Fixed Income Investments - Effective, Modified, and Macaulay Duration

Effective Duration
Duration is the approximate percentage change in price for a 100 basis point change in rates. To compute duration, you can apply the following equation that was presented earlier in the guide.

Price if yield decline - price if yield rise / 2(initial price)(change in yield in decimal)

Let's make:
y = change in yield in decimal (= "delta")
V1 = initial price
V2 = price if yields decline by y
V3 = price if yields increase by y

Duration = V2 - V3 / 2(V1)(? y)

Stone & Co 9% of 10 are option free and selling at 106 to yield 8.5%. Let's change rates by 50 bps. The new price for the increase in 50 bps would be 104 and the new price for a decrease in rates would be 109. Then:

Duration = 109 - 104 / 2 *(106) * (.005)
Duration = 5 / 1.06
Duration = 4.717

This means that for a 100 basis point change, the approximate change would be 4.717%

Price Change Given the Effective Duration and Change in Yield
Once you have computed the effective duration of a bond it is easy to find the approximate price change given at change in yield.

Formula 14.13

Approximate Percent Price change = - duration x change in yield x 100

Using the duration for 4.717% obtained from the previous example, let's see the approximate change for a small movement in rates such as a 20 bps increase.

Percentage Price Change = - 4.717 x (+0.0020) x 100 = -.943%
And for a large change, a 250 bps increase:

Percentage Price Change = -4.717. (+0.0250) x 100 = -11.79%

As noted before, these changes are estimates. For small changes in rates, the estimate will be almost dead on. For larger movements in rates, the estimate will be close but will underestimate the new price of the bond regardless of whether the movement in rates is up or down.

Modified Duration
Modified duration is the approximate percentage change in a bond's price for a 100 basis points change in yield, assuming that the bond's expected cash flow does not change when the yield changes. This works for option-free bonds such as Treasuries but not with option-embedded bonds because the cash flows may change due to a call or prepayment.

Effective Duration
Effective duration takes into account the way in which changes in yield will affect the expected cash flows. It takes into account both the discounting that occurs at different interest rates as well as changes in cash flows. This is a more appropriate measure for any bond with an option embedded in it.

Macaulay Duration
In order to better understand Macaulay duration, let's first turn to the modified duration equation:

Formula 14.14

modified duration = 1/(1+yield/k)[1 x pvcf1 + 2  x pvcf2 +...+n x pvcfn / k x Price

k = the number of periods: two for semi-annual, 12 for monthly and so on.
n = the number of periods to maturity
yield = YTM of the bond
pvcf = the present value of cash flows discounted at the yield to maturity.

The bracket part of the equation was developed by Frederick Macaulay in 1938 and is referred to as Macaulay Duration.

So Modified duration = Macaulay's Duration/ (1 + yield/k)

Macaulay's duration gives the analysis a short cut to measure modified duration. But because modified duration is flawed by not incorporating the change in cash flows due to an embedded option, so are Macaulay durations.

When is Effective Duration a Better Measure?
When a bond has an embedded option, the cash flows can change when interest rates change because of prepayments and the exercise of calls and puts. Effective duration takes into consideration the changes in cash flows and values that can occur from these embedded options.

Why is duration the best interpretation of a measure of the sensitivity of a bond or portfolio to changing interest rates?
As expressed throughout this guide, duration gives an approximate percentage change for a 100 basis point change in rates. Once you understand duration, it is a quick way to calculate the change in a bond's value. It also allows an investor to get a "feel" for the price change. For example, you can tell a client that the duration of measure of 7 for their portfolio would equal roughly a 7% change in their portfolio's value if rates change, plus or minus 100 basis points. It also allows a manager or investor a way to compare bonds regarding the interest rate risk under certain assumptions.

A portfolio's duration is equal to the weighted average of the durations of the bonds in the portfolio. The weight is proportional to how much of the portfolio consists of a certain bond.

Formula 14.15

Portfolio Duration = w1D1 + w2D2 ...+ wkDk

Let's take 3 bonds:

$6,000,000 market value of Stone & Co 7% of 10 with duration of 5.5
$3,400,000 market value of Zack Stores 5% or 15 with duration of 7.8
$1,535,000 market value of Yankee Corp. 9% or 20 with duration of 12

Total market valve of $10,935,000

First let's find the weighted average of each bond
Stone & Co. weighted average is 6,000,000 / 10,935,000 = .548
Zack Stores weighted average is 3,400,000 / 10,935,000 = .311
Yankee Corp. weighted average is 1,535,000 / 10935,000 = .14

So the portfolio duration = .548(5.5) + .311(7.8) + .14 (12)
= 7.119

This means that if rates change by 100 bps the portfolio's value will change by approximately by 7.119%. Keep in mind that the individual bonds will not change by this much because each will have their own duration.

You can also use this to figure out the dollar amount of the change. This is done by using the dollar duration equation and adding up the change for all of the bonds in the portfolio.

Going back to our example of those three bonds and a 50 bps yield change.

Percentage price change = -duration x change in yield x market value
Stone & Co = -5.5 x .005 x 6,000,000 = 165,000
Zack Stores = -7.8 x .005 x 3,400,000 = 132,600
Yankee Corp = -12 x .005 x 1,535,000 = 92,100

So the dollar change for a 50 bp change would be equal to approximately $389,700

Limitations of the Portfolio Duration Measure
The primary limitation of this measure is that each of the bonds in the portfolio must change by the 100 or 50bps, or there must be a parallel shift in the yield curve for the duration measure to be useful.

Related Articles
  1. Financial Advisors

    Tips on Passing the CFA Level I on Your First Attempt

    Obtain valuable tips and helpful study instructions that can help you pass the Level 1 Chartered Financial Analyst exam on your first attempt.
  2. Financial Advisors

    Putting Your CFA Level I on Your Resume

    Learn techniques for emphasizing your CFA Level I status in the Skills and Certifications or Professional Development section of your resume.
  3. Professionals

    Investment Analyst: Career Path and Qualifications

    Learn how to prepare for a career as an investment analyst, and read more about how many professionals in the field progress during their careers.
  4. Professionals

    CAIA Vs. CFA: How Are They Different?

    Find out how the CAIA and CFA designations differ, including which professionals should seek either title based on their career ambitions.
  5. Professionals

    Equity Investments: CFA Level II Tutorial

    Chapter 1: Equity Valuation: Its Applications and Processes Chapter 2: Return Concepts for Equity Valuation Chapter 3: Industry Analysis With Porter's 5 Forces
  6. Professionals

    What To Expect On The CFA Level III Exam

    The Chartered Financial Analyst Level III exam, which is only offered in June, is the last in the series of three tests that CFA candidates must pass.
  7. Professionals

    What To Expect On The CFA Level I Exam

    Becoming a chartered financial analyst requires the passing of three grueling exams covering an array of topics.
  8. Options & Futures

    The Alphabet Soup of Financial Certifications

    We decode the meaning of the many letters that can follow the names of financial professionals.
  9. Professionals

    How to Ace the CFA Level I Exam

    Prepare to ace the CFA Level 1 exam by studying systematically.
  10. Personal Finance

    How To Choose A Financial Advisor

    Many advisors display similar skillsets that can make distinguishing between them difficult. The following guidelines can help you better understand their qualifications and services.
  1. Personal Financial Advisor

    Professionals who help individuals manage their finances by providing ...
  2. CFA Institute

    Formerly known as the Association for Investment Management and ...
  3. Chartered Financial Analyst - CFA

    A professional designation given by the CFA Institute (formerly ...
  4. Security Analyst

    A financial professional who studies various industries and companies, ...
  1. What are the differences between a Chartered Financial Analyst (CFA) and a Certified ...

    The differences between a Chartered Financial Analyst (CFA) and a Certified Financial Planner (CFP) are many, but comes down ... Read Full Answer >>
  2. How do I become a Chartered Financial Analyst (CFA)?

    According to the CFA Institute, a person who holds a CFA charter is not a chartered financial analyst. The CFA Institute ... Read Full Answer >>
  3. What types of positions might a Chartered Financial Analyst (CFA) hold?

    The types of positions that a Chartered Financial Analyst (CFA) is likely to hold include any position that deals with large ... Read Full Answer >>
  4. Who benefits the most from prepaid expenses?

    Prepaid expenses benefit both businesses and individuals. Prepaid expenses are the types of expenses that are bought or paid ... Read Full Answer >>
  5. If I am looking to get an Investment Banking job. What education do employers prefer? ...

    If you are looking specifically for an investment banking position, an MBA may be marginally preferable over the CFA. The ... Read Full Answer >>
  6. Can I still pass the CFA Level I if I do poorly in the ethics section?

    You may still pass the Chartered Financial Analysis (CFA) Level I even if you fare poorly in the ethics section, but don't ... Read Full Answer >>
Hot Definitions
  1. Bar Chart

    A style of chart used by some technical analysts, on which, as illustrated below, the top of the vertical line indicates ...
  2. Bullish Engulfing Pattern

    A chart pattern that forms when a small black candlestick is followed by a large white candlestick that completely eclipses ...
  3. Cyber Monday

    An expression used in online retailing to describe the Monday following U.S. Thanksgiving weekend. Cyber Monday is generally ...
  4. Take A Bath

    A slang term referring to the situation of an investor who has experienced a large loss from an investment or speculative ...
  5. Black Friday

    1. A day of stock market catastrophe. Originally, September 24, 1869, was deemed Black Friday. The crash was sparked by gold ...
Trading Center