Fixed-Income Bond Valuation: Prices and Yields

Welcome to Bond Valuation!

Hello there! Today, we are diving into one of the most important chapters in the Fixed Income section: Fixed-Income Bond Valuation. If you have ever wondered how investors decide exactly how much a bond is worth today, you are in the right place.

Don't worry if the math looks a little intimidating at first. At its heart, bond valuation is just a "time travel" exercise for money. We are simply taking all the cash a bond promises to pay us in the future and calculating what that money is worth right now. Let’s get started!

1. The Fundamental Principle: Present Value

To value a bond, we use the Discounted Cash Flow (DCF) approach. A bond usually provides two types of cash flows:
1. Periodic Coupon Payments: The "interest" paid regularly.
2. Par Value (Principal): The "big check" at the end of the bond's life.

The price of a bond is the sum of the present values of all these expected future cash flows, discounted at the market's required rate of return (also called the yield-to-maturity).

The Bond Pricing Formula:

\( PV = \frac{PMT}{(1+r)^1} + \frac{PMT}{(1+r)^2} + ... + \frac{PMT+FV}{(1+r)^n} \)

Where:
PV = Present Value (Price of the bond)
PMT = Coupon payment per period
r = Market discount rate (yield per period)
n = Number of periods
FV = Face value or Par value of the bond

Real-World Analogy: Think of a bond like a fruit tree. The coupons are the fruit that falls every year, and the Par Value is the price you can sell the wood for at the end. To know what the tree is worth today, you have to value all that future fruit and the wood in today's dollars.

Quick Review: If market interest rates go up, we use a bigger number to "divide" (discount) those future cash flows, which makes the Price go down!

2. The Price-Yield Relationship: The See-Saw

One of the most critical "must-know" concepts for the CFA exam is the relationship between bond prices and yields. They move in opposite directions.

The Inverse Relationship:
- When the market discount rate (yield) increases, the bond price decreases.
- When the market discount rate (yield) decreases, the bond price increases.

The See-Saw Analogy: Imagine a see-saw. If Yield sits on one side and Price sits on the other, when one goes up, the other must go down. They are never on the same level unless the coupon rate equals the yield!

Key Relationships to Remember:

- Par Bond: Coupon Rate = Yield-to-Maturity. The Price is equal to the Par Value (\$1,000).
\n- Discount Bond: Coupon Rate < Yield-to-Maturity. The Price is less than Par Value.
\n- Premium Bond: Coupon Rate > Yield-to-Maturity. The Price is more than Par Value.

Did you know? Bond prices aren't a straight line; they are "convex." This means when yields drop, prices rise more than they fall when yields rise by the same amount. This is a "bonus" for bondholders!

Key Takeaway: If you see a bond trading at \$1,050 (above its \$1,000 par), you immediately know its coupon rate is higher than the current market interest rate.

3. Bond Prices Over Time: The Pull to Par

What happens to a bond's price as it gets closer to its maturity date? Assuming interest rates stay the same, the price will naturally move toward its Par Value. This is called the "Pull to Par."

- Premium Bonds: The price will gradually decrease over time toward par.
\n- Discount Bonds: The price will gradually increase over time toward par.
\n- Par Bonds: The price stays at par.

Step-by-Step Example: Imagine a 10-year discount bond priced at \$900. In year 5, it might be \$950. On the day it matures, it must be worth exactly \$1,000, because that is what the issuer is paying you back.

4. Yield Measures: Understanding Your Return

Not all "yields" are the same. You need to distinguish between these three:

1. Current Yield

This is a "quick and dirty" measure. It only looks at the annual coupon vs. the current price. It ignores the capital gain or loss as the bond moves to par.

\( Current \ Yield = \frac{Annual \ Cash \ Coupon}{Price} \)

2. Yield to Maturity (YTM)

This is the "Gold Standard." It is the internal rate of return (IRR) of the bond. It assumes you hold the bond to the end and reinvest all coupons at the same YTM rate.

3. Yield to Call (YTC)

Some bonds are "callable," meaning the issuer can pay them off early. For these bonds, investors calculate the YTC, which uses the call date instead of the maturity date and the call price instead of par.

Common Mistake: Don't confuse the Coupon Rate with the Yield. The Coupon Rate is fixed on the day the bond is born. The Yield (YTM) changes every second as the market price moves.

5. Matrix Pricing: When There is No Price

Sometimes, a bond doesn't trade often (it's illiquid), so we don't know its market price. How do we value it? We use Matrix Pricing.

Matrix Pricing is like looking at the prices of similar houses in a neighborhood to guess what your house is worth. We look at the yields of traded bonds with similar credit ratings and maturities to estimate the yield of our non-traded bond.

How to do it (Simplified):

1. Find the yields of traded bonds with slightly shorter and slightly longer maturities.
2. Use linear interpolation to find the "middle" yield for your bond's maturity.
3. Add a spread if the credit quality is different.
4. Use this estimated yield to calculate the price using the PV formula.

Key Takeaway: Matrix pricing is an estimation tool used for illiquid bonds or for pricing new bonds before they are issued.

6. Yield Spreads: The "Extra" Return

Investors usually want to know how much extra they are getting paid to take on risk compared to a "risk-free" government bond. This "extra" is the Spread.

G-Spread: The spread over a Government bond yield.
I-Spread (Interbank Spread): The spread over standard swap rates (like LIBOR or SOFR).
Z-Spread (Zero-volatility Spread): A constant spread added to each spot rate on the Treasury curve that makes the PV of the bond's cash flows equal its price.
OAS (Option-Adjusted Spread): This is the Z-spread minus the value of any embedded options (like a call option). It tells you the "real" spread for credit and liquidity risk alone.

Mnemonic for OAS: "OAS is the spread Off (after taking out) the Option." If a bond has no option, the Z-spread and OAS are exactly the same!

Key Takeaway Summary:
1. Bond Price = Present Value of Coupons + Present Value of Par.
2. Price and Yield move in opposite directions.
3. Premium bonds "pull down" to par; Discount bonds "pull up" to par.
4. Use YTM for the total return, and Spreads to measure risk relative to benchmarks.

Great job! You've just covered the core mechanics of bond valuation. Take a breath—Fixed Income is often considered one of the tougher sections, but once you master the "Time Value of Money" logic, it all starts to click. Keep practicing those calculator steps!