Duration

(Macaulay) duration measures the average length of time a bondholder must wait before receiving cash payments, as the average of all payments weighted by discounted cash flow present value. A higher duration naturally means greater risk exposure, as there is more time for something to happen that changes expected cash flows.

Finding duration requires 6 areas of information (Reilly and Brown, 2008). The first three are:

1. the yield to maturity (YTM);
2. the cash flow (CF) received for the year;
3. the number of years into the future (N) the cash flow will be received, assuming annual payments.
4. The remaining factors require calculation: the present value (PV) of the year’s cash flow after discounting with the yield rate due to the time value of money;
5. the cash flow present value for each year as a percentage (PV%) of the present value of the sum of all present value cash flows (∑PV) (i.e. a bond’s PV price);
6. each year’s cash flow present value as a percentage of price multiplied by the number of years into the future the cash flow is received (PV% * N), where the sum of these values is the duration:

PV = CF / (1 + YTM)^N 

PV% = PV / ∑PV 

Duration = ∑(PV% * N)

As calculating duration is relatively complicated it can be more easily explained using some examples. Annual interest payments are assumed, and a yield to maturity (YTM) of 8% is used to determine the duration of two bonds, bond A and bond B, with the following features:  

Bond A  

Face value: £1,000; Maturity: 3 years; Coupon: 4%  

Bond B  

Face value: £1,000; Maturity: 3 years; Coupon: 9%  

Bond A  

Looking first at bond A in-depth, it’s possible to calculate the values of CF (cash flow) for each of the 3 years until the bond reaches maturity. Each year will have a cash flow of 4% (the coupon rate) of the £1,000 face value of the bond, while the final year will also see the principal (face value) paid out as part of the cash flow. The CF values for bond A are:  

Year 1 CF = 1,000 * 0.04 = £40 

Year 2 CF = 1,000 * 0.04 = £40 

Year 3 CF = 1,000 + (1,000 * 0.04) = £1,040  

With these numbers, the PV (present value) of cash flows can then be found for bond A:

PV = CF / (1 + YTM)^N 

PV of Year 1 CF = 40 / (1.08)¹ = £37.04 

PV of Year 2 CF = 40 / (1.08)²  = £34.29 

PV of Year 3 CF = 1,040 / (1.08)³  = £825.59

From here the PV% can be found:  

PV% = PV / ∑PV 

∑PV = 37.04 + 34.29 + 825.59 = £896.92 

PV% for Year 1 PV = 37.04 / 896.92 = 0.0413 

PV% for Year 2 PV = 34.29 / 896.92 = 0.0382 

PV% for Year 3 PV = 825.59 / 896.92 = 0.9205  

And then (PV% * N) can be calculated and summed to give the duration of bond A:  

(PV% * N) for Year 1 = 0.0413 * 1 = 0.0413

(PV% * N) for Year 2 = 0.0382 * 2 = 0.0764 

(PV% * N) for Year 3 = 0.9205 * 3 = 2.7615  

Duration = ∑(PV% * N) = 0.0413 + 0.0764 + 2.7615 = 2.88 years  

Bond B  

Looking next at bond B, each of the 3 years until the bond reaches maturity will have a cash flow of 9% (the coupon rate) of the £1,000 face value of the bond, while the final year also sees the principal (face value) paid out as part of the cash flow. The CF values for bond B are:  

Year 1 CF = 1,000 * 0.09 = £90 

Year 2 CF = 1,000 * 0.09 = £90 

Year 3 CF = 1,000 + (1,000 * 0.09) = £1,090  

The PV of cash flows for bond B are:  

PV = CF / (1 + YTM)^N 

PV of Year 1 CF = 90 / (1.08)¹ = £83.33 

PV of Year 2 CF = 90 / (1.08)² = £77.16 

PV of Year 3 CF = 1,090 / (1.08)³ = £865.28  

From here the PV% can be found:  

PV% = PV / ∑PV 

∑PV = 83.33 + 77.16 + 865.28 = Price = £1,025.77 

PV% for Year 1 PV = 83.33 / 1,025.77 = 0.0812 

PV% for Year 2 PV = 77.16 / 1,025.77 = 0.0752 

PV% for Year 3 PV = 865.28 / 1,025.77 = 0.8435  

Then (PV% * N) can be calculated and summed to give the duration of bond B:  

(PV% * N) for Year 1 = 0.0812 * 1 = 0.0812 

(PV% * N) for Year 2 = 0.0752 * 2 = 0.1504

(PV% * N) for Year 3 = 0.8435 * 3 = 2.5305  

Duration = ∑(PV% * N) = 0.0812 + 0.1504 + 2.5305 = 2.76 years  

These two example bonds reveal some of the notable characteristics of (Macaulay) duration. First, while a zero coupon bond such as a Treasury bill (T-bill) will have a duration equal to its maturity, the duration of a bond with coupon payments is always less than its term to maturity as duration weights interim interest payments. This is visible here with 2.88 years duration for bond A and 2.76 years duration for bond B compared to maturity of 3 years for both bonds. 

Second, there is a generally positive relationship between duration and term to maturity, and a bond with a longer term to maturity will almost always have a higher duration, but duration increases at a decreasing rate as the term to maturity is longer. The relationship is only general and not direct, however, because as maturity increases the present value of the principal to be received at maturity at the end of a bond’s life declines in value due to greater discounting, and this will reduce duration. This second characteristic is related to the first noted above, and if the duration of a bond with coupon payments is always less than its term to maturity it follows that a longer term to maturity enables a longer duration. 

The third characteristic of duration is that there is an inverse relationship between coupon rate and duration, and a larger coupon ensures that a bond will have a shorter duration as a greater amount of total cash flows will come earlier in the bond’s life as interest payments. This is shown with the two bonds here as bond B’s 9% coupon has a duration of 2.76 and bond A’s lower 4% coupon gives a longer duration of 2.88 years. 

Finally, other things being equal, there is an inverse relationship between the yield to maturity (YTM) and duration. A higher yield to maturity reduces a bond’s duration as it ensures greater discounting of far-off cash flows relative to closer cash flows, to see cash flows received sooner play a greater role in average payments.

Do check related posts:

1. Debt and Leverage
2. Duration
3. Modified Duration & Risk Sensitivity
4. Convexity
5. Effective Duration & Effective Convexity