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IAS282 2022 Semester Test 2 Memo

This is used in IAS 282 an actuarial science module at TUKS/UP during...
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Actuarial Science (IAS 282)

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IAS 282

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Financial Mathematics 282

2022 Semester test 2 Memo

Question 1 [9 marks]

a) Working in millions:

𝑣@𝑖2 (1𝑎1|@𝑖 + 3𝑎1|@𝑖𝑣@𝑖 1 + 4𝑎1|@𝑖𝑣@𝑖 2 + 6𝑎1|@𝑖𝑣@𝑖 3 +... +21𝑎1|@𝑖𝑣@𝑖 13 + 22𝑎1|@𝑖𝑣@𝑖 14 )

− 35 − 12𝑣@𝑖0 = 0 𝑣@𝑖2(1)(𝑙𝑎)

15| @𝑖

− 35 − 12𝑣@𝑖0.

= 0 𝑣@𝑖2 × 1.

𝑎̈15|@𝑖 − 15 𝑣@𝑖 15

𝛿

− 35 − 12𝑣@𝑖0 = 0𝑓(𝑖) = 0𝑓(10%)

= 12... → 𝐴𝑓(15%) = −9... → 𝐵𝑈𝑠𝑖𝑛𝑔 𝑙𝑖𝑛𝑒𝑎𝑟 𝑖𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛: 𝑖

= 10% +

0 − 𝑓(10%)

𝑓(15%) − 𝑓(10%)

(15% − 10%)𝑖 ≈ 12%

Examiner comments: This question was answered reasonably well. A few students confused the

“once off increase being paid continuously” with a “continuous increasing rate of payment”. Some

students also confused the period at which the income started.

b) IRR is the interest rate at which NPV=0. A higher IRR corresponds with a higher NPV,

i. as IRR increases, NPV also increases.

Examiner comments: This question was not answered well. A surprising number of students stated

no relationship or that if the IRR increases the NPV will decrease. In this case students didn't get any

marks (irrespective of the rest of their answer) as they showed they do not know the answer we

required for this question. We were specifically interested in the students knowing that NPV increases

as IRR increases.

c) At t=8, NPV=0 so AV=0. Since the loan is repaid by a once-off payment, this implies

that the loan has already been paid and the only cashflow after 8 years is the

continuous income stream.

𝑃𝑉𝑡=8 = 9𝑎0|@𝑖

+ 𝑣@𝑖0 (10𝑎1|@𝑖 + 12𝑎1|@𝑖𝑣@𝑖 1 + 13𝑎1|@𝑖𝑣@𝑖 2 +... +21𝑎1|@𝑖𝑣@𝑖 7

+ 22𝑎1|@𝑖𝑣@𝑖 8 ) 𝑤ℎ𝑒𝑟𝑒 𝑖 = 8%

= 9𝑎0|@8% + 𝑣@8%0 (9𝑎9|@8% + (1)(𝑙𝑎)

9| @8%

) 𝑎0|@8% = 1 − 𝑣

0.

𝛿

= 0... → 𝐴𝑎9|@8% = 1 − 𝑣

9

𝛿

= 6... → 𝐵(𝑙𝑎)9| @8% =

𝑎̈9| − 9 𝑣 9

𝛿

= 29... → 𝐶 ∴ 𝑃𝑉𝑡=8.

= 9 × 𝐴 + 𝑣@8%0 (9 × 𝐵 + 1 × 𝐶) = 102...

𝑇ℎ𝑢𝑠 𝐴𝑉𝑡=17 = 𝑃𝑉𝑡=8 × (1)9 = 209 million

Examiner comments: Students didn’t perform well in this question. Stronger candidates managed to

get full marks. Students lost marks for using incorrect periods, amounts, annuity functions or interest

rates. Some students didn't attempt the question or gave over-complicated answers by trying to

calculate the PV at t=0 ignoring the information implicating that PV@10%=0 for t=8.

Question 3 [6 marks]

a) The existence of capital gains tax will not affect the decision of when to redeem the

bond. An investment which has a capital gain before allowing for capital gains tax must

still have a net capital gain after allowing for the capital gains tax liability, so that the

"worst case" for the investor is still the latest redemption.

Examiner comments: This question was not answered well. Many students did not understand the

question and a few made no reference to capital gains tax. A full mark was awarded for students

stating "no impact" while in some cases half a mark was given if the student stated that the investor

would choose the earliest possible date (this mark was not awarded if the student changed the

answer to say that tax would lead to the latest redemption date). Some students stated the investor

would choose the worst possible scenario which is not correct.

b) Worst case scenario for borrower will be the earliest redemption date = In 10 years’ time

Assuming tax is payable immediately when income is earned

𝑑(4) = 4(−1−0 + 1) = 0 ... 𝐴𝑄𝑢𝑎𝑟𝑡𝑒𝑟𝑙𝑦 𝑐𝑜𝑢𝑝𝑜𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 = 0(1000)

4

= 16𝑇ℎ𝑖𝑠 𝑖𝑛𝑐𝑜𝑚𝑒 𝑓𝑎𝑙𝑙𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑅10 − 𝑅20 𝑏𝑎𝑛𝑑 𝑠𝑜 𝑞𝑢𝑎𝑟𝑡𝑒𝑟𝑙𝑦 𝑡𝑎𝑥

= 𝑅0 + 6%(6) = 1𝑇ℎ𝑢𝑠 𝑞𝑢𝑎𝑟𝑡𝑒𝑟𝑙𝑦 𝑡𝑎𝑥 𝑟𝑎𝑡𝑒 =

1.

16.

= 8%

∴ 6%

100%

(1 − 8%) = 0 ... 𝐵𝑆𝑖𝑛𝑐𝑒 𝑑(4) < 𝑑

𝑔

(1 − 𝑡 1 ), 𝑡ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 𝑙𝑜𝑠𝑠.

Worst case scenario for borrower will be the earliest redemption date = In 10 years’ time

𝑃 = 65𝑎̈10|@6%

(4)

− 0(4)𝑎̈10|@6%

(4)

− 0(6)(4)𝑎̈10|@6%

(4)

+ 1000𝑣@6% 10 𝑎̈10|@6%

(4)

= 1 − 𝑣

10

𝑑(4)

= 7 ... 𝐶 ∴ 𝑃 = (65 − 3 − 1) × 𝐶 + 1000(1)−

= 𝑅1014.

Examiner comments: This question was answered poorly with stronger candidates managing to get

full marks. Many students lost marks as they didn't calculate the tax correctly. Some lost marks for

assuming capital gains or not performing/applying the capital gains test correctly. Some lost marks

for not using the correct term fitting the worst case scenario or because the final answer is not in line

with the capital gain/loss assumption made earlier on.

4.) Gross yield is unaffected. This is because the gross yield is unaffected by capital gains tax.

Net yield will increase. This is because the tax exemption will reduce the capital gains tax and

increase the net value of the capital gain. Therefore, it will increase the net yield earnt.

Examiner comments: This question was answered well, with majority of candidates scoring full

marks. Note that the half-mark for the gross yield portion was only awarded if it was justified why

the gross yield is unaffected – many candidates did not provide any justification.

4.) Gross yield is decrease. The gross value of rental income in the last 5 years will be lower, so the

gross yield earnt will decrease.

Net yield will decrease. The gross value of rental income in the last 5 years will be lower, so the net

value will also be lower, and so the net yield will decrease.

Examiner comments: Some candidates struggled with this question. Many assumed that the

reduction in income will be negated by the “saving” on income tax when considering the impact of

the net yield – however, this is flawed as the reduction in income is only partially offset by the tax

saving as only 30% of income is taxable.

4.)

𝐿𝑒𝑡 𝑖 = 𝑅𝑒𝑎𝑙 𝑛𝑒𝑡 𝑦𝑖𝑒𝑙𝑑.

𝑃𝑉 = 500,000 50

52.

𝑣 1 @𝑖 − 0(400,000) 50

52.

𝑣 1 @𝑖 − 0(35,000,000) 50

59.

𝑣 11 @𝑖

Examiner comments: Stronger candidates managed to score full marks for this question. The most

common mistake was to assume the third cashflow occurs at time 1 or 2 whereas it should occur at

time 1 since biannual cashflows in advance is considered.

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IAS282 2022 Semester Test 2 Memo

Course: Actuarial Science (IAS 282)

12 Documents
Students shared 12 documents in this course

University: University of Pretoria

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IAS282 Financial Mathematics Semester Test 2 2022 Memo
Financial Mathematics 282
2022 Semester test 2 Memo
Question 1 [9.5 marks]
a) Working in millions:
𝑣@𝑖
2.5 (1.5𝑎1|@𝑖 +3𝑎1|@𝑖𝑣@𝑖
1+4.5𝑎1|@𝑖𝑣@𝑖
2+6𝑎1|@𝑖𝑣@𝑖
3+...+21𝑎1|@𝑖𝑣@𝑖
13 +22.5𝑎1|@𝑖𝑣@𝑖
14)
−35−12𝑣@𝑖
0.5 = 0 𝑣@𝑖
2.5(1.5)(𝑙𝑎)15| @𝑖 −35−12𝑣@𝑖
0.5
= 0 𝑣@𝑖
2.5 ×1.5𝑎󰇘15|@𝑖 −15 𝑣@𝑖
15
𝛿−35−12𝑣@𝑖
0.5 = 0𝑓(𝑖)= 0𝑓(10%)
=12.7850...→ 𝐴𝑓(15%)= −9.2521...→ 𝐵𝑈𝑠𝑖𝑛𝑔 𝑙𝑖𝑛𝑒𝑎𝑟 𝑖𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛:𝑖
=10%+0−𝑓(10%)
𝑓(15%)−𝑓(10%)(15%−10%)𝑖 ≈ 12.9%
Examiner comments: This question was answered reasonably well. A few students confused the
“once off increase being paid continuously” with a “continuous increasing rate of payment”. Some
students also confused the period at which the income started.
b) IRR is the interest rate at which NPV=0. A higher IRR corresponds with a higher NPV,
i.e. as IRR increases, NPV also increases.
Examiner comments: This question was not answered well. A surprising number of students stated
no relationship or that if the IRR increases the NPV will decrease. In this case students didn't get any
marks (irrespective of the rest of their answer) as they showed they do not know the answer we
required for this question. We were specifically interested in the students knowing that NPV increases
as IRR increases.
c) At t=8.2, NPV=0 so AV=0. Since the loan is repaid by a once-off payment, this implies
that the loan has already been paid and the only cashflow after 8.2 years is the
continuous income stream.

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