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# GARP.FRM.PQ.P2Surplus Value GARP 2015 Question 5 (garp15-p2-5)

#### CYLoh

##### Member
Subscriber
This might be a stupid question, but I just gotta clarify!

5. An analyst reports the following fund information to the advisor of a pension fund that currently invests in government and corporate bonds and carries a surplus of USD 10 million:

Pension Assets Pension Liabilities
Amount (in USD million) 100 90
Expected Annual Growth 6% 7%
Modified Duration 12 10
Annual Volatility of Growth 10% 5%

To evaluate the sufficiency of the fund's surplus, the advisor estimates the possible surplus values at the end of one year. The advisor assumes that annual returns on assets and the annual growth of the liabilities are jointly normally distributed and their correlation coefficient is 0.8. The advisor can report that, with a confidence level of 95%, the surplus value will be greater than or equal to:

a. USD -11.4 million
b. USD -8.3 million
c. USD -1.7 million
d. USD 0 million

Rationale: The lower bound of the 95% confidence interval is equal to: Expected Surplus - (95% confidence factor * Volatility of Surplus). The required variables can be calculated as follows:
Variance of the surplus = 100^2 * 10%^2 + 90^2 * 5%^2 - 2 * 100 * 90 * 10% * 5% * 0.8 = 48.25
Volatility of the surplus = √48.25 = 6.94,
The expected surplus = 100 * 1.06 - 90 * 1.07 = 9.7.
Therefore, the lower bound of the 95% confidence interval = 9.7 - 1.645 * 6.94 = -1.725

Why is the variance of the surplus calculated as 100^2 * 10%^2 + 90^2 * 5%^2 - 2 * 100 * 90 * 10% * 5% * 0.8
and not 100^2 * 10%^2 + 90^2 * 5%^2 + 2 * 100 * 90 * 10% * 5% * 0.8.

Regards

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi
Surplus is given by
s=Assets(A)-Liabilities(L)=A-L
Take variance on both sides,
Var(s)=Var(A-L)=Var(A)+Var(L)-2*Cov(A,L)[we know that Var(a-b)= Var(a)+Var(b)-2*Cov(a,b)]
Var(s)=Var(A-L)=Var(A)+Var(L)-2*rho(A,L)*vol A*vol L[we know Cov(a,b)= rho(a,b)*vol a*vol b]
As Volatilities are expressed in dollars so, vol A=%vol of A*value A, vol L=%vol of L*value L
Var(s)=Var(A-L)=( %vol of A*value A )^2 + ( %vol of L*value L )^2-2*rho(A,L)* %vol of A*value A, * %vol of L*value L
Thanks

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#### Bester

##### Member
Subscriber
Hi,

Is the surplus value (lower bound of the 95% confidence interval) in this question not the same as absolute SaR. Also why in this question is the Expected Surplus set equal to 9.7 only?

If I calculate the absolute SaR then the calculation in my view would be as follow:

Expected Surplus - (95% confidence factor *Volatility of Surplus) = 19.7 - (1.645*6.94) = 8.3
where Expected Surplus is 19.7 (100 - 90 + 9.7) ---- not 9.7 as per answer below
and
Variance of the surplus = 1002 * 10%2 + 902 * 5%2 - 2 * 100 * 90 * 10% * 5% * 0.8 = 48.25
Volatility of the surplus = 48.25 = 6.94 (same as answer below)

Question:
An analyst reports the following fund information to the advisor of a pension fund that currently invests in government and corporate bonds and carries a surplus of USD 10 million:

Pension Assets = 100 Pension Liabilities = 90 Amount (in USD million)

Expected Annual Growth 6% for Assets and 7% for Liabilities

Modified Duration 12 for Assets and 10 Liabilities

Annual Volatility of Growth 10% for Assets and 5% for Liabilities

To evaluate the sufficiency of the fund's surplus, the advisor estimates the possible surplus values at the end of one year. The advisor assumes that annual returns on assets and the annual growth of the liabilities are jointly normally distributed and their correlation coefficient is 0.8. The advisor can report that, with a confidence level of 95%, the surplus value will be greater than or equal to:

a. USD -11.4 million
b. USD -8.3 million
c. USD -1.7 million
d. USD 0 million

Rationale: The lower bound of the 95% confidence interval is equal to: Expected Surplus - (95% confidence factor *Volatility of Surplus).
The required variables can be calculated as follows:

Variance of the surplus = 1002 * 10%2 + 902 * 5%2 - 2 * 100 * 90 * 10% * 5% * 0.8 = 48.25
Volatility of the surplus = 48.25 = 6.94,
The expected surplus = 100 * 1.06 - 90 * 1.07 = 9.7.
Therefore, the lower bound of the 95% confidence interval = 9.7 - 1.645 * 6.94 = -1.725

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#### Matthew Graves

##### Active Member
Subscriber
Not sure I follow your calculation of the expected surplus. Your calculation seems to assume that you would be receiving at the end of the year another 100*1.06 assets with additional liabilities of 90*1.07, giving total assets of 206 and total liabilities of 186.3 with a surplus of 19.7.

The question only specifies that the assets and liabilities increase by the respective growth rates, hence at the end of the year assets = 106 and liabilities = 96.3 for a surplus of 9.7, as in the answer.

#### Bester

##### Member
Subscriber
Hi thanks for the response.

How I calculated the Expected Surplus is as follow. Before any growth in the 1st year the surplus is 100 - 90 = 10. Then the assets and liabilities grow by the respective growth rates (106 - 96.3= 9.7). Thus the total new expected surplus (including growth) is 10 + 9.7 = 19.7. This is similar to the example in the Bionic Turtle study notes. See example page 24 Bionic Turtle notes on Jorion Portfolio Risk.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Bester The question gives expected annual growth of assets as 6% and expected annual growth of liabilities as 7%, so the expected dollar growth of assets is 6%*$100 = +6.0 and the expected dollar growth of liabilities is 7%*90 = +6.30, so the expected surplus at the end of the year = current surplus + change in surplus =$10 + (6 - 6.3) = $10 - 0.3 =$9.7 million. The situation here is:
• Let S = surplus = assets - liabilities, such that current S = 100 - 90 = +10 mm
• expected S at end of the year = expected end-of-year assets - expected end-of-year liabilities = (100*1.06) - (90 * 1.07) = 106 - 96.3 = 9.7 mm
• 95% relative surplus as risk; i.e., worst expected loss relative to expected surplus at end of year = 6.95*1.645 ~= 11.43
• 95% absolute SaR = 9.7 - 11.43 = -1.725, per the answer.
• However, there is a longstanding definitional ambiguity with respect to SaR: it is valid also to define SaR as -0.3 - 11.43 = -11.73, as this represents the worst expected loss in surplus relative to our initial surplus of +$10. #### Arnaudc ##### Member Hi all, This is probably a really dumb question but why in this formula: Variance of the surplus = 1002 * 10%2 + 902 * 5%2 - 2 * 100 * 90 * 10% * 5% * 0.8 = 48.25 do use the "-" sign? In all the VaR formula for 2 assets we have VaR² A + VaR² B "+" CorrAB x VaR A x VaR B Could someone explain what am I missing here? #### Nicole Seaman ##### Director of FRM Operations Staff member Subscriber Hi all, This is probably a really dumb question but why in this formula: Variance of the surplus = 1002 * 10%2 + 902 * 5%2 - 2 * 100 * 90 * 10% * 5% * 0.8 = 48.25 do use the "-" sign? In all the VaR formula for 2 assets we have VaR² A + VaR² B "+" CorrAB x VaR A x VaR B Could someone explain what am I missing here? Hello @Arnaudc Please notice that I moved your question here to this thread, where this has been discussed above Nicole #### Arnaudc ##### Member @Nicole Seaman , Thank you Nicole. Furthermore, I found my answer in a nice video made by David: This is because we are looking at a difference and not an addition as we usually do with VaR of 2 assets... Excellent to realize that before the exam #### bpdulog ##### Active Member 5. An analyst reports the following fund information to the advisor of a pension fund that currently invests in government and corporate bonds and carries a surplus of USD 10 million: Pension Assets Pension Liabilities Amount (in USD million) 100 90 Expected Annual Growth 6% 7% Modified Duration 12 10 Annual Volatility of Growth 10% 5% To evaluate the sufficiency of the fund's surplus, the advisor estimates the possible surplus values at the end of one year. The advisor assumes that annual returns on assets and the annual growth of the liabilities are jointly normally distributed and their correlation coefficient is 0.8. The advisor can report that, with a confidence level of 95%, the surplus value will be greater than or equal to: a. USD -11.4 million b. USD -8.3 million c. USD -1.7 million d. USD 0 million Correct Answer: c Rationale: The lower bound of the 95% confidence interval is equal to: Expected Surplus - (95% confidence factor * Volatility of Surplus). The required variables can be calculated as follows: Variance of the surplus = 1002 * 10%2 + 902 * 5%2 - 2 * 100 * 90 * 10% * 5% * 0.8 = 48.25 Volatility of the surplus = √48.25 = 6.94, The expected surplus = 100 * 1.06 - 90 * 1.07 = 9.7. Therefore, the lower bound of the 95% confidence interval = 9.7 - 1.645 * 6.94 = -1.725 Hi, Does any one know why the third term in the surplus variance formula is being subtracted? This is usually added. #### Nicole Seaman ##### Director of FRM Operations Staff member Subscriber 5. An analyst reports the following fund information to the advisor of a pension fund that currently invests in government and corporate bonds and carries a surplus of USD 10 million: Pension Assets Pension Liabilities Amount (in USD million) 100 90 Expected Annual Growth 6% 7% Modified Duration 12 10 Annual Volatility of Growth 10% 5% To evaluate the sufficiency of the fund's surplus, the advisor estimates the possible surplus values at the end of one year. The advisor assumes that annual returns on assets and the annual growth of the liabilities are jointly normally distributed and their correlation coefficient is 0.8. The advisor can report that, with a confidence level of 95%, the surplus value will be greater than or equal to: a. USD -11.4 million b. USD -8.3 million c. USD -1.7 million d. USD 0 million Correct Answer: c Rationale: The lower bound of the 95% confidence interval is equal to: Expected Surplus - (95% confidence factor * Volatility of Surplus). The required variables can be calculated as follows: Variance of the surplus = 1002 * 10%2 + 902 * 5%2 - 2 * 100 * 90 * 10% * 5% * 0.8 = 48.25 Volatility of the surplus = √48.25 = 6.94, The expected surplus = 100 * 1.06 - 90 * 1.07 = 9.7. Therefore, the lower bound of the 95% confidence interval = 9.7 - 1.645 * 6.94 = -1.725 Section: Risk Management and Investment Management Reference: Philippe Jorion, Value-at-Risk: The New Benchmark for Managing Financial Risk, 3rd Edition, Chapter 17, “VaR and Risk Budgeting in Investment Management.” Learning Objective: Distinguish among the following types of risk: absolute risk, relative risk, policy-mix risk, active management risk, funding risk, and sponsor risk. Hi, Does any one know why the third term in the surplus variance formula is being subtracted? This is usually added. Hello @bpdulog Please note that I have moved your question here to this thread, where this question is already being discussed. Please make sure to use our search box, and search for the first sentence in the question before posting a new thread somewhere else in the forum. Many times, these questions have already been posted in the forum. This helps to keep our forum organized and uncluttered so our members can search for answers to their questions easily without having to read through a bunch of different threads on the same question. Thank you, Nicole #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber @bpdulog because surplus = assets - liabilities, not asset + liabilities, so this is an application of variance(A-B) or in this case variance(S) = variance (A -L) = variance(A) + variance(L) - 2*σ(A)*σ(L)*ρ(A,L). I hope that explains! #### raghav159 ##### New Member @David Harper CFA FRM @Nicole Seaman A Similar Question appeared in 2018 Practice but the Z-Normal Deviate used in the solution was 2-tailed. Having difficulty understanding whether it should be a 1-tailed or 2-tailed CI for 95% Surplus . To make things worse the choices have options for both 1-tailed & 2-tailed deviates. Q. 24 An analyst reports the following fund information to the advisor of a pension fund that currently invests in government and corporate bonds and carries a surplus of USD 40 million: Assets - 180 million Growth 6% Volatility 25% Liabilities - 140 million Growth 10% Volatility 12% To evaluate the sufficiency of the fund's surplus, the advisor estimates the possible surplus values at the end of 1 year. The advisor assumes that annual returns on assets and the annual growth of the liabilities are jointly normally distributed and their correlation coefficient is 0.68. Assume that the volatility of surplus in dollar is USD 35.76 million, what is the lower bound of the 95% confidence interval for the expected end-of-year surplus that the advisor can report? A) -$ 76.4 million
B) - $58.2 million C) -$ 33.3 million