Duration and Convexity

5

(94)

Example:

An option-free bond has a remaining maturity of 5 years and a coupon of 4.5% which is paid semi-annually. The yield to maturity of the bond is 4.8%. Calculate the approximate modified duration and (based on the duration) the new bond price if interest raise by 50 basis points.

Interest Rate Risk

Duration and Convexity