Bond Pricing and Yield Fundamentals

Chapter 10: Bond Prices and Yields

• Bonds are an important part of portfolios
• Key learning objectives include calculating bond prices and yields, understanding yield to maturity, learning about interest rate risk and Malkiel's theorems, and measuring the impact of interest rate changes on bond prices.
• Goal is to understand the relationship between bond prices and yields, and the fundamental tools fixed-income portfolio managers use to assess bond risk.

Bond Basics, I

• A straight bond is an IOU obligating the issuer to pay a fixed sum of money (principal, par, or face value) at maturity, plus constant periodic interest payments (coupons) during the bond's life.
• U.S. Treasury bonds are a type of straight bond
• Special features like convertible, callable, and putable bonds can be added.

Bond Basics, II

• Two basic yield measures are coupon rate and current yield.
• Coupon rate = Annual coupon / Par value
• Current yield = Annual coupon / Bond price

Straight Bond Prices and Yield to Maturity

• Bond price is the sum of the present values of coupon payments and the face value.
• Yield to maturity (YTM) is the discount rate that equates the present value of future cash flows to today's bond price.

The Bond Pricing Formula

• Bond Price = C /2 (1 - 1/(1 + YTM/2)^2M ) + FV / (1 + YTM/2)^2M
  • C = annual coupon payments
  • FV = face value
  • YTM = yield to maturity
  • M = maturity of the bond

Example: Using the Bond Pricing Formula

• Example calculation for a straight bond with given values (face value, coupon rate, YTM, maturity).

Example: Calculating the Price of this Straight Bond Using Excel

• Excel function called PRICE calculates bond prices
• Function uses arguments like settlement date, maturity date, coupon rate, YTM, par value, frequency and actual / 365 day count to generate the bond price.

Spreadsheet Analysis, I

• Spreadsheet example using the PRICE function to calculate the price of a coupon bond.
• Spreadsheet use semi-annual coupon and 365/year day count to determine pricing.

Premium, Par, and Discount Bonds, I

• Bonds are categorized as premium, par, or discount based on the selling price vs. par value
• Premium bond: Coupon rate > YTM; Price > Face
• Discount bond : Coupon rate < YTM; Price < Face
• Par bond: Coupon rate = YTM; Price = Face

Premium, Par, and Discount Bonds, II

• Graph showing relationship between bond prices (% of par) and time to maturity for premium, par, and discount bonds.

Premium, Par, and Discount Bonds, III

• When coupon rate and YTM are constant, the longer the term to maturity, the greater the premium (for premium bonds) or discount (for discount bonds) over par.

Relationships among Yield Measures

• YTM > current yield > coupon rate for premium bonds
• YTM < current yield < coupon rate for discount bonds
• Current yield = Annual coupon payment / Current market price for par value bonds

Calculating Yield to Maturity, I

• Process of finding YTM given current price, coupon rate, and time to maturity.
• Methods include trial-and-error approach, financial calculators, and spreadsheets.

Calculating Yield to Maturity, II

• Excel function YIELD calculates YTM

A Quick Note on Bond Quotations, I

• Clean price : Bond price quoted without accrued interest.
• Dirty price : Acutal price the buyer pays, including accrued interest (also called invoice price).

A Quick Note on Bond Quotations, II

• Clean price = quoted price excluding accrued interest
• Dirty price = Clean price + accrued interest = actual price paid

A Quick Note on Bond Quotations, III

• Example calculation using clean price and accrued interest to find the dirty/invoice price for a bond.

Callable Bonds

• Callable bond gives the issuer the option to buy back the bond before maturity at a specified call price.
• Yield to call (YTC): Yield assuming the bond is called at its earliest possible call date. Formula for calculating callable bond price is provided.

Spreadsheet Analysis, III

• Examples using Excel functions DURATION and MDURATION calculate Macaulay's duration and Modified duration of a callable bond.

Interest Rate Risk

• Interest rate risk: possibility of losses due to interest rate changes.
• Realized yield: Actual return earned on a bond; almost never equal to the YTM.

Interest Rate Risk and Maturity

• Graph showing how bond prices decrease as yields increase, with longer maturity bonds showing higher sensitivity.

Malkiel's Theorems, I

• Bond prices and bond yields move in opposite directions.
• Larger change in bond price with longer maturity bonds for a given YTM change.

Malkiel's Theorems, II

• Size of price change decreases as maturity increases for a given YTM change.
• Percentage change in price is inversely related to coupon rate.
• Magnitude of price increase for a decrease is greater than magnitude of price decrease for an increase.

Bond Prices and Yields (numerical examples)

Duration

• Macaulay Duration measures the sensitivity of bond price to yield changes.
• Formula for duration is given; two bonds with same duration but not necessarily same maturity will have similar price sensitivity

Example: Using Duration

Modified Duration

• Modified duration is a variation of Macaulay duration for measuring bond price sensitivity to yield changes.

Calculating Macaulay's Duration, I

• Macaulay's Duration values are expressed in years.
• Duration for zero-coupon bond equals its maturity
• Duration for coupon bond is a weighted average of individual maturities of the bond's separate cash flows, weighted proportionately to their present values.

Calculating Macaulay's Duration for Par Bonds

• Duration formula for par-value selling bond given. Example calculation provided for the given values.

Calculating Macaulay's Duration, II

• General formula for Macaulay's duration with constant, semi-annual coupons.

Calculating Macaulay's Duration: Example

• Steps to calculate duration and modified duration for a given bond.

Calculating Macaulay Duration and Modified Duration, using Excel

• Using Excel functions DURATION and MDURATION to calculate Macaulay and Modified duration given bond data.

Calculating Duration Using Excel, Explanation

• Using DURATION and MDURATION excel functions to calculate bond duration and modified duration.

Duration Properties

• Longer maturity bond, longer duration
• Duration increases at a decreasing rate as maturity lengthens.
• Higher coupon rate, shorter duration.
• Higher yield to maturity, shorter duration.

Properties of Duration (Graph)

• Graph showing relationship between bond duration and maturity for differing coupon rates.

Bond Risk Measures Based on Duration, I

• Dollar Value of an 01: Change in bond price due to a one-basis-point change in yield
• Yield Value of a 32nd: Change in yield needed for a 1/32nd change in bond price. Bond prices are per $100 face value.

Bond Risk Measures Based on Duration, II

• Example problem to calculate the 01and(1)/32ndyieldvaluegivenamodifiedduration.Bondpricesareper01 and (1)/32nd yield value given a modified duration. Bond prices are per 01and(1)/32ndyieldvaluegivenamodifiedduration.Bondpricesareper100 face value

Dedicated Portfolios

• Dedicated portfolio tailored for a future cash outlay (e.g., pension fund), also known as a target date portfolio.

Dedicated Portfolios: Scenario

• Example of a pension fund needing to estimate future payout (e.g., $100 million in five years).

Dedicated Portfolios: Example

• Solving for the present value required to have an amount $100 million in the future.
• Using a specific interest rate and time period in an example, in calculating the amount that will be needed in present value
• Calculating investment amount to reach a future value given coupon rate, YTM, and time period

Dedicated Portfolios: How Does it Work?

• Illustrating the calculation and investment needed to generate the correct targeted payout amount for a dedicated portfolio.

Reinvestment Risk

• Risk of uncertainty regarding the future value of reinvesting bond coupon payments at unknown yields.

The Cost of Removing Reinvestment Risk Using STRIPS

• Illustrates the calculation to determine the present value of zero coupon bonds to reach a set future value at an expected yield.

Price Risk

• Risk that bond prices will decrease due to future uncertainty, which is an aspect of price risk in dedicated portfolios.

Price Risk versus Reinvestment Rate Risk

• Interest rate increases decrease bond prices but increase future value of coupons, which are reinvested
• Interest rate decreases increase bond prices but decrease future value of reinvested coupons.

Immunization

• Making a bond portfolio minimizing future value uncertainty surrounding target date values

Immunization by Duration Matching, I

• Duration matching strategy to immunize a portfolio, where its duration matches the portfolio's target date resulting in minimal price risk.

Immunization by Duration Matching, II

• Graph illustrating how matching portfolio duration and target maturity date minimizes price risk resulting from fluctuations in interest/coupon rates.

Dynamic Immunization

• Periodic rebalancing of a dedicated bond portfolio to maintain a duration that matches the target maturity date.

Chapter Review, I

• Brief overview of bond basics, straight bond pricing, premium/discount bonds, yield measures

Chapter Review, II

• More detail on yields, yield to call, interest rate risk, and Malkiel's theorems

Chapter Review, III

• Summarization of duration, dedicated portfolios, reinvestment risk, immunization, price risk, and dynamic immunization.