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)

Example:
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:

Answer:
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



Example:
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

Where:
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



Example:
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

Answer:
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.

Convexity
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