Analysis of surplus vs EV - Return on opening surplus

[curiousactuary]

Is the return on opening surplus calculation under Analysis of surplus the same as the return on net assets calculation under Embedded value.

AoS : Formula
The course notes seem to describe the AoS's calculation as:
Opening surplus (A - L) * actual return + Opening L (backing assets) * (actual return - expected return)

AoS : numerical example
For e.g. using the AoS's example from the course notes, where A(0) = 2200, L(0)= 2000, a = 1% e = 0.5%,

The opening surplus = A(0) - L(0) = 200

The return on opening surplus is calculated as

• (2200 - 2000) * (0.01) + 2000 * (0.01 - 0.005)
• = 200 * 0.01 + 2000 * (0.005)
• = 2 + 10 = 12

The closing surplus is calculated as :

• A(1) = A(0) * (1+a) = 2200 * 1.04 = 2222 less
• L(1) = L(0) * (1+e) = 2000 * 1.005 = 2010
• S(1) = A(1) - L(1) = 2222 - 2010 = 212

The surplus arising is calculated as:

• S(1) - S(0) = 212 - 200 = 12
• This ties in with the return on opening surplus calculation above (assumed to be the only source for the surplus arising)

EV: formula
The EV calculation for "return on net assets" should reflect the "actual return earned on the starting net assets" and is split into:

1. expected return and
2. excess of actual return over expected return

Would its formula therefore be:
expected return * (assets backing opening liabilities) + (actual return - expected return) * (net assets)?

EV: numerical example
Using my EV formula above each numbered point would be calculated as:

1. 0.005 * L(0) = 0.005 * 2000 = 10
2. (0.01 - 0.005) * {A(0) - L(0)} = 0.005 * 200 = 1

This adds to 10 + 1 = 1 which is not equal to the calculation from the AoS of 12.

My questions are:
1. Is my AoS's formula of "return on opening surplus" correct?
2. Is my EV's formula of "return on net assets" correct for each component (1) and (2)?
3. Should question 1 and 2 be equal to each other?

Thanks in advance

 

[Em Francis]

Is the return on opening surplus calculation under Analysis of surplus the same as the return on net assets calculation under Embedded value.

AoS : Formula
The course notes seem to describe the AoS's calculation as:
Opening surplus (A - L) * actual return + Opening L (backing assets) * (actual return - expected return)

AoS : numerical example
For e.g. using the AoS's example from the course notes, where A(0) = 2200, L(0)= 2000, a = 1% e = 0.5%,

The opening surplus = A(0) - L(0) = 200

The return on opening surplus is calculated as

• (2200 - 2000) * (0.01) + 2000 * (0.01 - 0.005)
• = 200 * 0.01 + 2000 * (0.005)
• = 2 + 10 = 12

The closing surplus is calculated as :

• A(1) = A(0) * (1+a) = 2200 * 1.04 = 2222 less
• L(1) = L(0) * (1+e) = 2000 * 1.005 = 2010
• S(1) = A(1) - L(1) = 2222 - 2010 = 212

The surplus arising is calculated as:

• S(1) - S(0) = 212 - 200 = 12
• This ties in with the return on opening surplus calculation above (assumed to be the only source for the surplus arising)

EV: formula
The EV calculation for "return on net assets" should reflect the "actual return earned on the starting net assets" and is split into:

1. expected return and
2. excess of actual return over expected return

Would its formula therefore be:
expected return * (assets backing opening liabilities) + (actual return - expected return) * (net assets)?

EV: numerical example
Using my EV formula above each numbered point would be calculated as:

1. 0.005 * L(0) = 0.005 * 2000 = 10
2. (0.01 - 0.005) * {A(0) - L(0)} = 0.005 * 200 = 1

This adds to 10 + 1 = 1 which is not equal to the calculation from the AoS of 12.

My questions are:
1. Is my AoS's formula of "return on opening surplus" correct?
2. Is my EV's formula of "return on net assets" correct for each component (1) and (2)?
3. Should question 1 and 2 be equal to each other?

Thanks in advance

You can only compare the two if there is no PVIF.

And if there is no PVIF, then there would be no expected return on inforce as this should represent the unwinding of the discount rate, ie the difference in profit due to the release of the future profits being closer in time.

If no PVIF then we have actual return earned on A-L which (using your example) should equal 0.01 * 200 = 2
Then the investment return variance step would include the actual versus expected on the assets backing the liabilities = 10.

However, note that it is not as simple as saying the AoS and Analysis of change in EV (without PVIF) should equal each other as the net assets of the EV may include the required capital component. You will need to look to how it is defined in the question.

Hope this helps.