Empirical vs analytical duration
Analytical duration is calculated from a formula or a pricing model and assumes benchmark yields and credit spreads move independently. Empirical duration is estimated from how prices actually moved in the past. For credit-risky bonds it is usually lower, because spreads tend to widen when government yields fall.
Before this: Effective duration and effective convexity
On the exam
- Describe the difference between empirical duration and analytical duration
Describe the difference between empirical and analytical duration. Expect definitions, and which is more relevant for a high-yield portfolio versus a government portfolio. Wrong options swap the two definitions, call effective duration empirical, or say empirical duration is higher for high-yield bonds.
How deeply to learn each part
- Know the idea
- Understand it in general. It frames other topics, but you will not calculate with it.
- Use it
- Understand it well enough to solve a question with it. Recognize when it applies.
- Memorize
- Learn it by heart. The exam will not give it to you.
You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.
You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.
What are analytical and empirical duration?
Every duration so far (Macaulay, modified, approximate, effective, key rate) is calculated from a formula or a valuation model. These are analytical duration measures. They give a reasonable picture of the price–yield relationship in many situations, but they carry a hidden assumption: that government bond yields and credit spreads are independent, so a change in the benchmark yield passes through to the bond’s price one for one.
Empirical duration takes the opposite route. Fixed-income professionals fit a statistical model to historical prices and yields, with the factors that affect bond prices, over time and in different interest rate environments. The estimate is whatever relationship the data show.
- Analytical duration Memorize formula or valuation model
- Macaulay and modified duration yield-based
- Effective duration curve-based, uses an option model
- Key rate duration curve-based, one maturity at a time
- Assumes spreads and benchmark yields are independent Memorize the weakness
-
- Empirical duration Memorize statistical model on historical prices
- Captures spread and yield correlation negative in stress
- Judgment calls which factors, which data period, how to weight old vs new data
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| Analytical duration | Empirical duration | |
|---|---|---|
| How it is found | Mathematical formula or valuation model (Macaulay, modified, effective, key rate) | Statistical model on historical prices and yields |
| Key assumption | Benchmark yields and spreads are independent | Uses whatever relationship the data show |
| Works best for | Government bonds (price driven by benchmark yields) | Bonds with credit and liquidity risk |
| Challenges | Misses the spread–yield correlation | Choosing factors, the data period and the weight of old versus new data |
Why do they differ for credit-risky bonds?
In a crisis, investors sell risky assets and buy government bonds. This flight to quality pushes government yields down and, because default risk is now a bigger worry, pushes credit spread s on risky bonds up. The two changes move in opposite directions, so the correlation of spreads and benchmark yields is negative, especially in stressed markets.
For a government bond with little or no credit risk, benchmark yields drive the price and the two duration measures agree. For a lower-quality corporate bond, wider spreads partly or fully offset the fall in the benchmark. The bond’s price gains less than analytical duration predicts, so its empirical duration comes out lower.
A bond has an analytical duration of 6. In a stress month the benchmark yield changes by −50 bp and the bond's own yield changes by −10 bp. Treating the bond's price change as its analytical duration times its own yield change, the duration the bond showed against the benchmark move is closest to:
(6 × −0.001) ÷ −0.005 = 1.2
Empirical duration (against the benchmark move)1.2
- 6×0.001+|-÷0.005+|-=
+|- change sign
- 6ENTER0.001CHS×0.005CHS÷
ENTER enter (on the HP, puts the number on the stack) · CHS change sign
Analysts must decide which to trust by looking at that correlation. For a portfolio of medium-term government bonds, analytical duration is fine. For a portfolio with high-yield bonds, empirical duration is the more accurate guide to how its value will respond to benchmark yield changes, particularly in a crisis.
Exam tip Spreads widen when government yields fall in a crisis. That is why credit-risky bonds behave as if their duration were shorter.
Go deeper Empirical duration is a regression slope
Behind the scenes, empirical duration is the slope of a regression of the bond’s price changes on benchmark yield changes. A slope of −1.2 means a 1 percentage point fall in the benchmark goes with about a 1.2% rise in price. See the simple linear regression module for how a slope is estimated.
Go deeper Why the estimate depends on the period chosen
Fit the model on a calm year and spreads barely move, so empirical duration lands close to analytical duration. Fit it on a crisis and the negative correlation dominates and the estimate falls. Analysts therefore choose the data window, and how heavily to weight recent data, with care.
Go deeper Hedging a corporate portfolio with government futures
A manager who hedges a corporate bond portfolio with government futures, sized on the portfolio’s analytical duration, hedges more benchmark exposure than the portfolio really has in a crisis. The corporate bonds gain less than the analytical duration implies, while the short futures position loses in full as government yields fall, so the hedge over-hedges.
Memorize this
MemorizeAnalytical vs empirical duration
| Analytical | Empirical | |
|---|---|---|
| How it is found | Formula or valuation model | Statistical model on historical data |
| Key assumption | Benchmark yields and spreads independent | Whatever the data show |
| Works best for | Government bonds | Bonds with credit and liquidity risk |
| Challenges | Misses spread–yield correlation | Choosing factors, period and weights |
| Credit-risky bond in a flight to quality | Higher | Lower |
Memory hook Analytical = A for algebra, from a formula. Empirical = E for evidence, from past prices. In a crisis the evidence shows less price gain than the algebra promises.
Know it cold Government bonds: the two are close. Credit-risky bonds: empirical is usually lower than analytical, often much lower for high yield.
- Analytical duration Memorize formula or valuation model
- Macaulay and modified duration yield-based
- Effective duration curve-based, uses an option model
- Key rate duration curve-based, one maturity at a time
- Assumes spreads and benchmark yields are independent Memorize the weakness
-
- Empirical duration Memorize statistical model on historical prices
- Captures spread and yield correlation negative in stress
- Judgment calls which factors, which data period, how to weight old vs new data
-
Quick example
Analytical duration 5.0. In a stress month the benchmark yield falls 60 bp and the credit spread widens 45 bp, so the bond’s own yield falls 15 bp and its price rises about 5.0 × 0.0015 = 0.75%. Against the benchmark move: 0.0075 ÷ 0.0060 = 1.25.
- 5×0.0015=÷0.006= 1.25, both yields fall so the signs cancel
- 5ENTER0.0015×0.006÷ 1.25, both yields fall so the signs cancel
ENTER enter (on the HP, puts the number on the stack)
How it is asked
-
Which is which
Duration estimated by applying a statistical model to historical bond prices is?
Empirical. Effective duration is model-based, so it is a type of analytical duration.
-
Flight to quality
During a flight to quality, a high-yield bond’s empirical duration is likely to be?
Lower than its analytical duration: wider spreads offset part of the government yield fall.
-
Government portfolio
Analytical and empirical duration for highly rated government bonds?
Broadly similar: benchmark yields drive their prices.
Exam traps
- Mixing up which is formula-based (analytical) and which is data-based (empirical).
- Calling effective or key rate duration empirical. Both are analytical, because they come from a valuation model.
- Saying empirical duration is higher for high-yield bonds. It is usually lower.
- Assuming the two differ much for government bonds.
Checkpoint
5 exam-style questions. About 90 seconds each.
1Duration estimated by applying a statistical model to historical bond price data is best described as:
2During a flight to quality, a high-yield bond's empirical duration is most likely:
3For a portfolio of highly rated government bonds, analytical and empirical duration estimates are most likely to be:
4A bond has an analytical duration of 7.0. In a stress month the benchmark yield falls 40 bp and the credit spread widens 30 bp, so the bond's own yield falls 10 bp. Using the analytical duration and the bond's own yield change, its price rises about 0.70%. Measured against the benchmark yield move, the duration the bond showed is closest to:
5Which of the following is a challenge in estimating empirical duration?
Key takeaways
- Analytical: formula or model. Empirical: historical data.
- Analytical assumes yields and spreads move independently.
- Credit-risky bonds: empirical duration is usually lower than analytical.
- Government bonds: the two are close.
Quick example
Analytical duration 5.0. In a stress month the benchmark yield falls 60 bp and the credit spread widens 45 bp, so the bond’s own yield falls 15 bp and its price rises about 5.0 × 0.0015 = 0.75%. Against the benchmark move: 0.0075 ÷ 0.0060 = 1.25.
- 5×0.0015=÷0.006= 1.25, both yields fall so the signs cancel
- 5ENTER0.0015×0.006÷ 1.25, both yields fall so the signs cancel
ENTER enter (on the HP, puts the number on the stack)
All Fixed Income formulas, every lesson
Practice exam-style questions on Effective Duration, Key Rate Duration and Empirical Duration
Questions about this lesson?
Where this comes back
- Credit Risk Fixed Income
- Introduction to Risk Management Portfolio Construction
Questions candidates ask
What is empirical duration?
A duration estimated from how a bond's price actually responded to benchmark yield changes in the past, using a statistical model on historical data.
Why is empirical duration lower for high-yield bonds?
When government yields fall in a crisis, credit spreads usually widen, offsetting part of the price gain. The bond reacts less to the benchmark than its analytical duration suggests.
Is effective duration analytical or empirical?
Analytical. It comes from a valuation model, not from a statistical fit to historical prices.
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