**Use This Simple Measure to Find Your Yield Exposure**

## How to calculate and interpret a bond’s duration

One of the most common and important questions investors ask when investing is: what am I getting exposure to? This question applies across all asset classes and may not be as straightforward as it first seems. There are a range of factors that affect the value of a security and, in combination, the value of your portfolio. These can be macroeconomic factors, which capture broad risks associated with the business cycle and its impact on financial markets, and style factors, which explain risk and return within an individual asset class.

As investors, we need to expose ourselves to risk in order to generate a return. Unfortunately, there is no escaping this fact. Factors that help us generate a return are therefore factors that expose us to risk, hence why they tend to be called risk factors. We want to know how much risk we are taking on to generate a given return. We also want to make sure that we are properly compensated by the market for taking on risk, and that the overall sensitivity of our portfolio to our target risk factors is manageable and in line with our risk tolerance and broader objectives.

This post will briefly cover the most important risk factor that bond investors must manage, namely yield exposure. Yield is the biggest determinant of a bond’s value, and movements in yield affect bonds and other fixed income instruments in different ways. We measure a bond’s exposure to yield by calculating its duration. There are different ways of calculating duration and different interpretations of what it means, which can make things confusing. Sometimes duration is quoted as a sensitivity, sometimes as the cash flow weighted time to maturity. Here, we will focus on one of the most fundamental measures of duration, known as modified duration, which is a measure of the price sensitivity of a bond to interest rate movements.

The point of departure for understanding duration is to understand how the cash flow of a bond is discounted to determine its value at a point in time. For a bond that pays a cash flow at a regular fixed interval, the value of the bond is determined by the size of the cash flows, the yield, and the time left until maturity. The value at time *t* of a generic fixed-income bond is described as:

Where the value (*V*) is given by the point in time at which the value is calculated and the prevailing market yield associated with the instrument. Here we are discounting the future cash flows that accrue to the holder of the bond to determine the present value. A higher yield (*y*) will result in a lower *V* because we are discounting the cash flows at a higher rate. Here the value of the bond is the sum of its discounted cash flows.

Clearly the yield has an important role to play in calculating a fixed-income security’s value. There is an inverse relationship between value in yield: when we increase *y*, we reduce the present value of the cash flows, which leads to a reduction in *V*. If you are not averse to a bit of calculus, this relationship can be expressed analytically as:

This is the first derivative of the bond’s value with respect to a change in yield. This is a sensitivity because it is describing how much the value changes based on a change in yield. What is remarkable about this expression is that, when we look inside the summation operator, we are summing the discounted values of cash flows that are somehow time weighted. We multiply each cash flow by the number of periods left until maturity. Even though we are calculating a sensitivity, we get a sense that time weightings play an important role in the outcome.

We are nearly at our goal of finding the modified duration. We need to express our sensitivity as a percentage gain or loss for our bond associated with a change in yield. We do this by simply dividing by the value of the bond:

Formally, we define modified duration as:

This is a popular metric for investors managing a bond portfolio. It conveys, in a single figure, the percentage gain or loss in the value of a single security — or a number of securities — for a given change in yield. However, this should only be used when calculating the effect of a judiciously small change in yield. Due to the convex relationship between a bond’s value and its price, modified duration can only provide us with a linear approximation. This means that for large changes in the yield, we will end up with results that deviate from the true relationship (see chart below).

Modified duration is clearly insufficient to capture the full exposure of a bond to the yield factor. We can correct for this by taking the second derivative of the value function to arrive at the convexity measure, but this is enough for the time being. Modified duration is not perfect, but it is a popular and relatively simple measure of your yield factor exposure.

I will delve into other factor exposures in subsequent posts which should hopefully provide you with an effective toolkit for understanding what you are really investing in. When we start to think of assets as providing exposure to different risks, we can start to formulate better ways of measuring our portfolio’s performance and ensure that what we are investing in provides us with adequate compensation in line with our objectives.