Welcome to the World of Duration!

Welcome, future Charterholders! If you’ve ever wondered exactly how much a bond’s price will wiggle when interest rates change, you’re in the right place. In this chapter, we explore Yield-Based Duration. Think of duration not just as "time," but as a measure of sensitivity. It tells us how "sensitive" a bond's price is to changes in its own yield-to-maturity (YTM). Understanding this is crucial because, in the fixed-income world, when interest rates move, bond prices move in the opposite direction—and duration tells us by how much.

1. Macaulay Duration: The "Wait Time"

The concept of duration started with Frederick Macaulay. He wanted to know the weighted average time an investor must wait to receive all the cash flows from a bond.

What it is:

Imagine a seesaw. On one side, you have the time until each coupon and the principal are paid. Macaulay Duration (MacDur) is the "balance point" of those cash flows. If you receive most of your money early (like a high-coupon bond), the balance point shifts closer to today. If you have to wait until the very end (like a zero-coupon bond), the balance point is much further out.

The Math (Don't panic!):

\( \text{MacDur} = \frac{\sum_{t=1}^{n} \frac{t \times CF_t}{(1+r)^t}}{PV} \)

In simple terms: We take each cash flow, find its present value, multiply it by the time it's received, and divide the whole sum by the bond's current price.

Key Rule to Remember:

For a Zero-Coupon Bond, the Macaulay Duration is exactly equal to its time to maturity. Why? Because there is only one cash flow at the very end, so the "average wait time" is just the maturity date!

Quick Review: Macaulay Duration is measured in years.

2. Modified Duration: The Price Mover

While Macaulay Duration is about time, Modified Duration (ModDur) is about money. Specifically, it measures the percentage change in a bond's price for a 1% change in yield.

The Formula:

\( \text{ModDur} = \frac{\text{MacDur}}{1 + r} \)

Where \( r \) is the yield per period. Note: If the bond pays semi-annually, you must divide the annual YTM by 2 before adding it to 1.

Why do we use it?

We use it to estimate price changes. The relationship is inverse (negative), meaning as yields go up, prices go down:
\( \% \Delta \text{Price} \approx -\text{ModDur} \times \Delta \text{Yield} \)

Example: If a bond has a Modified Duration of 5.0 and interest rates rise by 1% (100 basis points), the bond price will fall by approximately 5%.

Don't worry if this seems tricky at first: Just remember that ModDur is simply a "scaled" version of MacDur that makes it easier to talk about price volatility.

Key Takeaway: Modified Duration provides a linear estimate of how much the price will change for a small change in yield.

3. The Three Properties of Duration

How do bond characteristics affect duration? This is a favorite topic for exam questions! Think of these "levers" that make duration go up or down:

  • Time to Maturity: Generally, as maturity increases, duration increases. Long-term bonds are riskier and more sensitive to rate changes because the cash flows are further in the future.
  • Coupon Rate: As the coupon rate increases, duration decreases. Why? Because you are getting more of your money back sooner in the form of large coupon payments, which lowers the "average wait time."
  • Yield to Maturity (YTM): As the YTM increases, duration decreases. Higher yields mean future cash flows are discounted more heavily, making the distant payments "worth less" today compared to the near-term payments.

Mnemonic Aid: Think of "M.C.Y." (Maturity, Coupon, Yield).
- Maturity moves with duration (Direct).
- Coupon and Yield move against duration (Inverse).

4. Money Duration and PVBP

Sometimes, investors want to know the actual dollar change in a bond's price, not just the percentage. This is where Money Duration and PVBP come in.

Money Duration (MoneyDur)

This is the dollar change in price for a unit change in yield.
\( \text{MoneyDur} = \text{ModDur} \times \text{Full Price} \)

Price Value of a Basis Point (PVBP)

This is a very common industry term. It tells you exactly how many dollars the bond price will change if the yield moves by just one basis point (0.01%).
\( \text{PVBP} = \frac{\Delta \text{PV}_{up} + \Delta \text{PV}_{down}}{2} \) for a 1bp change.

Simplified way to think about it: \( \text{PVBP} = \text{Money Duration} \times 0.0001 \)

Did you know? Traders use PVBP to quickly size their positions. If a trader knows their PVBP is \$500, and rates move up 10 basis points, they know instantly they just lost \$5,000.

5. Limitations: The Problem with the "Straight Line"

Duration is a great tool, but it's not perfect. It assumes the relationship between bond prices and yields is a straight line. In reality, the price-yield relationship is curved (convex).

Common Mistake to Avoid: Using duration for large changes in interest rates. Duration is very accurate for a 0.1% change, but it becomes less accurate for a 2% change. This is why we eventually need "Convexity" to fix the error, but for now, just remember that duration is a linear approximation.

6. Summary Quick-Check

Before you move on, make sure you can answer these:

  • What is MacDur? The weighted average time to receive cash flows.
  • What is ModDur? The % change in price for a 1% change in yield.
  • Which bond has the highest duration? A long-term, zero-coupon bond with a low yield.
  • Inverse Relationship: If rates go UP, price goes DOWN. Duration tells you how much.

Key Takeaway: Yield-based duration measures (Macaulay and Modified) assume that a change in the bond's own YTM is the only thing driving the price change. They are the foundation of risk management in fixed income!