Welcome to Fixed Income: Interest Rate Risk and Return
Welcome! If you’ve ever wondered why bond prices seem to move in the opposite direction of interest rates, you’re in the right place. In this chapter, we explore how a bond’s value changes over time and how we can measure its sensitivity to interest rate movements. Don’t worry if this seems a bit technical at first—we’ll break it down into simple, manageable pieces that will help you master the exam!
1. Where Does the Return Come From?
When you invest in a fixed-rate bond and hold it until it matures, your return comes from three main sources:
1. Coupon and Principal Payments: These are the scheduled cash flows the issuer promised to pay you.
2. Interest on Interest: This is the "reinvestment" part. When you receive a coupon, you don’t just put it under your mattress; you reinvest it. The return you get from this reinvestment is crucial.
3. Capital Gains or Losses: If you sell the bond before it matures, the price you get might be higher or lower than what you paid.
Did you know? If you hold a bond until the very last day (maturity), you don't have to worry about capital gains or losses from market price changes because the issuer simply pays you the face value! The "price risk" only matters if you sell early.
Key Takeaway: Total return depends on the coupons, how much you earn by reinvesting those coupons, and the price you get if you sell before maturity.
2. The Great Tug-of-War: Price Risk vs. Reinvestment Risk
Interest rates are like a double-edged sword for bondholders. When interest rates change, two things happen that pull your return in opposite directions:
A. Market Price Risk: When interest rates rise, bond prices fall. This is bad if you want to sell your bond. (Think of a teeter-totter: Rates up, Price down).
B. Reinvestment Risk: When interest rates rise, you can reinvest your coupon payments at those newer, higher rates. This is good!
Quick Review:
• If rates rise: Bond price goes down (Bad), but reinvestment income goes up (Good).
• If rates fall: Bond price goes up (Good), but reinvestment income goes down (Bad).
Common Mistake: Students often think high interest rates are always bad for bonds. Remember, for a long-term investor who reinvests coupons, higher rates can actually increase the total return over time!
3. Measuring Risk: Macaulay and Modified Duration
How much will a bond's price move when rates change? We use a tool called Duration. There are two main types you need to know for the exam:
Macaulay Duration
Imagine a scale balancing the cash flows of a bond. Macaulay Duration is the "weighted average time" it takes to receive all the cash flows. It is measured in years.
• A Zero-Coupon Bond has a Macaulay Duration equal to its maturity (because there's only one payment at the very end).
• A Coupon Bond always has a Macaulay Duration shorter than its maturity because you get some cash (coupons) earlier.
Modified Duration (The "Price Sensitivity" measure)
This is what analysts usually mean when they say "duration." It tells us the percentage change in a bond's price for a 1% change in yield.
The formula to link the two is:
\( ModDur = \frac{MacDur}{1 + r} \)
(Where r is the yield per period)
To calculate the approximate price change:
\( \% \Delta Price \approx -ModDur \times \Delta Yield \)
Example: If a bond has a Modified Duration of 5 and interest rates rise by 1% (0.01), the bond price will fall by approximately 5%. Notice the negative sign! They move in opposite directions.
Key Takeaway: Higher duration means higher risk (the bond's price is more sensitive to rate changes).
4. Factors Affecting Duration
It’s easier to remember duration if you think of it as "riskiness." What makes a bond's price move more wildly?
1. Time to Maturity: Generally, the longer the maturity, the higher the duration. (More time for things to go wrong!)
2. Coupon Rate: The lower the coupon, the higher the duration. (If the coupons are small, you are waiting longer to get the bulk of your money back).
3. Yield to Maturity (YTM): The lower the YTM, the higher the duration.
Mnemonic: "Low and Long" = High Duration. (Low coupons, low yields, and long maturity lead to high interest rate risk).
5. Effective Duration (For Bonds with Options)
Some bonds have special features, like "Callable Bonds" (where the issuer can pay you back early). For these, Modified Duration doesn't work well because the cash flows themselves might change if rates drop. In these cases, we use Effective Duration.
Step-by-step: Effective duration is calculated using a formula that looks at what happens to the price if rates go up versus if they go down, using a pricing model. You don't need to do the complex model, just remember: Use Effective Duration when the bond has an embedded option.
6. Convexity: The "Safety" Curve
Duration is a straight-line estimate. However, the actual relationship between bond prices and yields is a curve. This curve is called Convexity.
Why is Convexity good?
For a "normal" (plain-vanilla) bond, convexity is positive. This means:
• When rates fall, the price rises more than duration predicts.
• When rates rise, the price falls less than duration predicts.
Analogy: Duration is like your car's speed, and Convexity is like the car's ability to hug a turn. Convexity makes the "bad" parts of rate moves less painful and the "good" parts even better!
The Big Formula:
\( \% \Delta Price \approx [-ModDur \times \Delta Yield] + [ \frac{1}{2} \times Conv \times (\Delta Yield)^2 ] \)
Summary: Duration is the first estimate; Convexity is the correction that makes the estimate more accurate.
7. Putting it All Together: The Duration Gap
How do you know if you are safe from interest rate risk? You look at your Investment Horizon (how long you plan to hold the bond) versus the bond's Macaulay Duration.
Duration Gap = Macaulay Duration - Investment Horizon
• Positive Gap (Dur > Horizon): You are more worried about Price Risk. If rates rise, you lose because the price drop hurts more than the reinvestment gain helps.
• Negative Gap (Dur < Horizon): You are more worried about Reinvestment Risk. If rates fall, you lose because you can't earn enough on your reinvested coupons.
• Zero Gap (Dur = Horizon): You are "immunized." The price risk and reinvestment risk roughly cancel each other out!
Key Takeaway: To protect yourself from interest rate changes, try to match the bond's Macaulay Duration to your expected holding period.
Great job! You've just covered the core mechanics of how interest rates impact bond returns. Take a break, and when you're ready, try a few practice problems to see these formulas in action!