Most investors recognize that there is a relationship between interest rates and bond prices. At their most simple level, bonds are simply loans that the issuing organization takes from investors.

And because of this, changes in interest rates have a significant impact on bond prices. However, the way that interest rates affect bond prices is incredibly interesting.

### Discounting Cash Flows

A bond’s price is the discounted value of it’s future cash flows. The prevailing interest rates provide the basis for how those cash flows are discounted back to their present value. However, the actual impact of changes in interest rates depends on how the bond’s cash flows are structured.

Bonds with longer maturity dates, and where more of the future cash flows will come in the form of the principal repayment will be more sensitive to interest rate changes. And bonds with shorter maturity dates, and higher coupon payments, will be less sensitive to interest rate changes. This is because bonds with more of their cash flows are impacted more by this discounting process than bonds with cash flows that are closer to the present.

### Measuring the Impact of Interest Rates

But we can see this in action.

Let’s consider a simple 10 year bond with a coupon of 3%. When the prevailing interest rate is 3%, the price of the bond is equal to the face value (we call this trading at par). However, if we were to increase the interest rate from 3 percent to 4 percent, this would cause the bond’s price to fall by 8.1 percent.

### Measuring Bond Duration

As you might expect, bond investors want a simply way of quantifying how sensitive a given bond is to changes in interest rate changes. The way that we do this is a measure called Bond Duration.

In this example, the bond we are looking at has a duration of 8.1, meaning that a one percentage point rise in interest rates leads to an 8.1 percent drop in price.

One of the (many) interesting – and initially confusing – things about bond duration is that it is measured in years. This is because you can actually find a bonds duration simply by calculating it’s dollar weighted average for when the bond’s cash flows are received.

Not only does our example bond’s price drop by 8.1% when interest rates increase by one percentage point, you will receive half of the bond’s future cash flows 8.1 years in the future.

### Digging Deeper Into Bond Prices

We’ve been looking at a pretty straight forward example. What if we expand our analysis?

If we keep the basic structure of the bond consistent (a 3% coupon rate, and currently trading at par) we can vary both the length of the bond and the prevailing interest rate to get a sense of how the bond’s price will change. In this case we will look at interest rates between one and 5 percent, and maturity dates from one to thirty years out.

In these examples, we are looking at the new price that bonds with different maturities could sell for after the rate change, followed by the percentage change in price resulting from the rate change. We can clearly see how bond prices move counter to interest rates, and how price fluctuations are more dramatic for longer-term bonds, demonstrating their higher duration. At the extreme, the thirty-year bond would experience a capital *gain* of 22.4 percent if interest rates *fell* by 1 percent, and a 17.3 percent capital *loss* if interest rates *rose* by 1 percent. Price risk increases with time to maturity. If interest rates rose to 5 percent from their base of 3 percent, the capital loss for a thirty-year bond would be 30.7 percent—comparable to a significant stock market drop. Despite their reputation as reliable and predictable, bonds can be risky.

While more complex bonds can have some unusual duration properties, the basic noncallable bonds that are typically considered for a retirement income plan define duration in a straightforward way. A bond’s duration is essentially the effective maturity of a bond—an average of when the bond’s payments are received, weighted by the discounted size of those cash flows.

At the extreme, a zero-coupon bond provides one payment at the maturity date, so its duration is the same as the time to maturity. The further away the maturity date, the higher the bond’s duration, making it more sensitive to interest rate changes. A bond that pays a coupon will have a shorter duration than the time to the maturity date because coupon payments are received before the maturity date. Higher coupon rates push relatively more cash flows sooner, which otherwise lowers the duration for a bond with the same maturity date. Also, lower interest rates mean the future cash flows from a bond are discounted less relative to nearer-term cash flows, and so bond duration increases when interest rates are low.

### Duration is Not Symmetric

One important point to note is that duration is not symmetric. For a thirty-year bond, a one percentage point increase in interest rates from our base of three percent to four percent results in a capital loss of 17.3 percent, while a one percentage point decrease in interest rates to two percent results in a capital gain of 22.4 percent. The duration measure works best for small interest rate changes because it is a linear approximation to a shape that is curved.

The term *convexity* describes price sensitivity to interest changes more precisely. Bond prices are more sensitive to rate decreases (prices rise more) than to equivalent rate increases (prices fall by less). These differences are accounted for by the fact that changing interest rates also impact duration. The duration for a given bond rises as interest rates fall and future cash flows are discounted by less. But for most individual investors, duration provides a close enough approximation to this relationship, and only those with a greater interest in the mathematics of bond pricing should worry about further adding bond convexity to their discussion.

### Duration in Practice

Though somewhat technical, this discussion of bond duration is important because the concept also applies to retirement spending liabilities and, therefore, the ability to meet retirement goals. Retirement spending has a duration that can be defined in the same way as an effective maturity for those cash flows. It is an average of when expenses must be paid, weighted by the size of the discounted values of those expenses.

Individual bonds have a duration. A bond fund, which is a collection of bonds, also has a duration equal to the average duration of each holding weighted by its proportion in the fund. Retirement liabilities have a duration, too. If the duration of the bonds and the spending liabilities can be matched to the same value, then the retiree has immunized his or her interest rate risk. Rising interest rates would lower the value of bond holdings, but rising rates also lower the present value of the future spending obligations. When durations are the same, both the asset and liability values are reduced by the same amount, and the retiree remains equally well-off in terms of the ability to meet liabilities. This is the meaning of immunizing interest rate risk. If the durations do not match, then the retiree is exposed to interest rate risk.

Bonds, at heart, are relatively simple. They are just loans, with known payments at known times. This property makes them very useful as retirement planning tools in a number of ways.

However, the math around their pricing, and especially how they move as interest rates change, is not particularly simple. While you may not need to go out and calculate a bond fund’s duration (thankfully it is one of the portfolio characteristics that is almost always reported by a portfolio manager), understanding what that duration means is helpful for targeting the appropriate level of risk in your bond holdings – and broader retirement income plan.

*To find out more about investing in retirement, read our eBook 8 Tips to Becoming a Retirement Income Investor.*