Duration

Everyone in the financial sector is getting used to the idea of low interest rates among countries like United States and United Kingdom and also the idea that the Federal Reserve (“Fed”) of the United States is planning on raising the interest rate. The question is not whether the rates will raise but rather when it would raise. Therefore, it is important to know what to do when the rates raises.

Most would agree that the bond markets would be impacted. How one would analyses the impact on the bond market? The qualitative answer would be higher interest rate would lower the bond prices which may causes the equity prices to increase. How about the quantitative answer? Is there a formula to measure the impact or at least a rough estimate of the impact?

Fortunately there is – DURATION

What is duration? A bond’s duration measures the sensitivity of the bond to changes in interest rate across all tenors, expressed in years. A high bond duration would cause the bond prices to shift more as the interest rate shifts which implies higher interest rate risk. For example, if the bond’s duration is 5, a 1% increment in the interest rate levels would mean that the bond price would decrease by 5%.

The computation of duration is multiplying the present value of each cash flows with their respective time. The formula is

Duration = \sum_{t=1}^{n} \frac{t \times C}{(1 + i)^{{t}}} + \frac{n \times M}{(1 + i)^{{n}}}}

Where

  • is time expressed in years
  • is the bond’s coupon
  • is the bond’s time to maturity expressed in years
  • is the bond’s notional
  • is the interest rate level

There are two downside of the duration. Firstly, the duration measures only the linear relationship between bond prices and interest rate levels. One should also use a non-linear metric to measure the impact on the bond markets with convexity being the most popular. Secondly, duration assumes a uniform increment of interest rate level across all tenors. This is a very unrealistic assumption as interest rates of all tenors would not move at the same rate. The workaround to this is to use key rate duration which measures the bond price sensitivity to the changes in only one interest rate tenor.

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