Why is Macaulay Duration of Bonds what it is ? (Part A — Building the blocks)
Hi there,
Complexity is the experience in the journey towards simplicity. I am a fan of intuitive explanations and committed to deconstructing the topics I study.
The formula of Macaulay Duration of a bond is usually taken by the face value coupled with appreciation that it differs from Maturity. However, what it is trying to tell intuitively and the concept that is sitting under it are worth visiting. In this article, I attempt to build the blocks to answer the question — “why we are doing what we are doing in Macaulay Duration?”
Macaulay Duration, roughly , asks a simple question — What is the time by which I recover my Principal ? If we are able to answer this question, we are done.
Named after Frederick Macaulay, Macaulay Duration is used to understand the sensitivity of the bond. The faster I recover my amount of bond, the lesser will be my exposure to future uncertainties. This is possible either by higher coupons or lower maturities. Let us see the intuition behind this concept.
Intuition
Consider the simplest example where you are giving a loan of $150 to “B” (loanee) at 10% p.a. The condition is that he/she has to pay $ 50 at the end of each year. So, the whole business is for 3 years. The schedule looks like this (Method I)-
This is the simplest and well-known way. As the goal of B is to see how much he/she has to pay towards interest, let us see the other way of arriving at the same (Method II).
In this method, we are looking at the periods for which each part of the principal is outstanding. For example, B is paying $50 at the end of the 1st year. That part is outstanding for 1 year, and the $50 paid at the end of the 2nd year is outstanding for 2 years, and so on. This is based on parts of Principal. In the end, we calculate the interest on that weighted principal, which is the same as in Method I.
As interest calculated under both the methods is same —
The question
Now, let us revisit our initial question -
“When am I going to recover my principal ?”
We all know that the formula for Simple interest is -
The above formula calculates the interest for the entire time period during which the principal amount is outstanding. So, somehow, this “T” element in the simple interest formula seems to be a potential candidate to answer my initial question about Macaulay duration. It indicates the time period for which that principal amount is outstanding ($ 150). That means, in one way, it is telling me as the lender the time by which I recover the amount lent to the loanee. Let us see.
Can I proceed with t = 3 as the loan term is 3 years ?
NO, because by doing so, I am essentially indicating that the entire lent amount of $150 is outstanding for the entire 3 years. Clearly, that is not the case, as portions are repaid each year. So, what is the correct period for which the entire $150 loan remains outstanding ?
The solution
One way to arrive at the answer is as follows. Total interest paid is $ 30. The amount I had lent is $ 150. Rate of Interest is 10% p.a. Applying the above Simple Interest formula, we get -
$ 30 = $ 150 (Principal or Amt. lent) × T × 10%(Rate of Interest) — —— — A
From Method — II, we can arrive at this interest using
[($ 50 × 1) + ($ 50 × 2) + ($ 50 × 3)] × 10% = $ 30 — — — — — — B
So, A = B
$ 150 × T × 10% = [($ 50 × 1) + ($ 50 × 2) + ($ 50 × 3)] × 10%
$ 150 × T = ($ 50 × 1) + ($ 50 × 2) + ($ 50 × 3)
T = [($ 50 × 1) + ($ 50 × 2) + ($ 50 × 3)] / $ 150
Solving for 𝑇 gives us 2 years.
Therefore, the effective period for which the lent amount remains outstanding is only 2 years. This is the Macaulay Duration of this loan case. The loan scheme is for 3 years, but the time by which I recover my loan is 2 years. Later on, in Part B, we shall see that the approach of calculating 𝑇 used above is the basis for the official formula of Macaulay Duration.
The moral is that scheme that involves intermediary cash inflows tends to recover the lent amount faster.
With these foundational blocks, let’s see how Macaulay’s Duration works in a real-world bond case with discounted cash flows in Part B.
Hope you enjoyed the article!
