HOME
Learn FM
5
LEARN
5.2
ADAPT
DISCUSSIONS
MENU
5.2.5 Portfolio Duration
Portfolio Duration
Recall a portfolio is a collection of assets, and an asset is a collection of cash flows. Thus, we can calculate a
portfolio's duration using basic principles by simply identifying all the cash flows in the portfolio and using the
duration formula.
However, there may be an easier method to calculate the portfolio duration if we know the duration of each
asset. The portfolio’s duration is the weighted average of the individual asset’s durations. The weights are
based on the asset’s price. The following example illustrates this idea.
EXAMPLE 5.2.10
You are given the following information for bonds X, Y and Z:
Bond
Price
Modified Duration
X
w
3
Y
2w
5
Z
15w
20
Larry creates a portfolio with three units of bond X, two units of bond Y, and one unit of bond Z.
Calculate the portfolio's modified duration.
SOLUTION
The portfolio's duration is the weighted average of the individual bond’s duration.
First, identify the portfolio's price:
3 ⋅ PX + 2 ⋅ PY + 1 ⋅ PZ
Then, calculate each bond’s weight:
wX =
wY
3 ⋅ PX
3 ⋅ PX + 2 ⋅ PY + 1 ⋅ PZ
2 ⋅ PY
=
wZ =
3 ⋅ w
=
3
=
3 ⋅ w + 2 ⋅ 2w + 1 ⋅ 15w
22
2 ⋅ 2w
=
3 ⋅ PX + 2 ⋅ PY + 1 ⋅ PZ
1 ⋅ PZ
4
=
3 ⋅ w + 2 ⋅ 2w + 1 ⋅ 15w
22
1 ⋅ 15w
=
3 ⋅ PX + 2 ⋅ PY + 1 ⋅ PZ
15
=
3 ⋅ w + 2 ⋅ 2w + 1 ⋅ 15w
22
Lastly, use these weights to calculate the portfolio’s modified duration:
M odD P ort = wX ⋅ M odD X + wY ⋅ M odD Y + wZ ⋅ M odD Z
3
=
4
⋅ 3 +
22
15
⋅ 5 +
22
⋅ 20
22
= 14.95
Let’s make sure this result is reasonable. Since the portfolio's duration is a weighted average of each asset's
duration, we expect the portfolio duration to be between 3 and 20. Note that there is a higher weight on the
bonds having a modified duration of 20. Thus, we expect the portfolio duration to be closer to 20 than 3. 14.95
appears to be a reasonable result.
■
EXAMPLE 5.2.11
Baron has a portfolio consisting of the following two bonds:
A five-year bond with an annual coupon rate of 4%.
A ten-year bond with an annual coupon rate of 6%.
Both bonds are redeemed at a par value of 1,000.
What is the portfolio's Macaulay duration, assuming an annual effective interest rate of 6%?
SOLUTION
We can solve for the portfolio duration by calculating the duration of each asset and finding the weighted
average. This approach is similar to the previous example.
Method 1:
Let
M acD A
bond.
be the Macaulay duration for the first bond, and
M acD B
be the Macaulay duration for the second
2
3
4
5
5
1 (40v) + 2 (40v ) + 3 (40v ) + 4 (40v ) + 5 (40v ) + 5 (1,000v )
M acD A =
40v + 40v 2 + 40v 3 + 40v 4 + 40v 5 + 1,000v 5
40(I a)¯¯¯¯¯
∣
5
=
40a¯¯¯¯¯
∣
5
∣
∣
6%
6%
+ 5,000v
5
+ 1,000v 5
4,222.1674
=
915.7527
= 4.6106
M acD B = 䯯¯¯¯¯¯¯∣
10
∣
6%
= 7.8017
Thus, the portfolio duration is:
M acD =
PA ⋅ M acD A + PB ⋅ M acD B
PA + PB
915.7527 (4.6106) + 1,000.00 (7.8017)
=
915.7527 + 1,000.00
= 6.2763
Method 2:
Alternatively, since the cash flows for each bond are given, we can combine the cash flows and calculate the
Macaulay duration.
2
5
6
100v + 2 (100v ) +. . . +5 (1,100v ) + 6 (60v ) +. . . +10 (1,060v
M acD =
10
)
100v + 100v 2 +. . . +1,100v 5 + 60v 6 +. . . +1,060v 10
12,023.8596
=
1,915.7527
= 6.2763
We can use this approach to calculate duration if we know each individual cash flow and use the same interest
rate for each asset.
■
Discussions
Search...
EXPAND
3
2
1
1
1
1
Duration formulas
ANSWERED
Ray Rosario-Cruz Sep 24 2019
calculating price weights
ANSWERED
Arya Bhogte Jun 16 2021
The portfolio's modified duration AND the portfolio's Macaulay duration
ANSWERED
Chloe May 28 2022
Example 5.2.1 Method 1
ANSWERED
Mark Menezes Nov 21 2022
Example 5.2.11 solution
Ettu Faruq Mar 16 at 12:52 AM
Formula
Ej Bei Jan 13 at 8:27 PM
Don't see what you're looking for?
ANSWERED
Our coaches are here to answer your questions, help clarify, or further discuss the above content. If you don't see a thread
that helps you, start one of your own!
START A DISCUSSION
Previous Lesson
Assignment 5.2.4 Passage of Time
Next Lesson
Watch 5.2.5 Portfolio Duration