1. What is Duration?
Duration represents the weighted average time to recover your cash flows from a bond. It differs from maturity, which simply shows the time until final repayment. Duration reflects the actual average period in which you recover your investment and is the most critical measure of a bond's price sensitivity to interest rate changes.
Why Should You Understand Duration?
When interest rates rise, bond prices fall. When rates decline, bond prices rise. However, different bonds experience different price changes from the same rate movement. Duration explains this differenceโit measures how sensitive a bond's price is to interest rate changes.
Duration of 5 years = 1% interest rate change results in approximately 5% bond price change
This is called "interest rate sensitivity."
What Duration Tells You
- Long-duration bonds: React more dramatically to rate changes. When rates rise, prices fall sharply; when rates decline, prices rise sharply.
- Short-duration bonds: React less dramatically. Rate changes have a smaller impact on price.
- High-coupon bonds (short duration): Due to larger early cash flows, the average recovery period shortens.
An investor places $100,000 in bonds.
- Bond A: Duration 3 years โ 1% rate rise causes ~3% price decline ($97,000)
- Bond B: Duration 7 years โ 1% rate rise causes ~7% price decline ($93,000)
Same rate movement, but Bond B with longer duration suffers a larger loss!
2. How to Calculate Duration
Macaulay Duration
The most basic way to calculate duration is Macaulay Duration. It weights each cash flow by its present value, multiplies by the time of occurrence, and divides by the total bond price.
t = Time period (years) when cash flow occurs
CF_t = Cash flow at time t
y = Yield to Maturity
P = Current bond price (PV of all cash flows)
ฮฃ = Sum across all cash flows
Step-by-Step Calculation Example
3-year Bond
- Par Value: $1,000
- Annual Coupon: $50 (5% coupon rate)
- Yield to Maturity (YTM): 4%
- Current Bond Price: approximately $1,028.04
< Calculate Present Value of Cash Flows > Year 1: $50 / (1.04)^1 = $48.08 Year 2: $50 / (1.04)^2 = $46.23 Year 3: $1,050 / (1.04)^3 = $933.73 Bond Price P = $48.08 + $46.23 + $933.73 = $1,028.04 < Calculate Weighted Cash Flows > Year 1: 1 ร $48.08 = $48.08 Year 2: 2 ร $46.23 = $92.46 Year 3: 3 ร $933.73 = $2,801.19 Total = $2,941.73 < Macaulay Duration > Duration = $2,941.73 / $1,028.04 = 2.86 years
This bond's duration of 2.86 years means you recover your initial investment on average after 2.86 years. Although the bond matures in 3 years, the intermediate coupon payments shorten the actual recovery period to 2.86 years.
Modified Duration
To directly calculate how bond prices change with interest rate movements, we use Modified Duration.
Modified Duration = 2.86 / 1.04 = 2.75
ฮP โ -Modified Duration ร ฮy ร P
If rates rise 0.5%: ฮP โ -2.75 ร 0.005 ร $1,028.04 = -$14.13 (about 1.37% decline)
3. Factors That Affect Duration
A bond's duration is determined by several factors. Understanding each helps you select appropriate bonds for various market conditions.
| Factor | Change | Duration Impact | Explanation |
|---|---|---|---|
| Maturity | Lengthens | โ Increases | Longer time to receive cash flows means longer average recovery period. |
| Coupon Rate | Increases | โ Decreases | Higher coupons provide more early cash, shortening the average recovery period. |
| Market Yield (YTM) | Rises | โ Decreases | Higher discount rates reduce the weight of distant cash flows relative to early flows. |
| Credit Quality | Deteriorates | โ Increases | Lower credit quality reduces confidence in distant cash flows, effectively increasing duration. |
| Embedded Options | Issuer-favorable | โ Decreases | Call options shorten effective maturity, reducing duration. |
Real-World Comparison
- 10-year 0% coupon bond: Duration ~9.4 years (close to maturity)
- 10-year 5% coupon bond: Duration ~7.7 years (higher early coupons shorten duration)
- 10-year 10% coupon bond: Duration ~6.8 years (even more early cash shortens duration)
4. Real-World Case Study: TLT vs SHY
Let's examine how duration actually impacts two popular U.S. Treasury bond ETFs.
TLT (Vanguard Extended Duration Treasury ETF)
Holdings: 20+ year Treasury bonds
Average Duration: ~14-15 years
Characteristics: Very long duration, high volatility
SHY (iShares 1-3 Year Treasury Bond ETF)
Holdings: 1-3 year Treasury bonds
Average Duration: ~1.8-2 years
Characteristics: Very short duration, low volatility
Scenario: 2% Interest Rate Increase
Expected TLT Price Change:
ฮP = -14.5 ร 2% = -29% ($300,000 โ ~$213,000)
Expected SHY Price Change:
ฮP = -1.9 ร 2% = -3.8% ($300,000 โ ~$288,600)
Result: A 2% rate increase causes TLT to fall 29% but SHY only 3.8%!
If rate increases are expected: Prefer short-duration bonds (SHY) to minimize losses
If rate decreases are expected: Prefer long-duration bonds (TLT) to maximize gains
5. Practice Problems
Apply what you've learned. Work through each problem and verify your answers.
Problem 1: Price Change with Rising Rates
A bond with 7-year duration has a $1,000,000 investment. If market rates rise 1%, what is the expected loss?
Modified Duration โ Macaulay Duration / (1 + YTM)
Simplified approximation: Modified Duration โ Duration
Price Change = -Duration ร Rate Change ร Bond Price
= -7 ร 1% ร $1,000,000
= -7 ร 0.01 ร $1,000,000
= -$70,000
Answer: Expected loss is approximately $70,000.
Problem 2: Comparing Bond Durations
Which of these three bonds would suffer the largest loss if rates rise 1%?
(1) 10-year maturity, 2% coupon, 3% YTM
(2) 10-year maturity, 5% coupon, 5% YTM
(3) 10-year maturity, 8% coupon, 4% YTM
For the same maturity, lower coupon means longer duration.
(1) 2% coupon: Duration ~9.5 years (longest)
(2) 5% coupon: Duration ~8.0 years (medium)
(3) 8% coupon: Duration ~7.0 years (shortest)
Also consider the YTM-coupon relationship:
- (1) has coupon(2%) < YTM(3%), maximizing duration
Answer: (1) 10-year, 2% coupon bond
The longest duration means the largest loss when rates rise.
Problem 3: Portfolio Interest Rate Risk
Portfolio composition:
โข Short-term bonds (2-year duration): $500,000
โข Long-term bonds (8-year duration): $500,000
If rates fall 0.5%, what is the expected portfolio gain?
Portfolio average duration:
= ($500,000 ร 2 + $500,000 ร 8) / $1,000,000
= ($1,000,000 + $4,000,000) / $1,000,000
= 5 years
Price gain from 0.5% rate decline:
= Duration ร Rate Change ร Portfolio Value
= 5 ร 0.5% ร $1,000,000
= 5 ร 0.005 ร $1,000,000
= $25,000
Answer: Expected gain is approximately $25,000.