Faced with the problem

Assuming annual payment of Coupon and Without using numbers and/or Excel, verify that the Duration D of a Zero Coupon Bond always Equals its Term to Maturity T

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what duration? Modified duration of zeros does not always equal to its maturity.

Macaulay duration of zeros is simply weighted average of timing of cash flow and since cash flow is only received once at maturity, then it must be the case that Mac duration of zeros equal to Maturity.

There are no annual (periodic) payments of coupons with a zero-coupon bond (hence 'zero-coupon'). The bond is purchased at a deep discount (usually inflation-based), and only one payment of the face-value is made at the end of the bond's term. Since the duration of a bond is the weighted-average of the time it'll be repaid, and since only one payment is made at T, D always equals T.

Admitwithschola: Faced with the problem

Assuming annual payment of Coupon and Without using numbers and/or Excel, verify that the Duration D of a Zero Coupon Bond always Equals its Term to Maturity T

Forumites, help!

Moderator, this is an opportunity for us all to learn..Fire to front page if you think it is reasonable.

Thank you for your question.

More formally, the formula for duration D, due to Frederick Macaulay 1938, is given by

D=Summation of W_t * t/m, where the summation starts from t=1 and ends at t=m.T, that is, t=1, 2, 3, ....., m.T

Here, m is the frequency of payment of coupon (we will take m=1 since your question says that the coupon is paid ANNUALLY. If you had said biannually, then it would have been m=2. If you had said quarterly, then m would have been m=4. Get it?)

T is the term to maturity, as clearly stated in your question.

What do we want to do? Well, as your question says, we want to analytically prove that D=T in the case of a zero coupon bond.

Before I start the proof proper, let me define the last variable in the duration formula, that is, variable W_t.

W_t=1/A * CF(t)*(1+y/m)^(-t), where A=Summation of CF(t)*(1+y/m)^(-t), where, again, the summation begins from t=1 and ends at t=m.T

Here, CF(t) is the cash flow to the bond holder in period t
y is the bond yield (I am sure you know what bond yields are)

Now, let me write out the summation for convenience. Writing out the summation gives
D=Summation of W_t * t/m = W1*1/m + W_2*2/m +.......+ W_T*T ----------> Equation 1

W_t=1/A * CF(t)*(1+y/m)^(-t) ----------->Equation 2a

A=Summation of CF(t)*(1+y/m)^(-t) = CF(1)*(1+y/m)^(-1) + CF(2)*(1+y/m)^(-2)+........+ CF(m.T)*(1+y/m)^(-m.T)------> Equation 2b

Okay, I have set up the framework, so let me begin with the proof proper.

Assumptions

1. You say annual payment of coupon. This therefore implies that m=1...(You should note that zeros pay no coupon. I am just trying to give you a broad picture of the proof. That is why I am even talking about coupon payment for a zero coupon)

2. You say the bond is zero coupon. This therefore implies that no interim payments are received on the bond until it matures. At maturity, payment is limited to 'only' principal, P. This is the unique feature of a zero coupon bond.

Assumptions 1 and 2 together imply that m=1 and CF(t)=0 for all t different from t=T, and CF(t)=P for t=T, where P=principal (that is, the only cash flow to be received at maturity T). I hope you get the flow.

Plugging assumptions 1 and 2 into Equation 2b gives

A= P*(1+y)^(-T)------> Equation 3

Plugging Equation 3 into Equation 2a gives

W_t= CF(t)*(1+y)^(-t) / P*(1+y)^(-T)------> Equation 4 (remember m=1)

Recall that Equation 1 says that

D= W_1*1/m + W_2*2/m +.......+ W_T*T

So, as you can see, we must plug in values for m and for W_1, W_2, ...., W_T

The value of m=1. This one follows from the story I earlier told. What about W_1, W_2, ...., W_T ? Well, we must turn to Equation 4 for the answer.

Using Equation 4, we see that W_1= W_2=...=0 because CF(t)=0 for all t=1, 2, 3, ....., T-1. The only variable that is not zero is W_T.

From Equation 4, put t=T to get W_T = P*(1+y)^(-T) / P*(1+y)^(-T), where CF(T)=P, i.e. cash flow to zero coupon bond holder at maturity is just the principal P.

Notice that terms cancel out and, finally, W_T= 1 !!!!!

Lastly, plug m=1, W_t=0 for all t=1, 2, 3, ...., T-1 and W_T=1 into Equation 1 to get

D=1*T
i.e. D=T

Thus, we have shown that for a zero coupon bond with maturity T and duration D, we must have that D=T essentially.

T22