fixed income 6: Fixed-Income Bond Valuation: Prices and Yields

What determines the price of a bond at issuance?

present value of its promised interest (coupon payments) and principal (face value) cash flows, discounted at the market discount rate (required yield).

Formula for the present value of a single bond coupon payment.

$$PV\left(coupon\right)=\frac{PMT_{t}}{\left(1+r\right)^{t}}$$

General formula for bond price using market discount rate.

$$PV=\frac{PMT_1}{\left(1+r\right)}+\frac{PMT_2}{\left(1+r\right)^2}+\cdots+\frac{PMT_{N}+FV_{N}}{\left(1+r\right)^{N}}$$

what is the bond called if

1. coupon = market rate

2. coupon < market rate

3. coupon > market rate

Coupon Rate vs Market Rate	Bond Price
Coupon = Market Rate	Par
Coupon < Market Rate	Discount
Coupon > Market Rate	Premium

texas ba ii plus - inputs for bond price

Meaning	button
periodic interest rate	I/Y
number of periods	N
coupon payment per period	PMT
face value	FV
bond price	CPT → PV

Definition of Yield-to-Maturity (YTM)

• internal rate of return (IRR) on a bond’s cash flows such that the present value equals the bond’s price.

• assumes the bond is held to maturity and all coupons are paid reinvested at the YTM.

how to calculate YTM from bond price on texas ba ii

Meaning	Cash Flow
number of periods	N
PV (negative)	PV
coupon payment per period	PMT
face value	FV
interest	CPT → I/Y

How are negative yields on bonds possible?

• if issued at prices above par

• or if previous high-yield bonds appreciate in price,

• especially seen in zero-coupon sovereign bonds during periods of very low or negative interest rates.

What is the difference between flat price, accrued interest, and full price of a bond?

• Flat price (clean price) = PV of bond excluding accrued interest

• Accrued interest = portion of next coupon earned by seller

• Full price (dirty price) = flat price + accrued interest

why are flat prices used for quotations?

If full prices were quoted by dealers, investors would see price rises each day even if the YTM did not change due to the accrual of interest.

Once a coupon payment is made, the quoted price would drop significantly

Formula for accrued interest.

$$AI=\frac{t}{T}\cdot PMT$$

• t = number of days from the prior coupon payment to the settlement date

• T = number of days in the coupon period

• t/T = fraction of coupon period that has passed since the prior payment

• PMT = coupon payment per period

formula for $PV^{full}$

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

What is the relationship between bond prices and yield-to-maturity (YTM)?

• Bond prices and yields move in opposite directions.

• Higher discount rate → lower PV → lower bond price.

• Lower discount rate → higher PV → higher bond price.

What is the coupon effect on bond price sensitivity?

• The size of bond coupon cash flows affects how much a bond’s price will change for a given yield change for bonds of the same maturity.

• Lower-coupon bonds have a larger price sensitivity to yield changes.

• Zero-coupon bonds are most sensitive because all cash flow is at maturity.

• Higher-coupon bonds have smaller percentage price changes.

What is the maturity effect on bond price sensitivity?

• Longer-maturity bonds are more sensitive to yield changes.

• Higher N in PV formula amplifies price changes.

• Exceptions: low-coupon long-term discount bonds (rare).

What is a constant-yield price trajectory?

• shows how a bond’s price changes over time when the market discount rate (YTM) remains constant.

• Prices converge toward par as maturity approaches.

What is convexity in bonds?

• Price–yield relationship is curved, not linear.

• Price increase from YTM drop > price decrease from same YTM rise.

• Positive convexity: bond price rises more for a yield decrease than it falls for the same yield increase.

What is matrix pricing in bond valuation?

• estimates bond prices using comparable bonds with similar:

  • Credit quality

  • Coupon rate

  • Time-to-maturity

• Comparable bonds are often actively traded and provide yields or prices to reference.

Why is matrix pricing used?

hen there is no current market price for a bond, such as for illiquid bonds or new issues, to estimate its price or yield-to-maturity.

Matrix Pricing Process

1. Identify comparable bonds with similar features.

2. Collect prices and yields-to-maturity (YTM) for those bonds.

3. Organise in a matrix (coupon vs. maturity) to visualize data.

4. Interpolate to estimate the YTM for the bond in question.

   • Linear interpolation is commonly used.

5. Calculate the estimated price using the interpolated yield and standard present value formulas.

How is matrix pricing used in bond underwriting?

It helps estimate the required yield spread over a benchmark rate for a new bond issue by comparing it with government bonds of similar maturity.