Macaulay Duration
Understanding Macaulay duration as the weighted average time to receive cash flows
What is Macaulay Duration?
Macaulay Duration is the foundational concept for measuring bond risk. It is not primarily a measure of price sensitivity (that's Modified Duration). Instead, Macaulay Duration measures the weighted-average time (in years) until an investor receives their bond's cash flows.
Think of it as the "center of gravity" or the "balancing point" of your bond's payments. A zero-coupon bond maturing in 10 years has a Macaulay Duration of exactly 10 years (all cash flow arrives at maturity). A coupon-paying bond maturing in 10 years will have a duration of *less* than 10 years, because the coupon payments shift the weighted-average time of cash flows closer to today.
The "weight" for each cash flow is its Present Value (PV) as a percentage of the bond's total price.
Calculation Formula
The formula for Macaulay Duration ($D_{mac}$) is the sum of all $t \times PV(CF_t)$ divided by the bond's price.
\[D_{mac} = \frac{\sum_{t=1}^{n} t \times PV(CF_t)}{P}\]Where:
- $t$ = The time period (in years) until the cash flow is received
- $PV(CF_t)$ = The present value of the cash flow at time t, discounted at the bond's YTM
- $P$ = The total price of the bond (the sum of all $PV(CF_t)$)
For JGBs, which pay semi-annual coupons, the formula is adapted. We calculate duration in *periods* first and then divide by 2 to get years.
\(D_{mac \text{ (periods)}} = \frac{\sum_{t=1}^{n} t \times \frac{C/2}{(1+y/2)^t} + \frac{n \times F}{(1+y/2)^n}}{\text{Price}}\) \(D_{mac \text{ (years)}} = \frac{D_{mac \text{ (periods)}}}{2}\)
Where:
- $n$ = Total number of semi-annual periods
- $t$ = The specific period (1, 2, 3... n)
- $C$ = Annual coupon rate
- $y$ = Yield to Maturity (YTM)
- $F$ = Face Value
Worked JGB Example
Let's calculate the Macaulay Duration for a simple 3-year JGB.
- Face Value (F): ¥1,000,000
- Annual Coupon (C): 1.0% (¥10,000 per year)
- Yield to Maturity (y): 0.8%
Step 1: List inputs (semi-annual basis)
- Number of periods (n) = 3 years × 2 = 6
- Semi-annual coupon = ¥10,000 / 2 = ¥5,000
- Semi-annual yield (y/2) = 0.008 / 2 = 0.004
Step 2: Build the cash flow table.
| Period (t) | Cash Flow ($CF_t$) | PV of $CF_t$ | $t \times PV(CF_t)$ |
|---|---|---|---|
| 1 | ¥5,000 | ¥4,980.08 | ¥4,980.08 |
| 2 | ¥5,000 | ¥4,960.24 | ¥9,920.48 |
| 3 | ¥5,000 | ¥4,940.48 | ¥14,821.44 |
| 4 | ¥5,000 | ¥4,920.80 | ¥19,683.19 |
| 5 | ¥5,000 | ¥4,901.19 | ¥24,505.97 |
| 6 | ¥1,005,000 | ¥981,200.74 | ¥5,887,204.44 |
| Total | ¥1,005,903.53 (Price) | ¥5,961,115.60 |
Step 3: Calculate Macaulay Duration.
$$D_{mac \text{ (periods)}} = \frac{¥5,961,115.60}{¥1,005,903.53} = \textbf{5.9262 periods}$$
$$D_{mac \text{ (years)}} = \frac{5.9262}{2} = \textbf{2.9631 years}$$
Interpretation
The bond's maturity is 3.0 years, but its Macaulay Duration is 2.9631 years. This means the weighted-average time to receive the cash flows is just under the final maturity date, which makes sense as the small coupon payments pull the "center of gravity" forward.
Key Drivers of Macaulay Duration
- Maturity: Longer maturity = Higher duration.
- Coupon Rate: Lower coupon = Higher duration (closer to a zero-coupon bond).
- Yield: Lower yield = Higher duration (distant cash flows are weighted more heavily).
Advanced Topic: Duration in Negative Yield Environments
The Japanese bond market provides unique insights into duration behavior because JGBs traded with negative yields for an extended period (2016-2024) during the Bank of Japan's ultra-accommodative monetary policy and Yield Curve Control (YCC) regime.
How Duration Changes at Negative Yields
Mathematically, Macaulay Duration remains well-defined even when yields are negative—the formula still works. However, several important phenomena occur:
1. Duration Increases as Yields Fall Below Zero
As yields become increasingly negative, future cash flows are discounted less heavily (or even "inflated" in present value terms), which pushes the weighted-average time further out. For a given bond:
- At y = 0.5%: 10Y JGB might have duration of 9.5 years
- At y = 0.0%: Same bond's duration approaches 9.7 years
- At y = -0.2%: Duration might reach 10.0+ years
This counterintuitive result means that as yields fall into negative territory, price sensitivity to further yield changes actually increases.
2. Premium Bonds Become the Norm
When yields are negative, nearly all bonds trade above par (premium bonds). Investors are paying more than ¥100 to receive ¥100 at maturity, compensated by ongoing coupon payments. This affects:
- Reinvestment risk: Coupons must be reinvested at negative rates
- Pull-to-par dynamics: Premium bonds experience capital losses as they approach maturity
- Total return calculations: Negative carry from price amortization offsets positive coupon income
3. Convexity at the Zero Bound
Near zero yields, bonds exhibit heightened convexity (see Section 5.4). This means:
- Price gains from yield decreases are larger than price losses from equivalent yield increases
- Duration itself becomes more sensitive to yield changes
- Portfolio managers must account for this non-linearity in risk models
JGB-Specific Examples from the Negative Rate Era
Example 1: 10-Year JGB in 2016
During July 2016, the 10-year JGB yield briefly touched -0.30%, an extreme level even by global standards.
- Bond: JGB 10Y, 0.1% coupon
- Yield: -0.30%
- Price: Approximately ¥103.80 (significant premium)
- Duration: ~10.2 years (higher than at positive yields)
An investor buying at this price would:
- Receive ¥0.10 annual coupon for 10 years = ¥1.00 total
- Receive ¥100 at maturity
- Pay ¥103.80 upfront
- Net loss of ¥2.80 if held to maturity (the cost of "safe haven" demand)
Example 2: Duration Risk During YCC Exit (2024)
When the BOJ began phasing out YCC in March 2024, yields rose sharply from negative to positive territory. Bonds with high duration suffered significant mark-to-market losses:
- 10Y JGB yield: -0.05% → +0.75% (80bp increase over 6 months)
- Price impact: Duration 10 bond lost ~8% in value
- 20Y JGB yield: 0.40% → 1.80% (140bp increase)
- Price impact: Duration 15 bond lost ~21% in value
This demonstrates that duration risk remains fully operational—and potentially more severe—in negative yield environments, especially during regime changes.
Portfolio Implications
For Japanese institutional investors managing portfolios during the negative rate period, duration took on special importance:
Banks
- Forced to hold JGBs for regulatory capital (Liquidity Coverage Ratio)
- Negative yields meant certain losses on hold-to-maturity positions
- Duration management became critical to limit capital losses when rates eventually normalized
- Many shifted to ultra-long bonds (20Y, 30Y, 40Y) to capture positive yields, accepting higher duration risk
Life Insurance Companies
- Asset-liability matching required long-duration assets to match long-dated policy obligations
- Negative yields on 10Y JGBs pushed insurers into 30Y and 40Y bonds, or even foreign bonds
- Higher duration increased interest rate risk but was necessary to generate any positive yield
Pension Funds
- Liability-driven investment (LDI) strategies require matching pension payment durations
- Negative yields made it nearly impossible to generate required returns (~3-4% annually)
- Increased allocation to alternative assets (equities, real estate, foreign bonds) to compensate
Regulatory Perspective: IRRBB Considerations
Basel III regulations require banks to measure Interest Rate Risk in the Banking Book (IRRBB) using duration-based metrics (see Section 5.6). In negative yield environments:
- Standard shocks: +/-200bp parallel shifts become asymmetric (can't go -200bp from -0.1%)
- Floor constraints: Regulators debate whether to impose yield floors in stress scenarios
- Economic Value of Equity (EVE): Duration-based EVE calculations become distorted
- Capital requirements: Higher duration = higher capital charges under some frameworks
Key Takeaways for Negative Yield Duration
- Duration increases at negative yields: Bonds become more sensitive to rate changes
- Premium pricing is universal: Nearly all bonds trade above par, creating pull-to-par capital losses
- Convexity matters more: Non-linear price behavior is pronounced near zero
- Regime change risk: Transitions from negative to positive yields can cause severe losses
- Portfolio strategy shifts: Institutions extend duration to find yield, accepting greater interest rate risk
Japan's experience with negative yields from 2016-2024 provides invaluable lessons for understanding duration behavior at the zero lower bound—lessons that may be relevant for other central banks in future crises.