## Duration: Predicting Bond Prices amidst Rate Hikes

Originally published on May 21, 2019

For the first time in two decades, the United States is in a rising interest rate environment. Janet Yellen, former Federal Reserve Chair, began the arduous process of raising interest rates back to normal levels in 2016. Jerome Powell, her replacement as of February 2018, has indicated he will continue this trend in coming years. Market analysts are now reading Federal Reserve minutes to estimate not “if” there will be a rate hike, but “when” and “how many” there will be in a given year. So, what does this mean for investors? All financial commodities are affected by the Federal Funds Rate, but fixed income securities are particularly sensitive to interest rate changes. However, if you understand how bonds work and the concept of duration, you need not fear the effects of unanticipated interest rate changes on your bond portfolio.

Most people think of a bond as a type of loan, with a notional amount and an interest rate. However, I want you to think of a bond in different terms: cash flows and timing. A bond is nothing more than a series of fixed cash payments at predictable intervals over a given time period. This means that the only variable, the reason there is any uncertainty about the price, is the market interest rate. Let’s illustrate this point with a brief example.

On March 16, our fictional investor Jane buys a 3-year 6% semi-annual bond with face value of $1000 at Par (price equal to face value). Jane knows that every 6 months, she will receive $30 (6% x 1000 x 0.5), and at the end of the 3 years, she will receive $1030, or the face value and one coupon payment. She also knows that the yield-to-maturity, or the rate of return if she holds the bond for three years, is 6%, because the bond is trading at Par. Since Jane knows the timing, the cash flows, and the interest rate, she can determine the correct price of this bond with a financial calculator or Microsoft Excel. Lo and behold, the market has priced it correctly at $1000; Jane buys the bond.

On March 21, Jane turns on CNBC and sees that Jerome Powell has announced an interest rate hike. She knows interest rates are inversely related to bond prices, so she checks the market price of her bond. She’s relieved to see it’s unchanged; the return if she sells it today is still 6%. However, this begs the question of “why is it unchanged”. Well, Jane unknowingly bought her bond just before the announcement, and the market had already “priced in” a 0.25% interest rate hike. Investors essentially could see this news coming from a mile away and planned accordingly. But what happens when investors cannot predict the future with near certainty?

Duration, in short, is the sensitivity of a fixed income security’s price to the interest rate. It is expressed in years, regardless of the security’s periodicity (annual, semi-annual, etc.). With a basic spreadsheet calculator and the information above, we can calculate the duration and the estimated change in bond price, given an unexpected interest rate change. Let’s examine the table below:

So, a 3-year 6% semi-annual bond has 6 six-month periods, and thus 6 cash flows. Each coupon, once again, is $30, and the face value is $1000. These 6 payments are summed in column “CF”. In order to find the present value of those cash flows, or what the future dollars are worth today, we need to use the Time Value of Money formula.

Present Value = Future Value / (1 + YTM / Periodicity)^Period

Since we have semi-annual periods, we must use a semi-annual YTM, in this case 6% / 2. The sum of these present values is $1000, or the price. Now, to find duration, the first step is to multiply the present value by its period, and sum the results, as shown in column “PV*Period”. Finally, the semi-annual duration is this sum divided by the price. However, we want annual duration, so we divide again by periodicity.

Finally, in order to create a simple mathematical formula, academia created the concept of a Modified Duration. This is the Duration divided by “1 + YTM / Periodicity”

This final figure allows a bondholder to estimate the price change due to a change in interest rates. So, referring to our previous example, let’s say the Federal Reserve raised the Federal Funds Rate by 0.5%. This is 0.25% higher than the market had priced in, meaning we will see a near-immediate drop in price of most fixed income securities. The price change on Jane’s bond would be:

Price Change = -Mod. x Price x Interest Rate Change = -2.71 x 1000 x 0.25% = -$6.77

Her bond would have lost nearly $7. Now, this 0.68% drop in price isn’t too dramatic, but there are three characteristics to consider here. First, the interest rate change was very small. If it were 1% instead, her bond would have lost $27, or 2.7% in value. That wipes out nearly half of her yield, assuming she sells before maturity. Second, her bond is short-term; let’s say it’s a 30 year bond rather than 3. The same 0.25% change in interest rate would cause a $35 decrease in price, or 3.5%. Third, her bond is a coupon bond. If she held a zero-coupon bond instead, a 0.25% interest rate change would cause a 0.73% drop in price.

This article focuses primarily on the quantification of duration, but there are additional qualitative aspects and strategies that can help small investors better manage their bonds. These include matching duration to holding period, modifying portfolio duration, and a number of others. Investopedia offers some great articles on these topics, so if you are currently invested in any fixed income securities, I recommend learning more about how you can protect your holdings.

Sources:

Fabozzi, Frank J. Bond Markets, Analysis, and Strategies. Pearson, 2016.

“Janet L. Yellen.” Federal Reserve History.

www.federalreservehistory.org/people/janet_l_yellen.