Lutions to the November 2005 Course FM/2 Examination 1 ...

Copyright ? 2005 by Krzysztof Ostaszewski. All rights reserved. Actuarial examinations are copyrighted by the Society of Actuaries and the Casualty Actuarial Society and are reproduced here with permission.

Krzys' Ostaszewski's (not official) solutions to the November 2005 Course FM/2 Examination

1. November 2005 Course FM/2 Examination, Problem No. 1. An insurance company

earned a simple rate of interest of 8% over the last calendar year based on the following

information:

Assets, beginning of year

25,000,000

Sales revenue

X

Net investment income

2,000,000

Salaries paid

2,200,000

Other expenses paid

750,000

All cash flows occur at the middle of the year. Calculate the effective yield rate.

A. 7.7%

B. 7.8%

C. 7.9%

D. 8.0%

E. 8.1%

Solution. You are supposed to guess that net investment income is not a cash flow, while only these accounting entries: sales revenue, salaries paid, and other expenses paid, contribute to the cash flow. On the other hand, you are supposed to include net investment income as income obtained at the end of the year (not before, as it apparently did not result in a cash flow). The net cash flow occurring in the middle of the year is

X 2, 200, 000 750, 000 = X 2, 950, 000.

Therefore, the rate of return based on simple interest, i.e., the dollar-weighted rate of

return, is

Year-end investment

Middle-year cashflow income

X 2, 950, 000 + 2, 000, 000

= 0.08.

( ) 25, 0 00, 000

Assets BOY, exposed for the whole year to earning interest

1+

X 2,950,000

Middle-year cash flow, exposed to earning interest for half a year

1 2

+

2, 000,000

Year-end investment income, exposed to earning interest for 0 years, i.e., not exposed

0

As 8% of 25 million is 2 million, this results in

2, 000, 000 + 0.04 ( X 2, 950, 000) = X 950, 000,

or

0.04 ( X 2, 950, 000) = 1 ( X 2, 950, 000).

But we know that 0.04 is not equal to one, so unless X = 2,950,000, the above equation results in a contradiction. Hence X = 2,950,000 and the middle-year cash flow is zero. This means that the initial 25,000,000 bring 2,000,000 of income during the year, resulting in the effective yield rate of 8%. Answer D.

2. November 2005 Course FM/2 Examination, Problem No. 2. Calculate the Macaulay duration of an eight-year 100 par value bond with 10% annual coupons and an effective

Study Manual for Course FM/2, Copyright ? 2005-2006 by Krzysztof Ostaszewski

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rate of interest equal to 8%.

A. 4

B. 5

C. 6

D. 7

E. 8

Solution.

The Macaulay duration of this bond equals

( ) 10 v81% + 20 v82%

10 v81% + 10 v82%

+ ... + 80 v88% + ... + 10 v88%

+ 800 v88% + 100 v88%

=

10 Ia 8 8% + 800 v88%

10

a 8

8%

+

100

v88%

=

=

10

a 8

8%

8v88%

0.08

+

800 v88%

10

a 8

8%

+ 100

v88%

667.74 111.49

5.989236702843.

Answer C.

3. November 2005 Course FM/2 Examination, Problem No. 3. An investor accumulates a fund by making payments at the beginning of each month for 6 years. Her monthly payment is 50 for the first 2 years, 100 for the next 2 years, and 150 for the last 2 years. At the end of the 7th year the fund is worth 10,000. The annual effective interest rate is i, and the monthly effective interest rate is j. Which of the following formulas represents the equation of value for this fund accumulation?

( ) A. s (1 + i) (1 + i)4 + 2(1 + i)2 + 3 = 200 24 i

( ) B. s (1 + j) (1 + j)4 + 2(1 + j)2 + 3 = 200 24 i

( ) C. s (1 + i) (1 + i)4 + 2(1 + i)2 + 3 = 200 24 j

( ) D. s (1 + i) (1 + i)4 + 2(1 + i)2 + 3 = 200 24 j

( ) E. s (1 + j) (1 + j)4 + 2(1 + j)2 + 3 = 200 24 i

Solution.

After 24 months, the account will accumulate to 50s .

24 j

After additional 24 months, the accumulation will be

50s (1 + i)2 + 100s .

24 j

24 j

After additional 24 months, the accumulated value will be

( ) 50s (1 + i)4 + 100s (1 + i)2 + 150s = 50s (1 + i)4 + 2 (1 + i)2 + 3 .

24 j

24 j

24 j

24 j

After one more year, at the end of the seventh year, the value will be

( ) 50s (1 + i) (1 + i)4 + 2(1 + i)2 + 3 , 24 j

and we know this to be 10,000. Therefore, the equation of value is

Study Manual for Course FM/2, Copyright ? 2005-2006 by Krzysztof Ostaszewski

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( ) s (1 + i) (1 + i)4 + 2(1 + i)2 + 3 = 10,000 = 200.

24 j

50

Answer C.

4. November 2005 Course FM/2 Examination, Problem No. 4. A ten-year 100 par value bond pays 8% coupons semiannually. The bond is priced at 118.20 to yield an annual nominal rate of 6% convertible semiannually. Calculate the redemption value of the bond.

A. 97

B. 100

C. 103

D. 106

E. 109

Solution.

Using the Frank formula

P = Fr a + K = Fr a + C vn,

n

n

so that with the values given

118.20 = 4 a + C 1.0320. 20 3%

This can be solved using a financial calculator with n = 20, i = 3%, PV = 118.20,

PMT = 4, resulting in C 106.00. Alternatively,

C = 118.20 1.0320 4 s 106.00. 20 3%

Answer D.

5. November 2005 Course FM/2 Examination, Problem No. 5. Alex is an investment analyst for a large fund management firm. He specializes in finding risk-free arbitrage opportunities in the stock market. His strategy consists of selling a specific number of call options for each share of stock selected in the fund. Which of the following best describes the technique used by Alex to achieve his goal?

A. Black-Scholes option pricing model B. Capital Asset Pricing Model C. Full immunization D. Short sales E. Hedge ratio

Solution. This question is very inappropriate. Nearly everything in the question and in the possible answers is not on the syllabus for this examination. Writing options against long stock position is not a risk-free arbitrage, unless the position is continuously adjusted (i.e., only in theoretical finance). Anyway, let's go over the answers: A. Black-Scholes option pricing model: This is not on the syllabus of Course FM/2. My guess is that the "original intent" of the question creator was that you would see this and realize it is not in the syllabus, and reject this answer. Same as if the answer were: The Easter Bunny. In any case, the Black-Scholes option pricing model could mean modeling

Study Manual for Course FM/2, Copyright ? 2005-2006 by Krzysztof Ostaszewski

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options stochastically (general concept of a Black-Scholes option pricing model), or the Black-Scholes option pricing formula. The formula can actually be used to derive a hedge ratio in hedging stock position with options and vice versa. So this answer does describe the technique used by Alex to achieve his goal, it just does not do it the best way. And it is an Easter Bunny answer, it was not covered in the syllabus. B. Capital Asset Pricing Model. This is yet another Easter Bunny answer. CAPM is not on the syllabus. CAPM is the theory that gives the expected rate of return on a stock in a one-period model of a market in relation to the risk-free rate of return and the rate of return on the entire market portfolio. Not here. C. Full immunization is on the Course FM/2 syllabus. It refers to the technique of interest rate risk management in which asset portfolio duration is set equal to the liabilities portfolio duration, while asset portfolio convexity exceeds the convexity of the liabilities portfolio, and assets value equals or exceeds the value of liabilities. Not here. D. Short sales. As short sales are presented in the Course FM/2 syllabus, you would not consider anything in this question to involve short sales. However, Alex has a short position in options, and he is selling them. I know, I know, these are not really "short sales," as defined in the approved Course FM/2 textbooks. So this is not the right answer, either. E. Hedge ratio. Buying a security, whose values move in opposite direction of that of the currently held portfolio, is called hedging. It is a risk-management strategy. It is mentioned, barely, in approved textbooks. The ratio of the number of units of the security bought to the number of units of the security currently held is called the hedge ratio. This concept is not anywhere in any of the textbooks approved, or in the syllabus. And it is the right answer here, as the short calls position will move in opposite direction of the movements of the underlying stock, and in order to specify the number of option contracts to be written, Alex must calculate the hedge ratio. So this answer makes the most sense. Answer E.

6. November 2005 Course FM/2 Examination, Problem No. 6. Consider a yield curve defined by the following equation:

ik = 0.09 + 0.002k 0.001k2, where ik is the annual effective rate of return for zero coupon bonds with maturity of k years. Let j be the one-year effective rate during year 5 that is implied by this yield curve. Calculate j.

A. 4.7%

B. 5.8%

C. 6.6%

D. 7.5%

E. 8.2%

Solution.

We need the effective rate of return over the fifth year, i.e.,

a(5) a(4)

1

=

(1 + (1 +

)i5 5 )i4 4

1.

This equals

Study Manual for Course FM/2, Copyright ? 2005-2006 by Krzysztof Ostaszewski

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(1 + 0.09 + 0.01 0.025)5 (1 + 0.09 + 0.008 0.016)4

1=

1.0755 1.0824

1

4.744994%.

Answer A.

7. November 2005 Course FM/2 Examination, Problem No. 7. A bank offers the following choices for certificates of deposit:

Term (in years)

1 3 5

Nominal annual interest rate convertible quarterly 4.00% 5.00% 5.65%

The certificates mature at the end of the term. The bank does NOT permit early withdrawals. During the next 6 years the bank will continue to offer certificates of deposit with the same terms and interest rates. An investor initially deposits 10,000 in the bank and withdraws both principal and interest at the end of 6 years. Calculate the maximum annual effective rate of interest the investor can earn over the 6-year period.

A. 5.09% B. 5.22% C. 5.35% D. 5.48% E. 5.61%

Solution.

The investor has the following options:

?Deposit the funds in one-year certificates every year for six years, producing an annual

effective rate of return of

1

1 +

0.04 4

64

6

1

=

1.014

1

4.060401%.

?Deposit the funds for three years at 5% per year convertible quarterly, followed by three

one-year deposits at 4% convertible quarterly, resulting in

1

1

+

0.05 34 4

1

+

0.04 4

4 3

6

1

4.576189%.

?Deposit the funds for three years at 5% per year convertible quarterly, followed by the

same deposit for another three years, resulting in an annual effective rate of return of

1

1

+

0.05 4

34

2

6

1

=

1.014

1

5.094534%.

?Deposit the funds for five years, and then for one year, or deposit funds for one year and then for five years, producing the same annual effective rate of return of

Study Manual for Course FM/2, Copyright ? 2005-2006 by Krzysztof Ostaszewski

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1

1

+

0.0565 54 4

1 +

0.04 4 4

6

1

5.483827%.

The last one is the best choice. Given the choices, this was to be expected: you could

have guessed intuitively that this is the highest effective rate you can get. The most

effective way to help develop that guess is to calculate the effective annual rate of return

on each of the available certificates:

?One-year

certificate:

1 +

0.04 4 4

1

4.060401%.

?Three-year

certificate:

1 +

0.05 4 4

1

5.094534%.

?Five-year

certificate:

1 +

0.0565 4 4

1

5.770841%.

Clearly, getting roughly 5.77% per year for five years makes up for getting 4.06% for just one year versus the second best of getting 5.09% per year. Answer D.

8. November 2005 Course FM/2 Examination, Problem No. 8. Matthew makes a series of payments at the beginning of each year for 20 years. The first payment is 100. Each subsequent payment through the tenth year increases by 5% from the previous payment. After the tenth payment, each payment decreases by 5% from the previous payment. Calculate the present value of these payments at the time the first payment is made using an annual effective rate of 7%.

A. 1375

B. 1385

C. 139

D. 1405

E. 1415

Solution.

The present value of the first ten payments is

100 + 100 1.05 1.07

+

...

+

100

1.05 1.07

9 9

=

100

1

1.05 1.07

1 1.05

10

.

1.07

The present value of the final ten payments is

100

1.059 0.95 1.0710

+

100

1.059 0.95 1.0711

2

+

...

+

100

1.059 0.9510 1.0719

=

=

100

1.059 0.95 1.0710

1

0.95 1.07

1 0.95

10

.

1.07

The sum of the two is

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