2 questions

[ExamTortoise]

Duration Gap

When we say the Duration Gap equals 1, what does that mean in words. If it is equal to 0 you are immune to changes in the interest rate. But if it equals 1, you are immune to changes in interest rate for 1 year?

Covariance

If you have two assets in your market (A & B with correlation between them p), is the formula for Cov(rA, rM) #1 or #2 below [chk the squaring of the weights]:

1) Cov (A,Market) =weight A^2σA^2 + (1 - weight A)^2ρABσAσB

2) Cov (A,Market) =weight AσA^2 + (1 - weight A)ρABσAσB

I've seen it both ways on some old CSM study manual solutions and am not sure if the weights should be squared.

[Colymbosathon ecplecticos]

The regulator doesn't look at you all the time. Suppose they come by once a year.

You are okay today and they will be back in one year. How should you invest your assets to protect yourself against interest rate movements?

Answer: Select a duration gap of surplus of 1. (Assuming parallel interest rate shifts, tiny movements, etc.) Your surplus in one year will not depend on interest rate movements if you do this. Your surplus in 6 months might be higher or lower, no matter, you are trying to make sure that you are okay when the regulator returns.

[ExamTortoise]

Thanks, and I like the regulator analogy, but I'm not there yet....

Lets say Assets = $100 with Duration 4, Liabs = $50 with Duration 7
Surplus = $100 - $50 = $50. Duration MVS = ($100 x 4 - $50 x 7) / $50 = 1

You are interested in immunizing your ROS over a one year period and you have indeed set your DG Total ROS = 0 (i.e. DG = D - H and here it is DG = 1 - 1)

So far, so good. Now interest rates go up 1%. What happens?

Are we saying both your return on Assets drops and your MVS drop an offsetting amount? So that the ROS doesn't change?

If so, what happens after a year, how do things get out of synch? After a year they won't move by offsetting amounts?

[Colymbosathon ecplecticos]

It is just capital gains.

Your example: after one year, assume zero-coupon bonds used.

Assets = 100, duration = 3
Liabilities = 50, duration = 6

Surplus = 50.0

100 basis point move. Say rates dropped.

Assets = 103
Liabilities = 53

Surplus = 50.0

[Scooterpye]

If you want to protect surplus against interest rate movements, you would select a duration gap of zero. Then the change in the value of assets will match the change in the value of liabilities.

If you had a duration gap of surplus of 1, then you would be exposed to interest rate moves and every one percent rise in interest rates would result in a one percent decline in your surplus. In the example you gave, a 1% rise in interest rates hits your assets by .01 * 4* $100 = $4, and it reduces your liabilities by .01 * 7 * $50 = $3.50. The reduction in surplus = $4 - $3.5 = $.5. As a percentage of surplus is .5/50 = 1%.

This duration gap measurement is done as a snapshot and has nothing to do with any horizon. It is really only good for small moves of interest rates that happen in a short amount of time. It ignores higher order effects such as convexity that would come into play with big moves in interest rates. And, over time, the duration of your assets and liabilities will tend to change -- not necessarily in tandem with one another -- so, in order to maintain your duration gap at a certain level you will need to rebalance your asset duration periodically.

[ExamTortoise]

I'm trying to reconcile your answer with this quote from Noris (pg 28 bottom)

"Another valuable feature of duration is that it indicates the holding period over which a rate of return can be immunized. If duration is set equal to the desired holding period, then the initial promised return can be realized independent of changes in the level of interest rates....Many insurers desire to manage annual ROS so that they are always positive. Provided the net yield on MVS is positive, this goal can be achieved by setting DG equal to one."

It's this point made by Noris that I'm really trying to understand. I mean I can spit it back to the examiner, but if they gave me a numerical problem, I'm concerned I'd get tripped up.

[Scooterpye]

The link between duration exposure and investment return over a certain horizon has to do with changes in reinvestment income offsetting changes in capital values.

Assume a positive duration gap. If interest rates rise, then you will experience a capital loss -- as I indicated --, but you will also be able to reinvest your cash flows at a higher interest rate. Over a horizon equal to your duration, you would expect the increase in income due to higher reinvestment rates to offset the capital loss you experience. Hence, you are "immunized" over this period.

[Colymbosathon ecplecticos]

Look at the example I worked above. I selected 0-coupon bonds to side-step the reinvestment issue.

Note the change in the remaining duration in your example.

[prashantsingh]

You are Right....
the weights should not be squared.
Cov(A ,W_AA+W_BB)=W_ACov(A,A)+W_B Cov(A,B)=W_A \sigma_A^2+(1-W_A) \rho \sigma_A \sigma_B

Duration Gap

When we say the Duration Gap equals 1, what does that mean in words. If it is equal to 0 you are immune to changes in the interest rate. But if it equals 1, you are immune to changes in interest rate for 1 year?

Covariance

If you have two assets in your market (A & B with correlation between them p), is the formula for Cov(rA, rM) #1 or #2 below [chk the squaring of the weights]:

1) Cov (A,Market) =weight A^2σA^2 + (1 - weight A)^2ρABσAσB

2) Cov (A,Market) =weight AσA^2 + (1 - weight A)ρABσAσB

I've seen it both ways on some old CSM study manual solutions and am not sure if the weights should be squared.

[ExamTortoise]

CE - Your response, though well intentioned, was cryptic. Using the numbers in Scooterpye's post, the duration gap is 1 and Noris says that with a duration gap of 1, the ROS should be immunized over a 1-year horizon.

Assuming Scooter was invested in zeros, and they did not mature during the year so their was no reinvestment at higher returns, can you explain to us how his ROS is immunized over the year? (Preferable using the numbers in his example)


Prashantsingh - Thanks.

[Colymbosathon ecplecticos]

At time zero your portfolio consists of a 4-year zero coupon bond with a face value of $131.08 and a 7-year zero coupon liability with a face value (nominal value) of 80.29.

Interest rates are 7%

Your discounted surplus is 50.

Duration gap of surplus is 1.

In one year your surplus is immunized to be 53.5 = 50 * (1.07).

Check it for yourself with a spreadsheet.

[ExamTortoise]

That helps, thanks.

[Scooterpye]

CE - Your response, though well intentioned, was cryptic. Using the numbers in Scooterpye's post, the duration gap is 1 and Noris says that with a duration gap of 1, the ROS should be immunized over a 1-year horizon.

Assuming Scooter was invested in zeros, and they did not mature during the year so their was no reinvestment at higher returns, can you explain to us how his ROS is immunized over the year? (Preferable using the numbers in his example)


Prashantsingh - Thanks.

If you are investing in one-year zero coupon bonds, you are immunized over a one-year horizon. You know exactly what your return will be. You are not subject to reinvestment rate risk since you have no cash flows to reinvest. At the horizon date, your capital value is also not a function of interest rates since all of your holdings will mature and you'll have nothing but cash.

[Colymbosathon ecplecticos]

While that is true, you generally have no control over the duration of your liabilities. In order to achieve the immunization that you want, you change the duration gap of surplus to 1. See my example above.