April 2009 Q3 (iv)

[BeckyBoo]

This should be a really basic question that I should know the answer too, but... I can't see how the semi-annual forward interest rates have been calculated from the annual interest rates in the solution to this question.

Please help!!!

 

[Colin McKee]

rates

Hi. We are told that (for example the second period) the rate is 4.25% and there are 181 days in the half year. We are also told that interest is actual/360. So, the rate that applies to the six month period is 4.25%*(181/360) = 2.1368%. To calculate a discount factor for a cashflow from a point (say) t=4 in the future, you simply use the 4 semi-annual rates and geometrically string them together (1/1.020333)*(1/1.021368) * ....) You will need 4 semi annual rates to do this. Hope this helps.

 

[BeckyBoo]

Ok, so I failed ST5 last time and am re-taking. I am just doing the April 2009 paper and got stuck on this question again... although for slightly different reason this time.

This time round I thought I was much cleverer... I "spotted" the "trick" straight away... the payments are made semi-annually and we've been given annual rates. Turns out I was too clever for my own good and was in fact wrong! The question is actually much simpler than I thought but my question is why. We're given annual forward interest rate and know that payments are calculated are paid semi-annually in arrears. So why don't we have to convert the annual rate to a semi-annual rate (rather than simply pro-rata by the number of days interest we want as has been done in the solution)?

Similarly for the discount rate. For example for period 1, I calculated the discount rate as 1.04^-(183/360) while the solution uses the approximation of 1/(1+(183/360)*4%).

I'm hoping someone can explain why these 2 calculation points are less complictaed than I was expecting!

Many thanks

 

[Colin McKee]

rates

I think the answer is that a rate quoted will usually be an annual rate, and it will usually compoun acording to the period of the interest payment. If you have a 30 year semi annual paying bond, and the GRY is 3%, then thats 1.5% per half year. If (as in this case) you have a LIBOR rate for 6 months, then it will be an annual rate which applies to the period of the investment. If there are 183 days in the half year you will get 3% (183/360) at the end of the period. (But always watch the wording of the qustion.) So (1+ 3%(183/360) ) is the right interest rate. Your method assumes that 3% is an annual effective rate and you have taken the (183/360)th root of it. Not a common approach unless it says the 3% is an annual effective rate. The 360 day convention for money maket rates is just a convention, but I seem to recall that it indicated 360 day year in the question.

 

[almost_actuary]

Hey, many years later I am stuck on the same question!

Thanks to your comments above, I can understand how to get to the semi-annual rates. However, I am now struggling to calculate the payments for each year. Could anyone explain how the first year's floating undiscounted payment of £1,016,667 is calculated, and the fixed undiscounted payment of £25,416,667 is calculated?

 

[almost_actuary]

Hey, many years later I am stuck on the same question!

Thanks to your comments above, I can understand how to get to the semi-annual rates. However, I am now struggling to calculate the payments for each year. Could anyone explain how the first year's floating undiscounted payment of £1,016,667 is calculated, and the fixed undiscounted payment of £25,416,667 is calculated?

I am now resitting this exam and still stuck on this! Can anyone help....?

 

[almost_actuary]

Hi there

I am stuck on this question, I have figured out how to calculate the discount factors but I don't know where the floating payments have come from. For example, in period one where has the floating payment of £1,016,667 in the answers come from?

 

[Gresham Arnold]

Hi Esme J

This question is difficult!

We are given forward rates in the question and the number of days in each period and we are told to assume that there are 360 days in a year.

So the examiners expected candidates to calculate semi-annual forward rates by pro-rating the annual forward rates by the number of days in a period.

So for the first period, the forward rate is 183/360 x 4% = 2.0333333%

You could calculate the undiscounted floating payments by multiplying the forward rate for each period by the notional principal of the swap.

So for Period 1, this would be 2.03333333% x £50m = £1,016,667, as you say.

Similarly, you can calculate the undiscounted fixed payment by pro-rating the notional principal of the swap by the number of days and multiplying by i (where i is the fixed rate of the swap we are trying to find in part (iv)(b))

So for Period 1 this would be 183/360 x £50m x i = 25416667i

I hope this helps?

Gresham

 

[almost_actuary]

Hi Esme J

This question is difficult!

We are given forward rates in the question and the number of days in each period and we are told to assume that there are 360 days in a year.

So the examiners expected candidates to calculate semi-annual forward rates by pro-rating the annual forward rates by the number of days in a period.

So for the first period, the forward rate is 183/360 x 4% = 2.0333333%

You could calculate the undiscounted floating payments by multiplying the forward rate for each period by the notional principal of the swap.

So for Period 1, this would be 2.03333333% x £50m = £1,016,667, as you say.

Similarly, you can calculate the undiscounted fixed payment by pro-rating the notional principal of the swap by the number of days and multiplying by i (where i is the fixed rate of the swap we are trying to find in part (iv)(b))

So for Period 1 this would be 183/360 x £50m x i = 25416667i

I hope this helps?

Gresham

Thankyou so much for going through this Gresham! That has really helped.