 # Duration of zero coupon bond

Duration formula = % change in bond price/ %change in yield price I read that if a zero coupon bond is maturing in 4 years its duration is 4 years. I did not understand how number of years is related to duration. Appreciate your help.

This is really simple. It is Macauley duration. Macauley duration is the weigthed average of the time that passes until the cash flows of a bond are made (coupon and prinicpal payments). The weight used for each cash flow is its present value divided by the total present value of the bond. In the very simple case of a zero coupon bond you just have one payment, so the weight of that payment is 100%. 100% x 4 years = 4 years…

I don’t think that answers his question - why does Macauley duration in the weighted cash flow sense in the case of a zero give interest rate sensitivity.

There are several durations. Duration formula = % change in bond price/ %change in yield price = - (derivative of the bond price)/bond price Macauley duration = weighted average maturity of a bond’s cash flows. Zero-coupon bond P = F/(1+r)^T, F - face value, r - interest rate. dP/dr = -T*F/(1+r)^{1+T} = -P*T/(1+r) Duration = T/(1+r) = Macauley duration/(1+r). For zero-coupon bond Macauley duration = T. So, zero-coupon bond interest rate sensitivity is equal to time to maturity over (1+r) - discrete case. For continuous compounding case P = F*exp(-rT), duration = -(dP/dr)/P = T*F*exp(-rT)/P = T*P/P = T = Macauley duration. So, in continuous compounding case duration is equal to Macauley duration (T), in discrete case, that’s not the case. I hope that helps.

Bet it doesn’t, but good job.

Thanks a lot for your excellent replies. I understood basics of it. I need to read couple of times on derivation. Your skills are deep. It is very much interesting. I don’t know what I might have done without the help of this forum.

chinni - instead of focusing on formulas, think about it this way: the longer it takes to get your principal back, the higher your sensitivity to rate movements, so the longer your duration thats how time and duration sort of go hand in hand a 10 year bond gets “average” principal back in around 8 years, if you think of getting a time weighted average of coupons and principal to get your starting money back. if rates move 1%, price moves 8%. time and sensitivity are closely linked if duration is time to “average money back”, and you dont have any interest payments to cloud the situation, you just wait for your principal, and duration would be 10 for a 10 year does that help?

But then what would be the formula with no interest payments? are you trying to say this:

PV = F/(1+r)^t ?

MacaulayDuration = \frac{1}{PV} \sum_{t=1}^T (t \times PV_t)

For a zero-coupon bond:

MacaulayDuration = \frac{1}{PV} (T \times PV_T) since there is only cash flow at maturity.

PV = PV_T = \frac{Face Value}{(1+r)^T}

So:

MacaulayDuration = \frac{1}{PV} (T \times PV) = T

Conclusion: Macaulay Duration of a zero-coupon bond is equals to its time to maturity.

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Hi guys,
I completely agree with your explanation and it seems pretty obvious that duration = maturity for a 0 coupon bond.

I am looking at effective duration here and when calculating the effective duration of my 0 coupon bond I don’t exactly get the maturity meaning there is some noise. (e.g. maturity is 15 whereas effective duration is 14.2). I suspect this is due to annual compounding instead of continuous compounding but I’d be curious to have your view on it .

Thanks