Bond Valuation

The Math Behind Fixed Income

Bonds are loans you make to governments or corporations. The borrower (issuer) promises to pay you periodic interest (coupon) and return your principal (par value, typically $1,000 per bond) at maturity. Bond investing requires understanding the relationship between price, yield, and interest rates. When interest rates rise, bond prices fall. When rates fall, bond prices rise. This inverse relationship is the single most important concept in fixed income. It exists because your existing bond's coupon becomes more or less attractive relative to newly issued bonds at current rates.

Bond Pricing and Yield to Maturity

A bond's price equals the present value of all future cash flows: each coupon payment discounted to today, plus the par value at maturity discounted to today. If you buy a 10-year bond with a 5% coupon ($50/year on a $1,000 par bond) when prevailing rates are also 5%, the bond trades at par ($1,000). If rates rise to 6%, your 5% coupon is less attractive than new 6% bonds. Buyers will only purchase your bond at a discount, maybe $925, so that the effective yield (accounting for the discount and par repayment at maturity) matches the new 6% rate. Yield to Maturity (YTM) captures the total return if you hold the bond to maturity. It accounts for the coupon payments, the difference between what you paid and the par value you receive at maturity, and the time value of money. A bond purchased at $925 with a $50 annual coupon and $1,000 par repayment in 10 years has a YTM of approximately 6%. YTM is the bond market's equivalent of the cap rate in real estate: it lets you compare different bonds on an equal footing.

• Bond price = present value of all future coupon payments + present value of par at maturity
• Rates rise = existing bond prices fall. Rates fall = existing bond prices rise.
• Trading at par: bond price = face value. Coupon rate = current market rate.
• Trading at a discount: bond price below par. Coupon rate below market rate.
• Trading at a premium: bond price above par. Coupon rate above market rate.
• YTM is the total return assuming you hold to maturity and reinvest coupons at the same rate.

Duration: Measuring Interest Rate Sensitivity

Duration measures how sensitive a bond's price is to interest rate changes. It is expressed in years but is really a sensitivity coefficient. A bond with a duration of 5 years will drop approximately 5% in price for every 1% increase in interest rates, and rise approximately 5% for every 1% decrease. Longer maturity bonds have higher duration (more sensitive to rates). Higher coupon bonds have lower duration (the larger periodic payments reduce your effective waiting time to get your money back). A 30-year Treasury bond has a duration around 20 years, meaning a 1% rate increase causes roughly a 20% price decline. A 2-year Treasury note has a duration around 1.9 years, meaning the same 1% rate increase causes only a 1.9% decline. This is why long-duration bonds are volatile when rates are moving. The 2022-2023 bond market saw 30-year Treasuries lose over 40% of their value as rates rose from near zero to above 4%. That is equity-like downside in an asset class many investors considered safe.

• Duration 5 years = ~5% price change for every 1% rate change
• Longer maturity = higher duration = more rate sensitivity
• Higher coupon = lower duration = less rate sensitivity
• 30-year Treasury: ~20 year duration. Extremely sensitive to rates.
• 2-year Treasury: ~1.9 year duration. Minimal rate sensitivity.
• 2022-2023: 30-year Treasuries lost 40%+ as rates rose from near zero to 4%+.

The Yield Curve

The yield curve plots the yields of Treasury securities across maturities, from the 3-month T-bill to the 30-year bond. A normal yield curve slopes upward: longer maturities pay higher yields to compensate investors for the risk of locking money up for longer periods. A flat yield curve shows similar yields across all maturities, typically during transitions in monetary policy. An inverted yield curve occurs when short-term rates exceed long-term rates. This inversion has preceded every U.S. recession since 1970. The 2-year/10-year spread (the difference between the 2-year and 10-year Treasury yields) is the most watched measure. When this spread turns negative, the bond market is signaling that it expects economic weakness and future rate cuts. The 2-year/10-year inverted in July 2022 and remained inverted for over two years, the longest inversion in modern history. The subsequent economic slowdown, while not fitting the technical definition of recession by some measures, validated the signal's directional accuracy.

• Normal curve: slopes up. 3-month yields less than 2-year, less than 10-year, less than 30-year.
• Flat curve: similar yields across maturities. Transition signal.
• Inverted curve: short rates above long rates. Recession warning.
• 2yr/10yr inversion has preceded every recession since 1970.
• July 2022: 2yr/10yr inverted. Longest inversion in modern history.
• The curve un-inverts (steepens) as the recession approaches or begins, typically through short rates dropping.

Credit Ratings and Credit Spreads

Credit rating agencies (Moody's, S&P, Fitch) assign letter grades to bond issuers based on their ability to repay. Investment grade ratings run from AAA (highest quality, lowest risk) through BBB- (lowest investment grade). Below BBB- is high yield, commonly called junk bonds. The spread is the yield premium above the risk-free rate (Treasury yield of the same maturity) that a corporate bond must offer to attract buyers. An AAA-rated corporate bond might trade at 0.5% above Treasuries. A BBB bond might trade at 1.5% above. A BB (junk) bond might trade at 3-5% above. A CCC bond (significant default risk) might trade at 8-15% above. Spread widening means the market is pricing in more risk. If BBB spreads widen from 1.5% to 3%, it means investors are demanding more compensation for corporate credit risk, typically during economic stress. Spread compression means the market is pricing in less risk, typically during expansion. Tracking the high yield spread (the average spread of all junk bonds over Treasuries) is one of the best real-time indicators of market stress. It typically sits at 3-4% during calm markets and spikes to 8-10%+ during crises.

• AAA: highest quality. Near-zero default probability. Spread: 0.3-0.7%.
• AA: very high quality. Spread: 0.5-1.0%.
• A: upper medium grade. Spread: 0.8-1.5%.
• BBB: lowest investment grade. Spread: 1.2-2.5%.
• BB: first level of junk. Spread: 2.5-5.0%.
• B to CCC: high default risk. Spreads: 5-15%+.
• High yield spread is a real-time stress indicator. 3-4% = calm. 8%+ = crisis.

Bond valuation is driven by the present value of future cash flows. Yield to maturity captures total return. Duration measures interest rate sensitivity and explains why long-term bonds can lose 20-40% when rates rise sharply. The yield curve, particularly the 2-year/10-year spread, is one of the most reliable recession indicators in finance. Credit ratings and spreads translate default risk into yield premiums. Understanding these mechanics lets you evaluate fixed income positions, interpret yield curve signals, and assess the risk-reward of different credit qualities.