# Help with understanding bond duration.

I'm currently studying for a pure maths degree, and so have no background knowledge of bonds. I'm reading through some material regarding actuarial work, and came across the following definition of duration which I cannot wrap my head around:

The duration is the time before the average payments are made of the bonds held.

I've been googling the definition of duration, and understand it to be some sort of measure of the sensitivity of the bonds value to changes in the interest rates. Could someone give me a simple explanation of what the duration is, and the meaning of the above (what are the average payments?) Thanks!

• thestreet.com/topic/46361/duration.html – saulspatz Dec 13 '19 at 15:41
• @saulspatz Thanks, but that doesn't really help with my confusion regarding the given definition in terms of average payments. – Chris Dec 13 '19 at 16:00
• I take it that "average payments" means half the present value of all future payments. I'll try to post an explanation, but first, please say what terms, if any, you don't understand. Do you know the meaning of yield, coupon, maturity, present value? – saulspatz Dec 13 '19 at 16:46
• Apparently, there is more than one definition of duration. I didn't know this. investinganswers.com/dictionary/d/duration – saulspatz Dec 13 '19 at 17:02

Consider a zero coupon bond-- it pays you 10 dollars next period. What will you pay for it? It depends on the interest rate, or conversely, the interest rate is determined by the price of the bond. $$P_1=\frac{10}{1+R}$$ If it pays you in 10 years, $$P_{10}=\frac{10}{(1+R)^{10}}$$ So this bond has a maturity of 10 years, and note that $$P_{10}$$ is more sensitive to R than is $$P_1$$. But if the bond has coupons, and along with paying you back 10 dollars in 10 years, it also pays you 1 dollar each year. Duration is a way of saying that the 10 year coupon bond is less sensitive to changes in rates than a zero coupon bond, since $$P_{c,10}=\frac{1}{1+R} +\frac{1}{(1+R)^2}+...\frac{10}{(1+R)^{10}}$$ It's like an average of the maturity of the different payments. That's how I've always considered it, anyways.
One, how sensitive the price of bonds change relative to interest rate change Suppose if we have a 3 year semi coupon bonds annual rate is 8% with a face value of $1000. If this bond has a duration of 2.5. It means that if there 1% interest rate increases, the price of bond will decrease 2.5%. Two, it is also the time that you can collect your principals. Suppose you invest 1000 dollars for the bond above. It has maturity of 3 years. A duration of 2.5 means that in 2.5 years you can successfully collect ur principal$1000 back (because of the semi-annual coupons you get).