A solid understanding of duration is essential to any fixed income education

Duration has a lot of different names but at the end of the day, it’s interest rate risk or sensitivity. When you think about duration, you should think “Effective Duration”. The “Effective” part just takes into consideration any optionality of a fixed income security. For example, if a bond can be called, you need to take this into consideration when thinking about duration. When I say duration going forward, I will be talking about Effective Duration.

## Interest Rate Movements Change Bond Prices

You should think of duration as interest rate risk because of the way you value a fixed income security. The value of a fixed income security is nothing more than a discounted cash flow (“DCF”). Figure 1 is the basic formula for pricing out a fixed income security. This is also just a simple DCF.

I know we all love doing DCFs and we love discounting Free Cash Flows to find the value of a company. A bond, or a fixed income security, is valued the exact same way. If you’ve ever done a company valuation and you created a sensitivity table for the value of the company and adjusted the value of the company by changing the interest rate, well, a fixed income security is the exact same.

Here’s an example: Let’s say you buy a bond with two years to maturity and a coupon payment of 5% yielding 4% (discount rate). What’s the price of the bond? In this 4% yield base case scenario, the price of this bond would be $101.89.

Now, let’s say you change the yield, or discount rate, up 100bps to 5%. What would the new price be in this up 100bps scenario? The price of the bond would now be $100.00 or par in bond jargon. If you changed the yield down 100bps from the base case to 3% the price would be $103.83. You can see the price differences based on adjusting the yield, or discount rate, in Figure 2.

Figure 2 also illustrates the inverse relationship between yield and bond prices. As yields go down, prices go up (-100bps and price jumped from $101.89 to $103.83), and as yields go up prices go down (+100bps and price decreased from $101.89 to $100.00).

The above example of the change in prices for a 100bp movement shows you how sensitive fixed income prices are to interest rate movements. This should clarify that the valuation of a fixed income security is directly related to the interest rates that are used to discount the security. So why is this important for duration. Well, because duration is essentially interest rate sensitivity!

## Duration Approximation

For a quick calculation of duration, you can use the Duration Approximation Formula in Figure 3. Just make sure you use the change in yield in decimal form.

Let’s look at our above example. The duration of the above bond in the example is 1.88. If you take the base case price of 101.89, then we should see a price movement of ~1.88% if yields go up or down 100bps (or 1%). If you want to see how we got this, just plug in the duration in the formula and 100bps movement in the denominator.

Let’s test this; 1.88% * 101.89 = $1.92. We should see a price of ~$103.8 ($101.89 + $1.92) if yields go down and price of $99.9 ($101.89 - $1.92) if yields go up. As you can see in Figure 1, this is a very close approximation. Our duration approximation gave us $103.8 vs $103.83 if yields go down and $99.9 vs $100.00 if yields go down.

The Duration approximation formula will get you by for a quick understanding of the change in price for a given change in yield. The smaller the yield change, the more accurate the approximation will be. If you want to understand more on why this is an approximation and why it works better for small changes in yields, see the section on __convexity__.

Obviously, as the duration gets larger, the % change will get larger. A very basic interview question is “if you have a duration of 5 years, and 100bps change in yield, and you start with a price of par, what is the price of the bond in an up & down scenario”. Well, if you know the duration approximation formula, you can just say “well, if duration is 5, 100bps is equal to 1% so you should expect a 5% change in price. Starting at par ($100) you should expect a price of $95 if yields go up and $105 if yields go down”. Boom! You got the job.

## Duration & Time

Let’s take this one step further and really solidify the understanding of duration as interest rate risk. Ok, so we made the point that duration should be thought of as interest rate risk or sensitivity, but why do people think of duration in years. The answer is because we are discount cash flows!!! As you discount further out the curve, you will be dividing by a much larger number, so any movement in the discount rate will have a larger impact on your present value. Also, in fixed income you typically back load the cash flow. You might receive a coupon payment intermittently, but you receive the par principal payment later in the future. This means the bulk of the cash load is later in the future and more sensitive to interest rate movements. Let’s look at an example:

Suppose you had a zero-coupon bond that paid you par in 5 years and is yielding 2.92%. What’s the price of the bond – let’s call it Bond A?

To make this extremely simple, let’s take this out to 30 years and not even change the discount rate (which should increase in an upward sloping yield curve environment). Let’s just change the time to 30 years and call it Bond B.

You can see the price drastically falls! So the million dollar question! What is the duration of Bond A and Bond B? Well, the Bond A has a duration of 4.86, and Bond B has a duration of 17. You can see changing nothing but the time to maturity drastically changes how sensitive each bond value is to interest rates. We didn’t even adjust the discount rate, time alone changed the value. Time and math I suppose. This is why people think of duration as time.

To sum up the above, when someone says duration, think of the duration approximation formula. This will allow you to think of duration as interest rate risk/sensitivity/exposure. The larger the duration, the larger your exposure to interest rate risk movement. Duration also has a time component because we are discount cash flow to get to a present value, and the further out you discount the more the interest rate exposure will impact your cash flow. The longer out the cash flow, the bigger the duration, the more exposed you are to interest rate risk.

## Coupon Payments & Duration

If you receive regular coupon payments your duration will be lower. The larger the coupon payment, the smaller the duration. Think about it logically. If you get more cash sooner, the less sensitive you are to interest rates. Now think about it mathematically. Let’s go back to bond A but put in a 5% annual coupon. The cash flows will change as seen in figure 7.

The duration of this bond is now 4.44 compared to the original duration of 4.86. We can see adding the 5% coupon lowered the duration. Note the price also increased since the coupon payment isn't free.

## Calculating vs Interpreting

All of the above assumes you are given a duration and you need to know how to interpret it. Interpreting duration correctly is probably more important than understanding how to calculate it. But understanding how to calculate it will help you interpret it. If you need to calculate the duration of a security, you can utilize the below formula.

For example, the bond that we started with has a duration of 4.88 which was calculated using the Effective Duration formula.

I hope all of this helps in your understanding of duration. I would recommend next checking out the section on convexity since the two go hand in hand.

## Duration Side Notes

There are a few bullets I’ll add here for duration that should make sense after reading this article.

• If a bond pays a coupon payment, the duration is always shorter than its maturity

(see adding a coupon to Bond A example)

• This article examines Effective duration which is more useful than Macaulay Duration (MACD) when analyzing interest rate risk. MACD is just a time figure.

• The longer a bonds maturity the longer the duration

• A Lower coupon bond has longer duration all else equal (you don’t get as much

cash as fast)

If you would like to see anything else on duration or if anything is unclear – please feel free to reach out. Thanks for reading!

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