Bond Risk Measures and Immunization › Portfolio Duration and Convexity
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Individual bond risk measures become even more powerful when extended to portfolios. A portfolio manager holding dozens or hundreds of bonds needs a way to summarize the aggregate interest rate exposure. Fortunately, duration and convexity combine in a straightforward way: they are computed as weighted averages, where the weights are each bond's market value as a proportion of total portfolio value.
For a portfolio with bonds, each with market value and modified duration :
where:
- : the portfolio's modified duration
- : the weight of bond in the portfolio, calculated as
- : the market value of bond
- : the modified duration of bond
- : the number of bonds in the portfolio
This formula follows from the linearity of the price change approximation. If each bond's percentage price change is given by , then the portfolio's percentage change is the value-weighted average of the individual changes. The same weighting applies to convexity:
where is the convexity of bond .
This weighted-average approach allows portfolio managers to adjust interest rate exposure by changing the mix of bonds. With positive portfolio value and a positive-value addition, holding the other positions fixed, a bond whose modified duration exceeds the current portfolio duration raises it; one below the current portfolio duration lowers it. These comparisons use the same yield-change coordinate.
def portfolio_duration_convexity(bonds_data, yield_rate):
"""
Calculate portfolio duration and convexity.
bonds_data: list of dicts with 'face_value', 'coupon_rate', 'maturity', 'quantity'
"""
results = []
total_value = 0
for bond in bonds_data:
price_i = bond_price(
bond["face_value"],
bond["coupon_rate"],
bond["maturity"],
yield_rate,
)
value_i = price_i * bond["quantity"]
dur_i = modified_duration(
bond["face_value"],
bond["coupon_rate"],
bond["maturity"],
yield_rate,
)
conv_i = convexity(
bond["face_value"],
bond["coupon_rate"],
bond["maturity"],
yield_rate,
)
results.append(
{
"price": price_i,
"value": value_i,
"duration": dur_i,
"convexity": conv_i,
}
)
total_value += value_i
# Calculate weighted averages
port_duration = sum(r["value"] * r["duration"] for r in results) / total_value
port_convexity = sum(r["value"] * r["convexity"] for r in results) / total_value
return port_duration, port_convexity, total_value, results# Create a sample portfolio
portfolio = [
{
"face_value": 1000,
"coupon_rate": 0.03,
"maturity": 2,
"quantity": 100,
"name": "2-year, 3%",
},
{
"face_value": 1000,
"coupon_rate": 0.04,
"maturity": 5,
"quantity": 150,
"name": "5-year, 4%",
},
{
"face_value": 1000,
"coupon_rate": 0.05,
"maturity": 10,
"quantity": 100,
"name": "10-year, 5%",
},
{
"face_value": 1000,
"coupon_rate": 0.055,
"maturity": 30,
"quantity": 50,
"name": "30-year, 5.5%",
},
]
yield_rate = 0.05
port_dur, port_conv, total_val, details = portfolio_duration_convexity(
portfolio, yield_rate
)
print("Portfolio Composition")
print("=" * 75)
print(
f"{'Bond':<15} {'Price':>10} {'Quantity':>10} {'Value':>12} {'Duration':>10} {'Weight':>10}"
)
print("-" * 75)
for bond, det in zip(portfolio, details):
weight = det["value"] / total_val * 100
print(
f"{bond['name']:<15} ${det['price']:>9.2f} {bond['quantity']:>10} ${det['value']:>11,.2f} {det['duration']:>10.2f} {weight:>9.1f}%"
)
print("-" * 75)
print(f"{'Total':<15} {'':<10} {'':<10} ${total_val:>11,.2f}")
print(f"\nPortfolio Duration: {port_dur:.2f}")
print(f"Portfolio Convexity: {port_conv:.2f}")Portfolio Composition =========================================================================== Bond Price Quantity Value Duration Weight --------------------------------------------------------------------------- 2-year, 3% $ 962.38 100 $ 96,238.03 1.91 24.5% 5-year, 4% $ 956.24 150 $ 143,435.95 4.46 36.4% 10-year, 5% $ 1000.00 100 $ 100,000.00 7.79 25.4% 30-year, 5.5% $ 1077.27 50 $ 53,863.58 15.16 13.7% --------------------------------------------------------------------------- Total $ 393,537.56 Portfolio Duration: 6.15 Portfolio Convexity: 74.98
The portfolio's modified duration is approximately 6.15 years, the market-value-weighted average of the individual modified durations. The thirty-year bond has the longest modified duration, about 15.16 years, but represents only 13.69% of portfolio value. Portfolio convexity is about 74.98. For a small common yield change, these measures give the portfolio's first- and second-order price approximation; positive convexity does not prevent a loss when yields rise.
