How do you work out DMT (duration)?

[Gbob1]

Specifically referring to question 14 part ii) of 2000 April paper:

'A pension fund has a liability to pay 100,000 pounds at the end of 1 year, 105,000 pounds at the end of 2 years, and so on, the amount increasing by 5000 pounds each year to 195,000 pounds at the end of 20 years, this being the last payment. The fund values these payments using an effective interest rate of 7% pa. This is also the interest rate at which the current prices of all bonds are calculated.

The fund invests an amount equal to the present value of these liabilities in the folowing 2 assets:

A) a zero coupon bond redeemable in 25 years, and

B) a fixed interest bond redeemable at par in 12 years' time which pays a coupon of 8% pa annually in arrears'

Calculate the present value and the duration of the liabilities'

Ok I've managed to calculate the present value of the liabilities, that was simple. But then I don't know what else to do next - I've looked up the formula for duration but I don't understand it. How do I work out the numerator of the duration. From the answer scheme I'm assuming the present value of the liabilities is the denominator. So what does: the sum of tk*Ctk*v(i)tk mean (i.e. the numerator)? In the answers it says it's: 95*increasing annuity for 20 yrs in arrears + 5[v + 2.2v^2 +...+ 20.2v^20] I don't understand this at all...

 

[John Lee]

The DMT (=duration) is the average time to a payment using the PVs of the cashflows as the weights.

So the mean formula is usually sum(fx) / sum(f)

So the DMT is sum(PV × time) / sum(PV)

Work from first principles for the numerator - and you should then notice you have an increasing annuity...

 

[Gbob1]

The DMT (=duration) is the average time to a payment using the PVs of the cashflows as the weights.

So the mean formula is usually sum(fx) / sum(f)

So the DMT is sum(PV × time) / sum(PV)

Work from first principles for the numerator - and you should then notice you have an increasing annuity...

Thanks! I figured it out in the end. It was because I hadn't done section 4 of the Q&A that I had no idea how to apply the formula. But DMT should be fine now

I wonder what I'll get stuck on next...

 

[Gbob1]

I had another look at the question last night and seems like I thought I knew what was going on but now am confused again. Here is my working:

DMT=Sum(t x PV)/Sum(PV)=[100v+2*105v^2+3*110v^3+...+20*195v^20]/1446.95 (PV worked out in previous part).

What I then did was say tth term is found by 95+5n so sub that in as Ct:

numerator becomes: Sum to n=20[t(95+5t)v^t] = Sum to n=20[95tv^t] +5*Sum to n=20[t^2v^t].

I know that Sum to n=20[95tv^t] =95(increasing annuity for 20 years) but I dont know how to value 5*Sum to n=20[t^2v^t]. How do you value this?

Am I doing the right thing or is there an easier way?

 

[John Lee]

DMT=Sum(t x PV)/Sum(PV)=[100v+2*105v^2+3*110v^3+...+20*195v^20]/1446.95 (PV worked out in previous part).

What I then did was say tth term is found by 95+5t so sub that in as Ct:

numerator becomes: Sum to n=20[t(95+5t)v^t] = Sum to n=20[95tv^t] +5*Sum to n=20[t^2v^t].

I know that Sum to n=20[95tv^t] =95(increasing annuity for 20 years) but I dont know how to value 5*Sum to n=20[t^2v^t]. How do you value this?

Am I doing the right thing or is there an easier way?

You've done exactly the right thing! But you've forgotten to use the result that they got you to prove in part (i) as this is the second part:

Sum to n=20[t^2v^t]