Fixed-Income Securities: Pricing and Trading

Chapter Overview

Learn how to calculate the price and yield of fixed-income securities.
Understand interest rates on bonds, including nominal vs. real rate of return.
Learn about yield curves and their determination based on three theoretical principles.
Understand how and why bond prices fluctuate based on fixed-income pricing properties.
Learn about bond trading, rules, regulations, delivery, and settlement.
Understand how bond indexes are utilized by portfolio managers for performance measurement and to construct bond index funds.

Learning Objectives

Perform calculations related to bond pricing and yield.
Describe factors determining the term structure of interest rates and the shape of the yield curve.
Explain how bond prices react to changes in interest rates, maturity, coupon, and yield.
Describe how bond trading is conducted.
Define bond indexes and their use in the securities industry.

Key Terms:

Accrued Interest
Bearer Bonds
Buy Side
Current Yield
Discount Rate
Duration
Expectations Theory
Inter-Dealer Broker
Liquidity Preference Theory
Market Segmentation Theory
Nominal Rate
Present Value
Real Rate of Return
Registered Bonds
Reinvestment Risk
Sell Side
Trade Ticket
Yield Curve

Introduction

Before recommending fixed-income securities, understand the risks and rewards.
Crucial to know how bond yields and prices are determined and the relationship between prices and interest rates.
Typical scenario: buy a bond, receive interest, hold to maturity, redeem at face value.
Bonds can be bought in secondary markets.
Price is a concern for investors seeking capital gains.
Economic conditions and interest rates affect bond and equity prices, but not identically.
Focus: methods to determine the fair price and fixed-income pricing properties.
Understand the impact of various events on markets and fixed-income security prices.

Calculating Price and Yield of a Bond

Present Value

The most accurate method to determine bond value.
Present value: amount an investor pays today for a guaranteed sum in the future.
Example: Receive $1,000$ in one year, current interest rate is 5%.
Present value will be less than $1,000$ .
How much must be invested today at 5% to get $1,000$ in the future?
$Present Value = { Future Value } / { 1 + Interest or Discount Rate }$
Present Value = $1,000 / 1.05 = $952.38
$952.38 invested today at 5% will grow to $1,000 in one year.
Verification: 952.38 + 5\% = $1,000 or 952.38 \times 1.05 = $1,000
Typical bonds provide regular coupon payments and principal at maturity.
Bond value equals the sum of the present values of all future cash flows.

Calculating Present Value of Bonds with Coupon Payments:

Choose the appropriate discount rate.
Calculate the present value of the income stream from coupon payments.
Calculate the present value of the principal at maturity.
Add these present values together.

General Formula:
$PV = \frac{C<em>1}{1+r} + \frac{C</em>2}{(1+r)^2} + … + \frac{C_n}{(1+r)^n} + \frac{FV}{(1+r)^n}$
Where:
- $PV$ = Present value of the bond
- $C$ = Coupon payment
- $r$ = Discount rate per period
- $n$ = Number of compounding periods to maturity
- $FV$ = Principal received at maturity (future value)
The math is not intended to be cumbersome; focus on interpretation.
Example: Four-year, semi-annual, 9% coupon bond, $10\%$ discount rate, $100$ principal used throughout examples.

The Discount Rate

Rate at which future values are discounted to present values.
Based on the risk of the bond.
Estimated using yields of bonds with similar coupon, term, and credit quality.
Yields are determined by the marketplace and change with market conditions.
Yields are often quoted as the Government of Canada bond with similar term plus a spread in basis points.
Discount rate and yield often used interchangeably.
Do not confuse discount rate with the coupon rate (set when the bond is issued and generally doesn't change).
If interest is paid more than once a year, adjustments are required.
Semi-annual adjustments (most bonds):
- Coupon = (9\% / 2 ) \times $100 = $4.50 per period
- Compounding periods = $4 \text{ years } \times 2 = 8$ periods
- Discount rate = $10\% / 2 = 5\%$ per period

Calculating the Fair Price of a Bond

Fair price: present value of principal plus present value of all coupon payments.
Shows the timing of cash flows on the example four-year, semi-annual, 9% bond
Coupon payments made twice a year; at maturity, bondholder receives the final coupon plus the return of principal.
Discounting these cash flows back to the present allows solving for the present value.
Present value: future amount divided by $(1 + \text{interest rate})^{\text{number of compounding periods}}$ .
This method is called discounting cash flows.
Calculation can be done manually or with a financial calculator (quicker and more precise).
Present Value of a Four-Year, Semi-Annual, 9% Bond:
$PV = \frac{4.50}{(1+0.05)^1} + \frac{4.50}{(1+0.05)^2} + … + \frac{4.50}{(1+0.05)^8} + \frac{100}{(1+0.05)^8}$
To calculate $(1 + r)^n$ , use the $y^x$ or $y^{\text{exp}}$ key on a calculator.
E.g., $(1.05)^8$ . Key in 1.05, press $y^x$ , enter 8, and press = to get 1.4775.

Present Value of the Income Stream

The present value of a bond's income stream is the sum of the present values of each coupon payment.
For the four-year, semi-annual, 9% bond ( $100$ par value), eight semi-annual coupon payments of $4.50$ each, totaling $36$ .
The present value of each $4.50$ coupon, added together, represents the present value of the bond's income stream.
Using a financial calculator:
- Type 8, then press N.
- Type 5, then press I/Y.
- Type 4.50, then press PMT.
- Type 0, then press FV (telling the calculator you are not interested in the principal).
- Press COMP, then press PV.
- Answer: −29.0845
A negative value denotes an outflow of money, whereas a positive value denotes an inflow of money.
This calculation tells us that the value of the stream of eight coupon payments totaling $36$ is worth $29.08$ today

Present Value of the Principal

Single cash flow to be received in the future.
Calculator steps:
1. Type 8, then press N.
2. Type 5, then press I/Y.
3. Type 0, then press PMT (to tell the calculator you are not interested in the coupons).
4. Type 100, then press FV.
5. Press COMP, then press PV.
6. Answer: −67.6839
Present value of the principal is approximately $67.68$ .
If you invested $67.68$ at a semi-annual rate of 5%, you would receive $100$ in four years.
Verification: 67.6839 + 5\% + 5\% + 5\% + 5\% + 5\% + 5\% + 5\% + 5\% = $100

Present Value of the Bond

Fair price: sum of present value of coupons and principal.
Coupons worth $29.08$ , principal worth $67.68$ .
At a $10\%$ discount rate, the present value is $96.77$ (29.0844 + $67.6839).
Calculator (one-step):
1. Type 8, then press N.
2. Type 5, then press I/Y.
3. Type 4.50, then press PMT.
4. Type 100, then press FV.
5. Press COMP, then press PV.
6. Answer: −96.7684
$96.77 indicates the price at which the bond will be quoted for trading in the secondary market, given current market conditions.
Bond value: sum of the present value of coupons plus the present value of principal, based on a discount rate reflecting risks.
Discount rate changes reflect the yield investors expect with changing economic conditions.
Financial calculator simplifies calculations, but knowing manual steps is important. The calculations are:
- Step 1: Present Value of the Principal
  $PV = \frac{FV}{(1+r)^n} = \frac{100}{(1+0.05)^8} = \frac{100}{1.47746} = 67.6839$
  Therefore, the present value of the principal is $67.68$ .
- Step 2 - Method 1: Present value of the income stream
  $PV = \frac{4.50}{(1+0.05)^1} = 4.2857$
  The present value of the first coupon to be received six months from now is approximately $4.29$ .
  In the same example, the present value of the second coupon is calculated as follows:
  $PV = \frac{4.50}{(1+0.05)^2} = 4.0816$
  Therefore, the present value of the coupon to be received a year from now is approximately $4.08$ . You can verify
  this with your calculator by entering 4.0816 + 5\% + 5\% = $4.50.
  Repeat this process for each of the coupon payments to be received, and add the present values together to
  obtain the present value of the income stream. In this example, the result is $29.08$ (calculated as 4.29 + $4.08 + $3.89 + $3.70 + $3.53 + $3.36 + $3.20 + $3.05).
- Step 2 - Method 2: Present value of the income stream
  A faster way to calculate the present value of a series of time payments is by using the formula for the present
  value of an annuity. With this formula, the sum of the present value of all coupons is found all at once.
  $APV = C \times \frac{[1 - (1+r)^{-n}]}{r}$
  Where:
```
*    $APV$  = Present value of the series of coupon payments
*    $C$  = Payment (the value of one coupon payment)
*    $r$  = Discount rate per period
*    $n$  = Number of compounding periods
```
  $APV = 4.50 \times \frac{[1 - (1 + 0.05)^{-8}]}{0.05} = 4.50 \times \frac{[1 - 0.676839]}{0.05} = 4.50 \times \frac{0.323161}{0.05} = 4.50 \times 6.4632 = 29.084$
- Step 3: Present Value of the Bond
  The fair price for a bond is the sum of its two sources of value: the present value of its principal and the
  present value of its coupons. Therefore, at a discount rate of $10\%$ , this bond has a value today of $96.77$
  (calculated as 29.0844 + $67.6839).

Calculating the Yield on a Treasury Bill

Treasury bill (T-bill): a very short-term security that trades at a discount and matures at par.
No interest is paid in the interim.
Return: difference between purchase price and sale/maturity value.
Earnings are treated as interest income for tax purposes.
Formula:
$\text{Yield} = \frac{\text{100 - Price}}{\text{Price}} \times \frac{\text{365}}{\text{Term}} \times 100$
Example: 89-day T-bill purchased for $99.5$
$\text{Yield} = \frac{\text{100 - 99.5}}{\text{99.5}} \times \frac{\text{365}}{\text{89}} \times 100 = 2.061\%$

Calculating the Current Yield on a Bond

Current yield considers cash flows and current market price, not original investment.
Applicable to any investment (bond or stock).
$\text{Current Yield} = \frac{\text{Annual Cash Flow}}{\text{Current Market Price}} \times 100$
Example: Four-year, semi-annual, 9% bond trading at $96.77$ .
$\text{Current Yield} = \frac{\text{9.00}}{\text{96.77}} \times 100 = 9.30\%$

Calculating the Yield to Maturity on a Bond

YTM: the most popular measure of yield in the bond market
Total return expected over the bond's life, assuming reinvestment of coupons at the same YTM.
YTM accounts for current market price, term to maturity, par value, and coupon rate.
Requires finding the implied interest rate (r) in the present value formula.
Assumes investor is repaid the par value at maturity.
YTM reflects coupon income and any capital gain (discount purchase) or loss (premium purchase).
Manual YTM calculation is difficult; use a financial calculator.
Example: four-year, semi-annual, 9% bond, trading at $96.77$ .
1. Type 8, then press N.
2. Type 4.50, press PMT.
3. Type 96.77, then press +/−, then press PV.
4. Type 100, then press FV.
5. Press COMP, then press I/Y.
6. Answer: 4.9997 (rounded to 5)
Semi-annual YTM is 5.0%. Annual YTM is $10\%$ ( $5\% \times 2$ ) (bond trading at a discount).
Buying at $96.77$ , holding to maturity, receiving eight payments of $4.50$ plus $100$ at maturity.
YTM factors in the $3.23$ gain (100 - $96.77), coupon income, and reinvestment of coupon income at this YTM.

Approximate Yield to Maturity—Manual Calculation:

$AYTM = \frac{\text{Interest Income} + \frac{\text{Par Value - Purchase Price}}{\text{Number of Years}}}{\frac{\text{Purchase Price + Par Value}}{2}}$

 $AYTM = \frac{\text{C + (FV - PV) / n}}{\frac{\text{FV + PV}}{2}}$

Use $+/−$ to show buying above or below par.
Buy a bond at a discount (e.g., 92), gain at maturity (add price appreciation to interest income).
Buy a bond at a premium (e.g., 105), loss at maturity (subtract the price decrease from the interest income).
Example: four-year, semi-annual, 9% bond, trading at 96.77 maturing at 100; semi-annual interest income is $4.50$ .
The annual price change is $3.23$ ( $100 - 96.77$ ).
$3.23 / 8 = 0.4038$ price gain per period
Average price: $(96.77 + 100) / 2 = 98.385$
$AYTM = \frac{\text{4.50 + 0.4038}}{\text{98.35}} = 4.9842\%$
Approximate annual $YTM = 9.9684\%$ ( $4.9842\% \times 2$ ).
Close to the calculator's result (calculator is more precise).
Bond quote includes price, maturity date, coupon rate, and YTM.
YTM is the most important measure.
YTM: average rate of return if bought today and held to maturity.
Assumes coupon payments are reinvested at the prevailing YTM at the time of purchase.
Bondholder will realize a return of $12.50\%$ over the term if held to maturity and coupons are reinvested at this YTM.
YTM is influenced by the difference between the purchase and maturity price, as well as reinvestment of coupon payments.
Current yield, approximate YTM, and YTM differ due to varying formulas and assumptions.
When a bond trades at par, the current yield, approximate YTM, and the YTM will be the same.

Reinvestment Risk

YTM: a good estimate, but market rate trends affect the true return on the bond, so it may differ from the YTM calculation.
Interest rate at purchase unlikely to be the same as the reinvestment rate of cash flows.
Longer term to maturity, less likely for interest rates to remain constant.
Reinvestment risk: coupons are reinvested at a lower rate than the bond's YTM at purchase.
If coupons are reinvested at a higher rate than the bond's YTM at purchase, overall return will be higher than the YTM quoted.
If coupons are reinvested at a lower rate than the bond's YTM at purchase, overall return will be lower than the YTM quoted.
Zero-coupon bond: no reinvestment risk (no coupon cash flows).
Purchased at a discount to face value, taking into account the compounded rate of return.

Term Structure of Interest Rates

Market forces (supply and demand) affect trading and YTM.
Excess demand, price increases, YTM decreases.
Market interest rates drive bond prices.
Need to understand:
- The general level of interest rates at any particular time
- The level of interest rates at different terms to maturity
Theories explain interest rate variations and their effects.
Interest rates: interaction between borrowers and lenders.
Fisher Effect: a theory that explains how interest rates are determined based on the interaction between the inflation rate, the nominal interest rate, and the real interest rate.

Real Rate of Return

Components:
- The real rate of return
- The inflation rate
Inflation reduces the value of a dollar, so the nominal rate must be reduced by the inflation rate to arrive at the real rate of return.
The real rate of return is determined by the level of funds supplied by investors and the demand for loans by businesses.
The supply of funds tends to rise when real rates are high because investors are more likely to earn higher returns on the funds they lend.
On the other hand, the demand for loans tends to rise when real rates are low because businesses that borrow to invest in their companies are more likely to earn returns that are higher than the costs of borrowing
The nominal rate includes the real rate plus expected inflation.
$\text{Nominal Rate} = \text{Real Rate + Inflation Rate}$
Factors affecting real rate forecasts:
- Real interest rate fluctuates throughout business cycle (falls during recession, rises during expansion).
- Unexpected inflation changes the real rate. Investors demand rates including inflation expectations.
- If inflation is higher than expected, the investor’s real rate of return will be lower than expected.

Yield Curve

Relationship between short-term and long-term bond yields.
Plotted on a graph for similar bonds to show a continually changing line.
Hypothetical yield curve:
Shows short-term Government of Canada bonds yield $1\%$ , long-term bonds around $4\%$ .
Three theories explain the shape of the yield curve:
- Expectations theory
- Liquidity preference theory
- Market segmentation theory

Expectations Theory

Long-term interest rates predict future short-term rates.
Investors buying a single long-term bond should expect the same interest as buying two equal short-term bonds.
Yield curve indicates investor expectations about future interest rates.
Investor with two-year horizon:
- Buy a two-year bond.
- Buy a one-year bond, then another one-year bond.
- Buy a six-month bond, then three more six-month bonds.
Efficient market: each choice is equally attractive.
Two-year rate equals two successive one-year rates.
One-year rate is an average of two consecutive six-month rates.
Example: 2-year bond at $5\%$ . Return is $1.05 \times 1.05 = 1.1025 = 1.05^2 = 1.1025$
1-year rate at $4\%$ . Need to roll over into another one-year bond.
$\text{2 Year Return} = \text{1 Year Return Year 1 } \times \text{ 1 Year Return Year 2}$
$(1 + 0.05)^2 = (1 + 0.04) \times (1 + r)$
$1.1025 = 1.04 \times (1 + r)$
$r = 0.06009 = 6 \%$
With one-year rates at 4% and two-year rates at 5%, rates on one-year bonds are expected to increase from 4% to 6% a year from now.
Achieve same result whether you buy a two-year bond today or two one-year bonds consecutively, with 1 year rate at $4\%$ and a future 1 year rate is expected to be at $6\%$ .
Upward sloping yield curve indicates expectation of higher rates; downward sloping indicates rates expected to fall.
Humped curve indicates rates expected to rise, then fall.
Yield curve reflects market consensus of expected future interest rates.
Yield curve sloping upward indicates investors expect interest rates to rise.

Liquidity Preference Theory

Investors prefer short-term bonds (more liquid, less volatile).
Venture into longer-term bonds only with additional compensation for lower liquidity and increased price volatility.
Upward sloping yield curve reflects additional return for additional risk.
Simplicity appealing but does not explain downward sloping yield curve.

Market Segmentation Theory

Institutional players concentrate in specific term sectors.
Banks invest short-term; insurance companies invest long-term.
Yield curve represents supply and demand for bonds of various terms, primarily influenced by the bigger players.
Explains all yield curves (normal, inverted, humped).

Fundamental Bond Pricing Properties

Important to know where the price is headed.
Current interest rates and understanding of term structure can help forecast the direction of bond prices.
Understand specific features of a bond.
Yields are calculated using present value techniques, including semi-annual compounding and full reinvestment of all coupons at the prevailing yield.

Relationship Between Bond Prices and Interest Rates

Inverse relationship: bond prices and bond yields; rise or fall with interest rates.
Interest rate and bond yield used interchangeably.
Interest rates rise, bond yields rise, bond prices fall.
Interest rates fall, bond yields fall, bond prices rise.
The coupon rate doesn’t change over the life of the bond.
When interest rates rise, the bond yield rises to keep pace
Only way to create additional yield is to lower the price.

Impact of Maturity

Longer-term bonds are more volatile in price than shorter-term bonds.
If interest rates rise, both the five-year and the 10-year bond will drop in price, but to different degrees.
The five-year bond drops 4.49%, and the 10-year bond drops 8.18%.
Similar pattern when interest rates and therefore yields drop.
Uncertainty about the markets and interest rates increases as we forecast farther into the future.
The longer the term of the bond, the more volatile its price will be.
Longer-term rises more sharply (9.02% if yields drop to 2%) than shorter terms rises (4.74%).
As bonds approach maturity, they become less volatile.
Originally 10-year, seven years later, it has a three-year term; will be priced and traded as a three-year bond.

Impact of Coupon

Lower-coupon bonds are more volatile in price percentage change than high-coupon bonds.
Same factors (credit quality and liquidity).
Difference: the coupon rate.
When yields rise, both drop in price. The lower coupon bond drops more than the higher-coupon bond.
Significant when differences between coupons are great or when large sums are invested.

Impact of Yield Changes

Relative yield change is more important than absolute yield change.
A drop from $12\%$ to $10\%$ has a smaller impact than from $4\%$ to $2\%$ . The former is a $17\%$ change, and the latter is $50\%$ .
Bond prices are more volatile when interest rates are low.
Yield rises or falls by the same percentage, price is impacted more by the fall in yield.
$1\%$ drop in yield leads to greater price change than a $1\%$ rise in yield.
Price rises by 4.74% in the first scenario and falls by 4.49% in the second.

Duration as a Measure of Bond Price Volatility

Value changes inversely to interest rates.
Higher coupon is usually less volatile than lower coupon (same term and yield).
Longer-term is usually more volatile than shorter-term (same coupon and yield).
Difficult to compare bonds with different coupons and maturities.
Changes in interest rates affect different bonds differently, depending on features.
Need to determine the impact of interest rate changes on the prices of different types of bonds to make sound decisions.
Duration: a calculation that combines coupon rate and term to maturity impact.
Duration: approximate percentage change in price for a $1\%$ change in interest rates.
Higher duration: more reaction to interest rate changes.
Duration helps determine bond's or fund's volatility.
Single duration figure can be compared directly.
Example: Bond priced at 105 with 12 years left to maturity, but you are concerned that interest rates are going to rise by 1% over the next year. The duration of the bond is 10, its price will change by approximately 10% for each 1% change in interest rates. You determine that the price of the bond could drop from 105 to 94.50, if your expectations about the interest rate change are correct. $\text{105 − (10\% × 105) = 94.50. }$
Higher duration, higher percentage of price change when interest rates decline.
When rates are expected to rise, invest in bonds with low duration (protect from decline}.
The same interest rate change has a greater impact on the price of one bond as compared to price change on other bond.

Bond Market Trading

Fixed-income trading in investment banking in two areas:
- Sell side
- Buy side
Sides may be separate institutions; some have both.

Sell Side

Investment dealer side.
Trading (buying and selling) investment products for their own accounts.
Creating, producing, distributing, researching, marketing, and trading fixed-income products.
Medium-to-large firms have three roles:
- Investment banker: structures new debt issues and brings new issues to the primary market. Their clients are firms that need to raise funds for working capital and to fund asset acquisitions.
- Trader: trades securities that exist in the secondary market. They typically trade on a proprietary basis.
- Sales representative: markets new and exisiting products, performs research, and provides market analysis, credit analysis, and commentary. They also take client orders, which are then relayed to traders for pricing.

Buy Side

Investment management side.
Asset management (buying and holding securities) on behalf of institutional clients.
Clients: entities such as mutual funds, insurance companies, and pension funds.
Two occupational roles:
- Portfolio manager
- Trader

Buying Bonds Through an Investment Dealer

Based on the bond-trading capacity of their firm.
Firms with a large institutional dealing desk are served by a retail trading desk. Its primary function is to help advisors by sourcing products providing market commentary.
Advantage: access to range of securities in inventory.
Proprietary trading system linked to inventory; automatic execution of trades.
Large trades and illiquid securities are executed over the phone.
Firms without a large institutional dealing desk are served by a trading desk as the source of product.

Role of Inter-Dealer Brokers

Participants in the wholesale bond market (institutional buy side and sell side).
Act as agents: bring together institutional buyers and sellers.
Perform price discovery: determine the correct price by studying demand and supply.
Perform trade execution, clearing, and settlement.
Provide public transparency of prices.
Similar function to a market exchange.
Key advantage: anonymity.
Help sell large position to other institutions without revealing position and trading against it.

Mechanics of the Trade

Non-electronic trades: over the phone.
Legal responsibility of a full commercial agreement.
Calls are recorded; clear language is used.
Trader and advisor commit: the party agrees to deliver the full amount of sold bonds to the other party on the settlement date, either from the trading book or from the client’s account.

Trade Ticket

Electronic confirmation sent through secure, proprietary systems.
Includes:
- Specific details of counterparties includes the name and address of the investment advisor’s employer the holding the account
- Full identification of the bond, including the issuer’s name, the maturity date, and the coupon.
- The bond’s Committee on Uniform Security Identification Procedure (CUSIP), or other electronic settlement identification number
- The nominal, par, or face amount of the transaction.
- The price, and often the yield.
- The settlement date.
- The name of the custodian where the trade will settle.
- The total settlement amount, sometimes with the amount of accrued interest shown separately.

Clearing and Settlement

Change in legal ownership happens immediately; payment is later.
Securities are delivered at the end of the settlement period.
T-bills settle on the day of the transaction.
Other securities settle on the first clearing day after the transaction.
Recognition of ownership has varied:
- Bearer bonds: A certificate is produced, and detachable coupons are attached to the residual principal payment. Owners is signified by physical possession. The risk of losing certificates was a concern because they could be sold by anyone who had physical possession, whether or not the seller was considered the rightful owner.
- Registered bonds: Bear name of rightful owner; can be sold or transferred only when the owner signs the back of the certificate. Coupon payments mailed to the registered owner. Protection solved theft and loss of certificates.
- Bonds registered in book-based format: Book- based format is an electronic record keeping system used by