Please consider/clear my doubts for SP6 CMP of year 2021

[NewStudent]

I recently completed studying through SP6 CMP for year 2021.

Please consider my list of doubts/questions and explain wherever necessary.


List of Doubts

1. SP6-09: Exotic Derivatives, Page 31 Q. 9.2 iii)

Old Values : r = In(1.05) = 4.879016417% , σ *√(1) = σ = In(1.25) = 22.31435513% , St = 480, Δ = 53.7

Now question states that r increases by 0.25%, σ increases by 1% and share price drops by 2.

In Acted Answer it considers % changes as it is without converting them into absolute changes.

If we assume that changes in r and σ are % changes then :

δr = +0.25% * 4.879016417% = +1.219754101 * 10^(-4)

δσ = 1% * 22.31435513% = +0.2231436%

δS = -2

δf = ΔδS + ρδr + Vδσ + ½*(δS)^2 * Γ (As per SP6-08 Page 14 Section 2.6 Box )

So my question why is Γ and its effect is ignored here? Surely we must incorporate it ?

If we ignore Γ term and continue, we get:

δf = 126.8634 and New f = Old f + δf = 11113.86 ≅ 11114


2. SP6-15 : Risk Management – Page 52, Q. 15.5 ii)

In 6th line from bottom, (Merton model to value shares), shouldn’t Theoretical share price at time 10 be denoted by S10 ? In answer it is denoted by S0.


3. SP6-15 : Risk Management – Page 57, Q. 15.9 ii)

On SP6-15 Page 45, Question states that £ 1 million is held in each bond. Portfolio has 2 corporate zero coupon bonds. So isn’t the total portfolio value is £ 2 million and

95 % VaR is (1-0.3) * £2m = £1.4m ? Why is Acted Answer (1-0.3) * £1m = £0.7m


4. SP6-16 : Credit Derivatives – Page 22, 23 Part ii)

Here, Probability and discount factors both are applied to Premiums, Claims and Accrual. But it is denoted by PV. Shouldn’t it be denoted by EPV (Expected Present Value) ?

Also, can you please explain (E)PV Accrual adjustment, and when and why it is required?


5. SP6-16 : Credit Derivatives – Page 44, Q. 16.5 ii)

Here, please mention that Assumption : Premium paid annually in advance.

Because in similar question (IFoA ST6 S2008 Q. 8 ii)) on SP6-16 Page 22, it is mentioned that

Assumption : Premiums P are paid annually in arrears.


6. SP6-17: Practical Derivative Management and Problem Solving – Page 31

In last line I did not understand the partial derivative.

Why it is -T* B(t, T*) instead of (t - T*) B(t, T*) ?


7. SP6 – X Assignments Questions : X5 – Page 2 Q. X 5.4)

The question numbering should be i), ii), iii) instead of only i) and ii)


8. SP6 – X Assignments Solutions : X2 – Page 7, Q. X 2.4) i)

In third line I did not understand why integral of Ys dWs from t1 to t2 is 0 given F t1 . Please explain


9. SP6 – X Assignments Solutions : X4 – Page 6, Q. X 4.5) ii)

The numerical accuracy of final answer (18.9%) is low. Accurate answer is 17.850369%.

I am pasting values from Excel for reference

St Log Return

298 0.010016778 =LN(A3/A2) Answer 17.85036904%

301 -0.006666691 =LN(A4/A3) =STDEV.P(B2:B10)*SQRT(250)

299 0.006666691 =LN(A5/A4)

301 0.029462033 =LN(A6/A5)

310 0.016000341 =LN(A7/A6)

315 -0.009569451 =LN(A8/A7)

312 0 =LN(A9/A8)

312 0 =LN(A10/A9)

312 0.003200003 =LN(A11/A10)

313

Note : In 1-Var Stat mode in calculator, I had entered data values as In(S / S[i-1] ) and used statistical function σ directly to calculate daily standard deviation.


The list is long as I have aggregated all my doubts in a single post.
Thanks in advance for your help.

 

[NewStudent]

Please reply

 

[Steve Hales]

Sure thing
Probably easiest if we break the queries down into individual replies.

 

[Steve Hales]

1. SP6-09: Exotic Derivatives, Page 31 Q. 9.2 iii)

Old Values : r = In(1.05) = 4.879016417% , σ *√(1) = σ = In(1.25) = 22.31435513% , St = 480, Δ = 53.7

Now question states that r increases by 0.25%, σ increases by 1% and share price drops by 2.

In Acted Answer it considers % changes as it is without converting them into absolute changes.

If we assume that changes in r and σ are % changes then :

δr = +0.25% * 4.879016417% = +1.219754101 * 10^(-4)

δσ = 1% * 22.31435513% = +0.2231436%

δS = -2

δf = ΔδS + ρδr + Vδσ + ½*(δS)^2 * Γ (As per SP6-08 Page 14 Section 2.6 Box )

So my question why is Γ and its effect is ignored here? Surely we must incorporate it ?

If we ignore Γ term and continue, we get:

δf = 126.8634 and New f = Old f + δf = 11113.86 ≅ 11114

The changes in the parameters are always quoted as absolute changes. So an increase in r of 0.25% means that r_new = r_old + 0.25%.
If the value of gamma had been given in the question then we would have used it in the solution. So you're correct, using gamma would have been more accurate. However, this is only a one-step binomial tree and so gamma can't be estimated anyway.

 

[Steve Hales]

2. SP6-15 : Risk Management – Page 52, Q. 15.5 ii)

In 6th line from bottom, (Merton model to value shares), shouldn’t Theoretical share price at time 10 be denoted by S10 ? In answer it is denoted by S0.

Yes, that should be S10 and not S0. Thanks for pointing it out.

 

[Steve Hales]

3. SP6-15 : Risk Management – Page 57, Q. 15.9 ii)

On SP6-15 Page 45, Question states that £ 1 million is held in each bond. Portfolio has 2 corporate zero coupon bonds. So isn’t the total portfolio value is £ 2 million and

95 % VaR is (1-0.3) * £2m = £1.4m ? Why is Acted Answer (1-0.3) * £1m = £0.7m

At the 5% level, only one of the bonds will have defaulted, therefore the £1m is used rather than the £2m.

 

[Steve Hales]

4. SP6-16 : Credit Derivatives – Page 22, 23 Part ii)

Here, Probability and discount factors both are applied to Premiums, Claims and Accrual. But it is denoted by PV. Shouldn’t it be denoted by EPV (Expected Present Value) ?

Also, can you please explain (E)PV Accrual adjustment, and when and why it is required?

Yes, these are the expected present values. The first paragraph of the solution uses the word "expected", but the left hand side of the equations just uses "PV".

 

[Steve Hales]

5. SP6-16 : Credit Derivatives – Page 44, Q. 16.5 ii)

Here, please mention that Assumption : Premium paid annually in advance.

Because in similar question (IFoA ST6 S2008 Q. 8 ii)) on SP6-16 Page 22, it is mentioned that

Assumption : Premiums P are paid annually in arrears.

Not sure about the phraseology, but from the calculation of the premium factor the "1" is paid at the start of year one, and the "0.98" is the expected premium to be paid at the start of year two, and so on. So premiums are paid in advance. Hope that helps.

 

[Steve Hales]

6. SP6-17: Practical Derivative Management and Problem Solving – Page 31

In last line I did not understand the partial derivative.

Why it is -T* B(t, T*) instead of (t - T*) B(t, T*) ?

So "(t - T*) B(t, T*)" is the general result, but in this question the current time t=0. Therefore "- T* B(t, T*)" would suffice, although it would be clearer to have "- T* B(0, T*)".

 

[Steve Hales]

7. SP6 – X Assignments Questions : X5 – Page 2 Q. X 5.4)

The question numbering should be i), ii), iii) instead of only i) and ii)

Yes, thank you. That'll be easy to fix in the next edition.

 

[Steve Hales]

8. SP6 – X Assignments Solutions : X2 – Page 7, Q. X 2.4) i)

In third line I did not understand why integral of Ys dWs from t1 to t2 is 0 given F t1 . Please explain

Between time 0 and t1 the increments of dWs are fully known (given F t1), therefore the expected value of the integral is equal to the integral itself. That is to say that there's nothing random about it.
Between time t1 and t2 the increments of dWs are independent of F t1, and so the expectation of the Ito integral is zero by the standard result. Watch out for the stochastic integrand here.

 

[Steve Hales]

9. SP6 – X Assignments Solutions : X4 – Page 6, Q. X 4.5) ii)

The numerical accuracy of final answer (18.9%) is low. Accurate answer is 17.850369%.

I am pasting values from Excel for reference

St Log Return

298 0.010016778 =LN(A3/A2) Answer 17.85036904%

301 -0.006666691 =LN(A4/A3) =STDEV.P(B2:B10)*SQRT(250)

299 0.006666691 =LN(A5/A4)

301 0.029462033 =LN(A6/A5)

310 0.016000341 =LN(A7/A6)

315 -0.009569451 =LN(A8/A7)

312 0 =LN(A9/A8)

312 0 =LN(A10/A9)

312 0.003200003 =LN(A11/A10)

313

Note : In 1-Var Stat mode in calculator, I had entered data values as In(S / S[i-1] ) and used statistical function σ directly to calculate daily standard deviation.

You've used Excel's function for finding the population standard deviation (STDEV.P) rather than the sample standard deviation function (STDEV.S). When the data set is large the difference between the two is very small, but here we've only got nine returns and so that would account for the relatively large discrepancy.

 

[NewStudent]

Thanks for your guidance.
For point no. 9:
In exam, should we calculate population standard deviation or sample standard deviation?
What if I calculated population sd in place of sample sd? Will I get marks?

 

[Steve Hales]

It depends what you want to calculate
Use STDEV.S to estimate the standard deviation of the population based on only a sample of it. Use STDEV.P when you're processing the entire population.
In questions like the one in Assignment X4, the data points are only a subset of the entire performance of the security and yet we want to infer something about the standard deviation of the whole - therefore use STDEV.S.