Instead of introducing the equation for duration right away, we will work through an example. We have a 3 year, 5% coupon bond with face value of 1000. If r = 8%, what is the present value?

Well, let’s first get the cash flows for each period.

Let’s get the present value of each.

flows = [50,50,1050]
period = [1,2,3]
flowsPV = [x/(1.08)**y for x,y in zip(flows,period)]
print(flowsPV)
print(sum(flowsPV))

Now, what we want to see is what the present value of each cash flow is in terms of the total present value. This weight will be a factor in the duration equation.

weights = [x/922.6870903825636 for x in flowsPV]
print(weights)

The final step is multiplying these weights by the number of periods until each cash flow is realized. We will then sum these numbers to get the duration.

weightedYears = [x*y for x,y in zip(weights,period)]
print(weightedYears)
print(sum(weightedYears))

We get a duration of 2.853 from this bond. This duration is called Macaulay duration (other forms of duration will be covered in the fixed income course) and its equation is: