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LM01 Alternative Investment Features, Methods, and Structures

LM01 Categories, Characteristics, and Compensation Structures of Alternative Investments

LM01 Corporate Structures and Ownership

LM01 Derivative Instrument and Derivative Market Features

LM01 Ethics and Trust in the Investment Profession

LM01 Fixed-Income Instrument Features

LM01 Fixed-Income Securities: Defining Elements

LM01 Introduction to Financial Statement Analysis

LM01 Market Organization & Structure

LM01 Market Organization and Structure

LM01 Organizational Forms, Corporate Issuer Features, and Ownership

LM01 Portfolio Management Overview

LM01 Portfolio Management: An Overview

LM01 Rates and Returns

LM01 The Firm & Market Structures

LM01 Time Value of Money

LM01 Topics in Demand and Supply Analysis

LM02 Alternative Investment Performance and Returns

LM02 Analyzing Income Statements

LM02 Code of Ethics and Standards of Professional Conduct

LM02 Code of Ethics and Standards of Professional Conduct Profession

LM02 Financial Reporting Standards

LM02 Fixed Income Markets - Issuance Trading and Funding

LM02 Fixed-Income Cash Flows and Types

LM02 Forward Commitment and Contingent Claim Features and Instruments

LM02 Introduction to Corporate Governance and Other ESG Considerations

LM02 Investors and Other Stakeholders

LM02 Organizing, Visualizing, and Describing Data

LM02 Performance Calculation and Appraisal of Alternative Investments

LM02 Portfolio Risk & Return: Part I

LM02 Portfolio Risk and Return Part I

LM02 Security Market Indexes

LM02 The Firm and Market Structures

LM02 Time Value of Money in Finance

LM02 Understanding Business Cycles

LM03 Aggregate Output, Prices and Economic Growth

LM03 Analyzing Balance Sheets

LM03 Business Models & Risks

LM03 Corporate Governance: Conflicts, Mechanisms, Risks, and Benefits

LM03 Derivative Benefits, Risks, and Issuer and Investor Uses

LM03 Fiscal Policy

LM03 Fixed-Income Issuance and Trading

LM03 Guidance for Standards I-VII

LM03 Guidance for Standards I–VII

LM03 Introduction to Fixed Income Valuation

LM03 Investments in Private Capital: Equity and Debt

LM03 Market Efficiency

LM03 Portfolio Risk & Return: Part II

LM03 Portfolio Risk and Return Part II

LM03 Private Capital, Real Estate, Infrastructure, Natural Resources, and Hedge Funds

LM03 Probability Concepts

LM03 Statistical Measures of Asset Returns

LM03 Understanding Income Statements

LM04 An Introduction to Asset-Backed Securities

LM04 Analyzing Statements of Cash Flows I

LM04 Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

LM04 Basics of Portfolio Planning & Construction

LM04 Basics of Portfolio Planning and Construction

LM04 Capital Investments

LM04 Common Probability Distributions

LM04 Fixed-Income Markets for Corporate Issuers

LM04 Introduction to the Global Investment Performance Standards (GIPS)

LM04 Monetary Policy

LM04 Overview of Equity Securities

LM04 Probability Trees and Conditional Expectations

LM04 Real Estate and Infrastructure

LM04 Understanding Balance Sheets

LM04 Understanding Business Cycles

LM04 Working Capital and Liquidity.

LM05 Analyzing Statements of Cash Flows II

LM05 Capital Investments and Capital Allocation

LM05 Company Analysis: Past and Present

LM05 Fixed-Income Markets for Government Issuers

LM05 Introduction to Geopolitics

LM05 Introduction to Industry and Company Analysis

LM05 Monetary and Fiscal Policy

LM05 Natural Resources

LM05 Portfolio Mathematics

LM05 Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

LM05 Pricing and Valuation of Forward Contracts.

LM05 Sampling and Estimation

LM05 The Behavioral Biases of Individuals

LM05 Understanding Cash Flow Statements

LM05 Understanding Fixed-Income Risk and Return

LM05 Working Capital & Liquidity

LM06 Analysis of Inventories

LM06 Capital Structure

LM06 Cost of Capital-Foundational Topics

LM06 Equity Valuation: Concepts and Basic Tools

LM06 Financial Analysis Techniques

LM06 Fixed-Income Bond Valuation: Prices and Yields

LM06 Fundamentals of Credit Analysis

LM06 Hedge Funds

LM06 Hypothesis Testing

LM06 Industry and Competitive Analysis

LM06 International Trade

LM06 Introduction to Geopolitics

LM06 Introduction to Risk Management

LM06 Pricing and Valuation of Futures Contracts

LM06 Simulation Methods

LM07 Analysis of Long-Term Assets

LM07 Business Models

LM07 Capital Flows and the FX Market

LM07 Capital Structure

LM07 Company Analysis: Forecasting

LM07 Estimation and Inference

LM07 International Trade and Capital Flows

LM07 Introduction to Digital Assets

LM07 Introduction to Linear Regression

LM07 Inventories

LM07 Pricing and Valuation of Interest Rate and Other Swaps

LM07 Pricing and Valuation of Interest Rates and Other Swaps

LM07 Technical Analysis

LM07 Yield and Yield Spread Measures for Fixed-Rate Bonds.

LM08 Currency Exchange Rates

LM08 Equity Valuation: Concepts and Basic Tools

LM08 Exchange Rate Calculations

LM08 Fintech in Investment Management

LM08 Hypothesis Testing

LM08 Long Lived Assets

LM08 Measures of Leverage

LM08 Pricing and Valuation of Options

LM08 Topics in Long-Term Liabilities and Equity

LM08 Yield and Yield Spread Measures for Floating-Rate Instruments

LM09 Analysis of Income Taxes

LM09 Income Taxes

LM09 Option Replication Using Put-Call Parity

LM09 Option Replication Using Put–Call Parity

LM09 Parametric and Non-Parametric Tests of Independence

LM09 The Term Structure of Interest Rates: Spot, Par, and Forward Curves

LM10 Financial Reporting Quality

LM10 Interest Rate Risk and Return

LM10 Non-current (Long-Term) Liabilities

LM10 Simple Linear Regression

LM10 Valuing a Derivative Using a One-Period Binomial Model

LM11 Financial Analysis Techniques

LM11 Financial Reporting Quality

LM11 Introduction to Big Data Techniques

LM11 Yield-Based Bond Duration Measures and Properties

LM12 Applications of Financial Statement Analysis

LM12 Introduction to Financial Statement Modeling

LM12 Yield-Based Bond Convexity and Portfolio Properties

LM13 Curve-Based and Empirical Fixed-Income Risk Measures

LM14 Credit Risk

LM15 Credit Analysis for Government Issuers

LM16 Credit Analysis for Corporate Issuers

LM17 Fixed-Income Securitization

LM18 Asset-Backed Security (ABS) Instrument and Market Features

LM19 Mortgage-Backed Security (MBS) Instrument and Market Features

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IFT Notes for Level I CFA^® Program

LM03 Introduction to Fixed Income Valuation

Part 3

3. Prices and Yields: Conventions For Quotes and Calculations

3.1. Flat Price, Accrued Interest, and the Full Price

When a bond is between coupon dates, its price has two parts: flat price and accrued interest. The sum of the two parts is called the full price or dirty price.

Full price or dirty price = Flat price + Accrued interest
${PV}^{full}=\ {PV}^{flat}+\ AI$
Flat price = Full price – Accrued interest
Accrued interest = $\frac{t}{T}\times \ PMT$
where:
t = number of days since the last coupon payment
T = number of days between coupon payments
PMT = interest payment per period

As indicated in the above equation, the accrued interest is based on the number of days since the last coupon payment and the number of days between coupon payments. Here are two conventions for calculating the number of days:

Actual convention: Uses the actual number of days, including weekends, holidays and leap days as in our example above.
30/360 convention: Assumes 30 days in a month and 360 days in a year.

Illustration of Flat Price and Dirty Price through an Example

Assume you own a two-year bond issued on 10^th January, 2017 with a par value of $100. The bond pays a coupon of 8% half-yearly. The first coupon is due on 10^thJuly 2017, the next on 10^th January 2018, the third on 10^th July 2018, and the final one is paid along with principal at maturity date on 10^th January 2019. The market discount rate is equal to the coupon rate and so the flat price is 100. An investor wants to purchase the bond from you on 31^stMay, 2017. What is the full price he must pay?

You have held the bond for 140 days (since the bond was issued) and earned an interest for this period even though it has not been paid. The next coupon payment is $4. Assuming each period is 180 days the accrued interest is 4 x (140/180) = 3.11. The full price is 100.00 + 3.11 = 103.11. This is the amount you will receive on selling the bond.

Note that it is the flat (or clean) price which is quoted. The accrued interest increases every day until the coupon payment. So, if the accrued interest was included there would be a variation in bond quotes on a daily basis and a sudden drop after the coupon payment is made, which is misleading.

The full price of a bond can also be calculated using the equation shown below.

$PV_{full}=\ PV\ x\ {\left(1+r\right)}^{\frac{t}{T}}$

where:
t = number of days since the last coupon payment
T = number of days between coupon payments
PV = present value of the bond (not the flat price)

Example

A 5% French corporate bond is priced for settlement on 14 July 2015. The bond makes semi-annual coupon payments on 1 March and 1 September of each year and matures on 1 September 2020. Assuming a 30/360 day count convention for accrued interest, calculate the full price, the accrued interest, and the flat price per EUR 100 of par value for the following three yields-to-maturity: (I) 4.75%, (II) 5.00% and (III) 5.10%.

Solution:

Given the 30/360 day-count convention, there are 134 days between the last coupon on 1 March 2015 and the settlement date on 14 July 2015 (120 days for full months of March, April, May and June plus 14 days in July). Therefore, the fraction of the coupon period that has gone by is assumed to be 134/180. At the beginning of the period, there are 11 periods to maturity.

I) Stated annual YTM of 4.75%

The price at the beginning of the period is 101.198 per 100 of par value.

PV = $\frac{2.5}{1.02375}+\frac{2.5}{{1.02375}^2}+\dots +\frac{102.5}{{1.02375}^{11}}$ = 101.198

The full price on 14 July is PV(full) = $101.198\ \times \ \left({1.02375}^{\frac{134}{180}}\right)$ = 102.979

The accrued interest is AI = $2.5\ \times \ \left(\frac{134}{180}\right)$ = 1.861

The flat price is PV(flat) = 102.979 – 1.861 = 101.118

II) Stated annual YTM of 5.00%

The price at the beginning of the period is 100.00 per 100 of par value.

PV = $\frac{2.5}{1.025}+\frac{2.5}{{1.025}^2}+\dots +\frac{102.5}{{1.025}^{11}}$ = 100.00

The full price on 14 July is PV(full) = $100.00\ \times \ \left({1.025}^{\frac{134}{180}}\right)$ = 101.855

The accrued interest is AI = $2.5\ \times \ \left(\frac{134}{180}\right)$ = 1.861

The flat price is PV(flat) = 101.855 – 1.861 = 99.9942

III) Stated annual YTM of 5.10%

The price at the beginning of the period is 99.5256 per 100 of par value.

PV = $\frac{2.5}{1.0255}+\frac{2.5}{{1.0255}^2}+\dots +\frac{102.5}{{1.0255}^{11}}$ = 99.5256

The full price on 14 July is PV(full) = $99.5256\ \times \ \left({1.0255}^{\frac{134}{180}}\right)$ = 101.4088

The accrued interest is AI = $2.5\ \times \ \left(\frac{134}{180}\right)$ = 1.861

The flat price is PV(flat) = 101.4088 – 1.861 = 99.5478

The flat price of the bond is a little below par value, even though the coupon rate and the yield-to-maturity are equal, because the accrued interest does not take into account the time value of money. The accrued interest is the interest earned by the owner of the bond for the time between the last coupon payment and the settlement date. However, that interest income is not received until the next coupon date. The calculation of accrued interest in practice neglects the time value of money. Therefore, compared to theory, the reported accrued interest is a little “too high” and the flat price is a little “too low.”

3.2. Matrix Pricing

One can easily get information at what price a stock is trading. But, it is not easy to determine the price of most bonds as they do not trade as often as equity. Matrix pricing is an estimation process to find the market price of a not-so-frequently traded bond based on the prices of comparable bonds with similar times-to-maturity, type of issuer, coupon rates, and credit quality. This is also applicable for bonds that have not been issued yet.

For example, consider a three-year, 4% semi-annual bond X that you are trying to value. Comparable bonds whose prices are known are shown in the matrix below. Based on the four bonds, we can determine the price of bond X.

Matrix Pricing
	2% coupon	3% coupon	4% coupon	5% coupon
Two years		98.5		102.25
		3.786%		3.821%
Three years			Bond X
Four years
Five years	90.25		99.125
	4.181%		4.196%

There are five steps to determine the value of a bond using the matrix pricing method:

Determine the YTM of comparable bonds, which is given to us already. If it were not given, you can calculate the YTM using the price and coupon. For example, YTM of a two-year, 3% coupon is: N = 4; PMT = 1.5; FV = 100; PV = -98.5; CPT I/Y. I/Y = 3.786%.
Determine the average yield for two-year bonds:
$\frac{0.03786+0.03821}{2}=0.038035$
Determine the average yield for five-year bonds:
$\frac{0.04181+0.4196}{2}=0.041885$
Calculate the three-market discount rate using linear interpolation. We have the data (yield) for two-year and five-year bonds, but not a three-year bond. From the graph you can see that the YTM of a three-year bond = 0.038 + x.

x can be calculated as $\frac{3-2}{5-2}$ x (0.041885 – 0.038035) = 0.001283. Discount rate for the three-year bond = = 3.39%.

Using as the discount rate, compute the price for the three-year bond.

N = 6; I/Y = 1.966; PMT = 2; FV = 100; CPT PV. PV = -100.1906.

A few points to be noted on matrix pricing:

Matrix pricing is used when underwriting new bonds to estimate the required yield spread over the benchmark rate.
Benchmark rate is Libor, or a government bond with the same maturity as the bond being priced.
Assume the YTM for a new bond is calculated using the matrix pricing method as 2.2% and a comparable government bond has a yield of 2.0%. The difference between the yield on a new bond over its benchmark rate is called the required yield spread or spread over the benchmark. In this case, it is 0.2%.
Yield spreads are always specified in basis points where 1 basis point is one-hundredth of a percentage point. In this case, it is 20 basis points.

Example

An analyst decides to value an illiquid 3-year, 3.75% annual coupon payment corporate bond. Given the following information and using matrix pricing, what is the estimated price of the illiquid bond? Additional information: 2-year, 4.75% annual coupon payment bond priced at 105.60 and 4-year, 3.5% annual coupon payment bond priced at 103.28.

Solution:
The required yield on the 2-year, 4.75% bond priced at 105.60 is 1.871%.
The required yield on the 4-year, 3.50% bond priced at 103.28 is 2.626%.
The estimated market discount rate for a 3-year bond having the same credit quality is the average of two required yields: $\frac{1.871\%\ +\ 2.626\%}{2}$ = 2.249%.
The estimated price of the illiquid 3-year, 3.75% annual coupon payment corporate bond is 104.3078 per 100 of par value.