Fixed-Income Bond Valuation: Prices and Yields

Fixed Income

Bond Pricing and the Time Value of Money

Learning Outcome Statement:

calculate a bond’s price given a yield-to-maturity on or between coupon dates

Summary:

Bond pricing involves calculating the present value (PV) of future cash flows, which include periodic coupon payments and the bond's face value at maturity. The market discount rate, or yield-to-maturity (YTM), is used to discount these cash flows. The bond's price can be affected by whether it is trading at a discount, par, or premium, depending on the relationship between the coupon rate and the market discount rate. Additionally, bond pricing between coupon dates involves calculating the flat price and accrued interest to determine the full price.

Key Concepts:

Market Discount Rate

The market discount rate is the required rate of return by investors given the risk of the bond. It is used to discount the bond's future cash flows to their present value.

Yield-to-Maturity (YTM)

YTM is the internal rate of return on the bond's cash flows, assuming the bond is held to maturity and all coupon payments are reinvested at the YTM rate. It equates the present value of the bond's future cash flows to its current market price.

Flat Price

The flat price of a bond is its price excluding any accrued interest. It represents the clean price quoted in the market.

Accrued Interest

Accrued interest is the interest earned on the bond since the last coupon payment. It is added to the flat price to determine the full price or dirty price, which is the actual market price of the bond.

Full Price

The full price of a bond is the total price including accrued interest. It represents the actual cost to the buyer and the amount received by the seller.

Formulas:

Present Value of Bond

PV = \sum_{t=1}^{N} \frac{PMT_t}{(1 + r)^t} + \frac{FV}{(1 + r)^N}

This formula calculates the present value of a bond based on its periodic coupon payments and face value, discounted at the market discount rate.

Variables:

$PV$ :: Present Value of the bond
$PMT_t$ :: Coupon payment at time t
$r$ :: Market discount rate per period
$FV$ :: Face value of the bond
$N$ :: Total number of periods until maturity

Units: currency units

Full Price Calculation

PV_{Full} = PV_{Flat} + AI

This formula determines the full price of a bond by adding the accrued interest to the flat price.

Variables:

$PV_{Full}$ :: Full price of the bond
$PV_{Flat}$ :: Flat price of the bond
$AI$ :: Accrued interest

Units: currency units

Accrued Interest

AI = \frac{t}{T} \times PMT

This formula calculates the accrued interest based on the fraction of the coupon period that has passed since the last payment.

Variables:

$AI$ :: Accrued interest
$t$ :: Number of days from the last coupon payment to the settlement date
$T$ :: Total number of days in the coupon period
$PMT$ :: Coupon payment per period

Units: currency units

Relationships between Bond Prices and Bond Features

Learning Outcome Statement:

identify the relationships among a bond’s price, coupon rate, maturity, and yield-to-maturity

Summary:

This LOS explores how various features of a bond such as coupon rate, maturity, and yield-to-maturity influence its price. It discusses the inverse relationship between bond prices and yields, the impact of coupon size on price sensitivity, how maturity affects price volatility, the concept of constant-yield price trajectory, and the convexity effect in bond pricing.

Key Concepts:

Inverse Relationship

Bond prices and yields move in opposite directions. A higher yield results in a lower present value of future cash flows, thus lowering the bond price, and vice versa.

Coupon Effect

Bonds with lower coupons are more sensitive to changes in yields compared to higher coupon bonds, as a larger proportion of their cash flows are received at maturity.

Maturity Effect

Longer-term bonds exhibit greater price volatility in response to yield changes compared to shorter-term bonds. This is due to the greater impact of discounting on distant cash flows.

Constant-Yield Price Trajectory

Bond prices change over time even if market rates remain constant. As maturity approaches, the price of a bond will converge towards its par value, assuming no default.

Convexity Effect

The relationship between bond prices and yields is convex, meaning that price increases due to yield decreases are greater in magnitude than price decreases due to yield increases.

Formulas:

Present Value of Bond

PV = \frac{C}{(1+r)^1} + \frac{C}{(1+r)^2} + ... + \frac{C + F}{(1+r)^N}

This formula calculates the present value of a bond based on its coupon payments, face value, discount rate, and time to maturity.

Variables:

$C$ :: Annual coupon payment
$r$ :: Discount rate or yield
$F$ :: Face value of the bond
$N$ :: Number of periods until maturity

Units: Currency units (e.g., USD)

Percentage Price Change

\text{Percentage Price Change} = \frac{\text{Price Change}}{\text{Original Price}} \times 100

This formula calculates the percentage change in the price of a bond in response to changes in yield.

Variables:

$Price Change$ :: Difference between the new price and the original price
$Original Price$ :: Initial price of the bond

Units: Percentage (%)

Matrix Pricing

Learning Outcome Statement:

describe matrix pricing

Summary:

Matrix pricing is a method used to estimate the price of bonds that are not actively traded by using the prices of comparable, more frequently traded bonds. These comparable bonds share similar characteristics such as times-to-maturity, coupon rates, and credit quality. The process involves identifying comparable bonds, calculating their yields-to-maturity, interpolating these yields to find the yield closest to the bond in question, and then using this yield to calculate the bond's price.

Key Concepts:

Matrix Pricing Process

Matrix pricing involves several steps to estimate the price of a bond. First, comparable bonds with similar features are identified. Then, yields-to-maturity for these bonds are calculated and averaged for each maturity year. Using linear interpolation, the yield for the bond's specific maturity is estimated. Finally, this yield is used to calculate the bond's price by discounting all future coupon and principal payments.

Comparable Bonds

In matrix pricing, comparable bonds are those that are actively traded and have similar characteristics to the bond being priced, such as maturity, coupon rate, and credit quality. These bonds serve as a reference for estimating the yield and price of less liquid bonds.

Linear Interpolation

Linear interpolation is a method used in matrix pricing to estimate the yield-to-maturity for a bond's specific maturity by using the yields of bonds with nearby maturities. This involves calculating a weighted average of these yields based on the proximity of their maturities to the maturity of the bond being priced.

Yield-to-Maturity (YTM)

Yield-to-maturity is the total return anticipated on a bond if the bond is held until it matures. It is a critical factor in determining a bond's price using matrix pricing, as it reflects the discount rate used to calculate the present value of all future cash flows from the bond.

Formulas:

Linear Interpolation Formula

YTM_{estimated} = YTM_{lower} + \left( \frac{maturity_{target} - maturity_{lower}}{maturity_{upper} - maturity_{lower}} \right) \times (YTM_{upper} - YTM_{lower})

This formula is used to estimate the yield-to-maturity for a bond's specific maturity by taking a weighted average of the yields of bonds with the nearest maturities above and below the target maturity.

Variables:

$YTM_{estimated}$ :: The estimated yield-to-maturity for the target bond's maturity
$YTM_{lower}$ :: The yield-to-maturity for the lower reference maturity
$YTM_{upper}$ :: The yield-to-maturity for the upper reference maturity
$maturity_{target}$ :: The maturity year of the bond being priced
$maturity_{lower}$ :: The lower reference maturity year
$maturity_{upper}$ :: The upper reference maturity year

Units: percentage

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