Portfolio Duration and its Limitations

Portfolio Duration and its Limitations

Like equities, bonds are typically held in a portfolio. Therefore, bond portfolio managers need to measure the whole portfolio duration.

There are two methods for calculating the duration of a bond portfolio:

  • the weighted average of time to receipt of the aggregate cash flows; and
  • the weighted average of the individual bond durations comprising the portfolio.

The first method is the theoretically correct approach which is, nevertheless, practically hard to implement. The second method is commonly used by fixed-income portfolio managers but has its own limitations.

Portfolio Duration

Assume that an investor holds the following portfolio of two zero-coupon bonds.

\textbf{Bond} & \textbf{Maturity} & \textbf{Price} & \textbf{Yield} & \textbf{Mac. Dur} & \textbf{Mod. Dur} & \textbf{Par Value} & \textbf{Market Value} & \textbf{Weight} \\ \hline
\text{X} & \text{1 year} & \text{98.00} & \text{2.0408%} & \text{1.000} & \text{0.980} & \text{\$1,000,000} & \text{\$980,000} & \text{16.81%} \\
\text{Y} & \text{2 years} & \text{97.00} & \text{1.5346%} & \text{2.000} & \text{0.984} & \text{\$5,000,000} & \text{\$4,850,000} & \text{83.19%} \\

Portfolio Duration Using Method 1: The Weighted Average of Time to Receipt of the Aggregate Cash Flows

The prices are per 100 of par value. The yields-to-maturity are effective annual rates. The total market value for the portfolio is 5,830,000. The portfolio is unevenly weighted in terms of market value between the 2 bonds (with weights of 16.81% and 83.19% for X and Y, respectively). Assume the yield per period to be 12.366%.

The first approach views the portfolio as a series of aggregated cash flows, in which case the aggregate cash flow yield is 1.5811% (the solution to the following equation).

$$5,830,000=\frac{1,000,000}{(1+r)^1} +\frac{5,000,000}{(1+r)^2}; \quad r = 1.5811\%$$

There are just 2 future cash flows in the portfolio, which are the redemption of the principal of the 2 zero-coupon bonds. However, in more complex portfolios, a series of coupon and principal payments may occur on various dates.

Using a discount rate of 1.5811%, we can, therefore, find the portfolio’s Macaulay duration as the weighted average of time to receipt of aggregated cash flow.

$$MacDur=\frac{\frac{1×1,000,000}{1.015811^1} +\frac{2×5,000,000}{1.015811^2}}{\frac{1,000,000}{1.015811^1}+\frac{5,000,000}{1.015811^2 }}=\frac{8,810,040}{4,850,000}=1.8311$$

The modified duration for the portfolio is 1.8026


The modified duration indicates the percentage change in the market value given a change in the cash flow yield. For example, if the cash flow yield increases or decreases by 100 bps, the portfolio’s market value is expected to increase or decrease by about 1.8026%.

Although this approach is theoretically correct, it is difficult to use in practice. To begin with, the cash flow yield is not commonly calculated for bond portfolios. Aside from this, the amount and timing of future cash payments are uncertain. Lastly, interest rate risk is not usually expressed as a change in benchmark interest rates. In practice, the second approach to portfolio duration is commonly used, and there is a stronger likelihood you will have to use the second formula in your CFA level 1 exam.

Portfolio Duration Using Method 2: The Weighted Average of the Durations of Individual Bonds That Make Up the Portfolio

Using this approach:

$$\text{Portfolio duration} =w_1 D_1+w_2 D_2+⋯+w_n D_n$$


wi  = market value of bond i / market value of portfolio

Di  = duration of bond i

n = number of bonds in portfolio

Using the above data, we can calculate portfolio duration as follows:

$$\text{Portfolio Macaulay duration} =0.1681×1+0.8319×2=1.8319$$

$$\text{Modified duration} =\frac{1.8319}{1+0.12366}=1.6303$$

One of the advantages of this approach is that it can be an effective measure of interest rate risk. For instance, if the yields-to-maturity on the bonds in the portfolio increase by 100 bps, the estimated drop in the portfolio value is 1.6303%. However, its main disadvantage is that it assumes a parallel shift in the yield curve where all rates change by the same amount in the same direction. As is usually observed, interest rates change in an irregular manner, resulting in a steeper or flatter yield curve.

Question 1

When using the weighted average of time to receipt of the aggregate cash flows method (method 1), you would most likely use:

  1. The price of the bonds calculate the weights.
  2. The par value of the bonds calculate the weights.
  3. The market value of the bonds calculate the weights.


The correct answer is B.

According to method 1, you would use the par values of the bonds to come up with the weights used to calculate the aggregated cash flow yield. After that, you would compute the Macaulay duration and finally, the modified duration.

Question 2

A bond portfolio consists of the following three fixed-rate bonds. Assume annual coupon payments and no accrued interest on the bonds. Prices are per 100 of par value.

\textbf{Bond} & \textbf{Maturity} & \textbf{Price} & \textbf{Yield} & \textbf{Coupon} & \textbf{Modified Duration} & \textbf{Market Value} \\
\text{1} & \text{6 years} & \text{85.00} & \text{4.95%} & \text{2.00%} & \text{5.42} & \text{170,000} \\
\text{2} & \text{10 years} & \text{80.00} & \text{4.99%} & \text{2.40%} & \text{8.44} & \text{850,000} \\
\text{3} & \text{9 years} & \text{85.78} & \text{5.00%} & \text{3.00%} & \text{7.54} & \text{180,000} \\

The bond portfolio’s modified duration is closest to:

  1. 6.55
  2. 7.54
  3. 7.88


The correct answer is C.

The portfolio’s modified duration is closest to 7.62. Portfolio duration is commonly estimated as the market-value-weighted average of the yield durations of the individual bonds in the portfolio.

The total market value of the bond portfolio is 170,000 + 850,000 + 180,000 = 1,200,000.

The portfolio duration is 5.42 × (170,000/1,200,000) + 8.44 × (850,000/1,200,000) + 7.54 × (180,000/1,200,000) = 7.88

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

    Subscribe to our newsletter and keep up with the latest and greatest tips for success
    Shop Actuarial Exams Prep Shop Graduate Admission Exam Prep

    Sergio Torrico
    Sergio Torrico
    Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar.
    So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings.
    Kriti Dhawan
    Kriti Dhawan
    A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep.
    nikhil kumar
    nikhil kumar
    Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures.
    Great support throughout the course by the team, did not feel neglected
    Benjamin anonymous
    Benjamin anonymous
    I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend
    Daniel Glyn
    Daniel Glyn
    I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
    michael walshe
    michael walshe
    Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.