Bond investing is more complex than it appears. Although bonds have fixed cash flows that are straightforward, the value of these investments fluctuates. When the interest rate goes up, bond prices fall. However, how much the price of a bond changes depends on a whole range of factors. To cut through the complexity and measure this interest rate sensitivity, duration was invented.
What is bond duration?
Bond duration is a tool used by investors to measure how bond prices change with the interest rate. Bond price and the interest rate are inversely correlated. When the interest rate goes up, bond prices fall and vice versa.
The reasoning behind this relationship is simple. When the interest rate goes up, new bonds are issued with larger coupons. For example, an issuer, who issued a 5% coupon bond in the past, now issues a similar bond with a 6% coupon, due to the higher interest rate. This means that the old bonds, with the 5% coupons, are now less attractive to buyers. To compensate for the smaller coupons and attract investors, bondholders have to reduce the price of bonds with the 5% coupon. Consequently, an increase in the interest rate leads bond prices to fall.
By the same principle, when the interest rate goes down, old bonds become more attractive and, as a result, more valuable.
The next question investors need to answer is how much the bond price would move when the interest rate changes. The answer to this question directly affects our profit and determines the risks we are taking.
Duration gives investors the tools to precisely measure potential changes in bond price from interest rate fluctuation. With duration, investors can measure the interest rate risk they are taking and compare the interest exposure for different bonds. It even allows us to calculate exact changes in bond price given different interest rate levels.
Macaulay and modified duration
There are two duration concepts that look at interest rate sensitivity from different angles: the Macaulay duration and modified duration. Some investors use the terms interchangeably, but it is always good to be clear in your thinking and terminology. Both duration concepts are useful and shed different lights on interest rate exposures.
The math behind duration can be daunting but don’t stress. Most websites and brokers provide duration data. The key is to understand the intuitions behind these concepts and apply them to your investing. The calculation is only secondary.
The Macaulay duration
When the interest rate changes, bonds with longer maturities experience more price movements than short-term bonds. Short-term bonds mature quickly. When they mature, investors can reinvest the money in new bonds at the current interest rate. Thus, they offer more flexibility to adjust to changing interest rates.
On the other hand, long-term bonds lock in investor capital for a long period of time. So when the interest rate changes, long-term bondholders will feel the impact for a long time. They can only reinvest at the current interest rate after the bond matures. As a result, the price of bonds with long maturities is more heavily affected by the interest rate than short-term bonds.
This intuition shows that the sensitivity of bond price to interest rate changes must be related to the “length” of the bond. But what exactly is the length of a bond and how do we measure it? It turns out the best way to estimate a bond’s “length” is by calculating the weighted average time to receive all future cash flows from a bond. This weighted average is the Macaulay duration.
You can calculate the Macaulay duration using the formula below, where is the present value of a coupon or principal payment and is the time (in years) you must wait to receive that payment. This calculation can be tedious. Fortunately, the Macaulay duration is so popular that you can easily find this data online or from your broker without the need to sweat over its calculation.
The\ Macaulay\ duration=\sum_{n=1}^{m}\frac{PV_{n}*t}{Bond\ price}While the formula seems intimidating, the intuition is straightforward. The formula essentially multiplies the present value of each coupon and principal payment by the time the investor needs to wait to receive it. Then it sums up these products and divides the sum by the total present value of the bond, which is also the bond price.
| Present Value (PV) | PV of coupon 1 | PV of coupon 2 | PV of coupon 3 | PV of coupon 4 | PV of coupon 5 | PV of principal |
| Time to payment | 0.25 yrs | 0.5 yrs | 0.75 yrs | 1 yr | 1.25 yrs | 1.5 yrs |
Applying the formula to our example:
\frac{PV(coupon1)*0.25yrs+\ …\ \ PV(principal)*1.5yrs}{Total\ PV/bond\ price}The Macaulay duration calculates the weighted average time to receive all future payments of a bond. The weight is the present value. Thus, the formula gives more weight to larger PVs and discounts smaller PVs. Because this variable measures the weighted “length” of a bond, the Macaulay duration is measured in years.
The Macaulay duration is a precise measure that shows you effective interest rate exposure of a bond. This calculation applies to any bond and allows the investors to compare their interest rate sensitivity. Bonds with longer Macaulay duration are more sensitive to interest rate changes.
Modified duration
While the Macaulay duration quantifies a bond’s sensitivity to the interest rate, it doesn’t provide an exact estimate of how much the bond price will change given a different interest rate. To investors, this estimate is important as it directly affects their profits.
Modified duration allows us to calculate the exact percentage change in bond price when the interest rate moves by 1%. For example, if a bond has a 2.5 modified duration, the bond price will increase by 2.5% when the interest rate drops by 1%. Bonds with higher modified duration will experience larger price changes when the interest rate moves.
Modified duration is a handy tool to forecast investment losses and gains given different interest rate changes. It converts the abstract into quantifiable dollars. With this information, investors can understand their potential gains and losses given different interest rate scenarios.
You can calculate the modified duration of a bond from its Macaulay duration using the formula below. Calculating modified duration can be tedious. Fortunately, most brokers and investment websites provide modified duration. Investors rarely need to calculate it themselves.
Modified\ duration=\frac{Macaulay\ duration}{1+\frac{YTM}{nu}}n= number of coupon payments each year
YTM = yield to maturity
Properties of bond duration
Both the Macaulay duration and modified duration share a few interesting properties that could come in handy when you need to compare bonds quickly without calculation.
Bonds with longer maturities have longer durations
Duration is directly affected by the “length” of the bond. Given the same coupon rate, bonds with shorter maturities also have shorter durations, as the investors don’t need to wait long to receive all the payments. On the flip side, bonds with longer maturities have longer durations, because the payments are more spread out into the future.
Bonds with larger coupons have shorter durations
A bond’s duration is often close to its maturity. For example, a bond that matures in three years might have a Macaulay duration of 2.79 years. This is because duration is a weighted average of time and the weights are the PV of each payment. The principal payment received at maturity is usually the largest payment of a bond and it gives the maturity date a large weight in duration calculation. Thus, duration often is close to maturity.
When bonds have larger coupons, these large coupon payments increase the weights of other payment periods, shifting the weighted average away from the maturity. Thus, large coupons can reduce bond duration.
Zero-coupon bond has the same maturity and duration
Following the same principle, we can understand why bonds with no coupon payments have the same duration and maturity. A zero-coupon bond has only one cash flow, a principal repayment at maturity. Because the weighted average of one number is just itself, a zero-coupon bond’s duration equals to its maturity.
How to apply duration in your investing
The purpose of learning the details of duration is to add a new tool to your investment toolbox. Both Macaulay and modified durations can help investors cut through the market noises and gain more clarity on their cash flows and the overall interest rate risk with accuracy.
Managing interest rate risks
Duration provides a clear picture of how the interest rate affects your portfolio. This allows bond investors to update their portfolios based on their outlooks on the interest rate.
For example, when the interest rate has been kept low for a long time, the central bank might raise it in the near future, just like what happened in 2016. With this view, bond investors should reduce the duration of their bond portfolio to avoid losses when the interest rate goes up. To reduce duration, investors can invest in short-term bonds instead of long-term bonds that could lock in your investment for years.
On the flip side, if an investor believes that the interest rate will decrease in the future, they can also increase their portfolio duration to benefit from the price increase.
Advanced investing strategies
The application of duration goes beyond basic risk management for bond portfolios. It is an important variable used in building more complex investing strategies. For example, fixed-income derivatives such as fixed-floating swaps also have durations. Investors can increase and reduce their portfolio duration by purchasing and selling swap contracts.
Duration is a crucial variable in building these advanced strategies as the key is to produce a portfolio that achieves the optimal level of interest rate exposure, often a specific portfolio duration.
A quantitative method to bond investing
While bond investing seems simple, becoming a good bond investor requires research and precise analyses. Mastering duration is the first step to become a savvier and more quantitative bond investor. Duration takes investors out of the dark and gives them a clear view of their risk exposure. This knowledge is critical to risk management and advanced investing strategies.
