FM SOA Exam May 05 questions fall 05 - bpptraining.com SOA Exam May 05.pdfExam FM May 2005 Exam - Questions 2 Question 4 An estate provides a perpetuity with payments of X at the end

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SOA Exam FM / CAS Exam 2

May 2005 Exam Questions Question 1 Which of the following expressions does NOT represent a definition for na ?

A + −

(1 ) 1nn iv

i

B −1 nvi

C + + +2 nv v v

D −

−

11

nvvv

E −+ 1(1 )n

ns

i

Question 2 Lori borrows $10,000 for 10 years at an annual effective interest rate of 9%. At the end of each year, she pays the interest on the loan and deposits the level amount necessary to repay the principal to a sinking fund earning an annual effective interest rate of 8%.

The total payments made by Lori over the 10-year period is X.

Calculate X.

A 15,803 B 15,853 C 15,903 D 15,593 E 16,003

Question 3 A bond will pay a coupon of $100 at the end of each of the next three years and will pay the face value of $1,000 at the end of the three-year period. The bond’s duration (Macaulay duration) when valued using an annual effective interest rate of 20% is X.

Calculate X.

A 2.61 B 2.70 C 2.77 D 2.89 E 3.00

Exam FM May 2005 Exam - Questions

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Question 4 An estate provides a perpetuity with payments of X at the end of each year. Seth, Susan, and Lori share the perpetuity such that Seth receives the payment of X for the first n years and Susan receives the payments of X for the next m years, after which Lori receives all the remaining payments of X.

Which of the following represents the difference between the present value of Seth’s and Susan’s payments using a constant rate of interest?

A − n

n mX a v a

B − n

n mX a v a

C + − 1n

n mX a v a

D − − 1n

n mX a v a

E + − 1n

n mX va v a

Question 5 Susan can buy a zero-coupon bond that will pay $1,000 at the end of 12 years and is currently selling for $624.60. Instead, she purchases a 6% bond with coupons payable semi-annually that will pay $1,000 at the end of 10 years. If she pays X, she will earn the same annual effective interest rate as the zero-coupon bond.

Calculate X.

A 1,164 B 1,167 C 1,170 D 1,173 E 1,176

Question 6 John purchased three bonds to form a portfolio as follows:

• Bond A has semi-annual coupons at 4%, a duration of 21.46 years, and was purchased for $980.

• Bond B is a 15-year bond with a duration of 12.35 years and was purchased for $1,015.

• Bond C has a duration of 16.67 years and was purchased for $1,000.

Calculate the duration of the portfolio at the time of purchase.

A 16.62 years B 16.67 years C 16.72 years D 16.77 years E 16.82 years

May 2005 Exam - Questions Exam FM

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Question 7 Mike receives cash flows of $100 today, $200 in one year, and $100 in two years. The present value of these cash flows is $364.46 at an annual effective rate of interest i.

Calculate i.

A 10% B 11% C 12% D 13% E 14%

Question 8 A loan is being repaid with 25 annual payments of $300 each. With the 10th payment, the borrower pays an extra $1,000, and then repays the balance over 10 years with a revised annual payment. The effective rate of interest is 8%.

Calculate the amount of the revised annual payment.

A 157 B 183 C 234 D 257 E 383

Question 9 The present value of a series of 50 payments starting at $100 at the end of the first year and increasing by $1 each year thereafter is equal to X. The annual effective rate of interest is 9%.

Calculate X.

A 1,165 B 1,180 C 1,195 D 1,210 E 1,225

Question 10 Yield rates to maturity for zero-coupon bonds are currently quoted at 8.5% for one-year maturity, 9.5% for two-year maturity, and 10.5% for three-year maturity. Let i be the one-year forward rate for year two implied by current yields of these bonds.

Calculate i.

A 8.5% B 9.5% C 10.5% D 11.5% E 12.5%


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Question 11 A $1,000 par value bond pays annual coupons of $80. The bond is redeemable at par in 30 years, but is callable any time from the end of the 10th year at $1,050.

Based on her desired yield rate, an investor calculates the following potential purchase prices, P:

• Assuming the bond is called at the end of the 10th year, = $957P

• Assuming the bond is held until maturity, = $897P

The investor buys the bond at the highest price that guarantees she will receive at least her desired yield rate regardless of when the bond is called.

The investor holds the bond for 20 years, after which time the bond is called.

Calculate the annual yield rate the investor earns.

A 8.56% B 9.00% C 9.24% D 9.53% E 9.99%

Question 12 Which of the following are characteristics of all perpetuities?

I. The present value is equal to the first payment divided by the annual effective interest rate.

II. Payments continue forever.

III. Each payment is equal to the interest earned on the principal.

A I only B II only C III only D I, II, and III E The correct answer is not given by (A), (B), (C), or (D).

Question 13 At a nominal interest rate of i convertible semi-annually, an investment of $1,000 immediately and $1,500 at the end of the first year will accumulate to $2,600 at the end of the second year.

Calculate i.

A 2.75% B 2.77% C 2.79% D 2.81% E 2.83%


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Question 14 An annuity-immediate pays $20 per year for 10 years, then decreases by $1 per year for 19 years. At an annual effective interest rate of 6%, the present value is equal to X.

Calculate X.

A 200 B 205 C 210 D 215 E 220

Question 15 An insurance company accepts an obligation to pay $10,000 at the end of each year for 2 years. The insurance company purchases a combination of the following two bonds at a total cost of X in order to exactly match its obligation:

(i) 1-year 4% annual coupon bond with a yield rate of 5%

(ii) 2-year 6% annual coupon bond with a yield rate of 5%.

Calculate X.

A 18,564 B 18,574 C 18,584 D 18,594 E 18,604

Question 16 At the beginning of the year, an investment fund was established with an initial deposit of $1,000. A new deposit of $1,000 was made at the end of 4 months. Withdrawals of $200 and $500 were made at the end of 6 months and 8 months, respectively. The amount in the fund at the end of the year is $1,560.

Calculate the dollar-weighted (money-weighted) yield rate earned by the fund during the year.

A 18.57% B 20.00% C 22.61% D 26.00% E 28.89%

Question 17 At an annual effective interest rate of i, the present value of a perpetuity-immediate starting with a payment of $200 in the first year and increasing by $50 each year thereafter is $46,530.

Calculate i.

A 3.25% B 3.50% C 3.75% D 4.00% E 4.25%


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Question 18 A store is running a promotion during which customers have two options for payment. Option one is to pay 90% of the purchase price two months after the date of sale. Option two is to deduct X% off the purchase price and pay cash on the date of sale.

A customer wishes to determine X such that he is indifferent between the two options when valuing them using an effective annual interest rate of 8%.

Which of the following equations of value would the customer need to solve?

A + =

0.081 0.90100 6X

B − + =

0.081 1 0.90100 6X

C ( ) =

1 61.08 0.90100X

D =

1.08 0.90100 1.06X

E ( ) − =

1 61 1.08 0.90100X

Question 19 Calculate the nominal rate of discount convertible monthly that is equivalent to a nominal rate of interest of 18.9% per year convertible monthly.

A 18.0% B 18.3% C 18.6% D 18.9% E 19.2%

Question 20 An investor wishes to accumulate $10,000 at the end of 10 years by making level deposits at the beginning of each year. The deposits earn 12% annual effective rate of interest paid at the end of each year. The interest is immediately reinvested at an annual effective interest rate of 8%.

Calculate the level deposit.

A 541 B 572 C 598 D 615 E 621


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Question 21 A discount electronics store advertises the following financing arrangement:

“We don’t offer you confusing interest rates. We’ll just divide your total cost by 10 and you can pay us that amount each month for a year.”

The first payment is due on the date of sale and the remaining eleven payments at monthly intervals thereafter.

Calculate the effective annual interest rate the store’s customers are paying on their loans.

A 35.1% B 41.3% C 42.0% D 51.2% E 54.9%

Question 22 On January 1, 2004, Karen sold stock A short for $50 with a margin requirement of 80%. On December 31, 2004, the stock paid a dividend of $2, and an interest amount of $4 was credited to the margin account. On January 1, 2005, Karen covered the short sale at a price of X, earning a 20% return.

Calculate X.

A 40 B 44 C 48 D 52 E 56

Question 23 The stock of Company X sells for $75 per share assuming an annual effective interest rate of i. Annual dividends will be paid at the end of each year forever.

The first dividend is $6, with each subsequent dividend 3% greater than the previous year’s dividend.

Calculate i.

A 8% B 9% C 10% D 11% E 12%


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Question 24 An annuity pays $1 at the end of each year for n years. Using an annual effective interest rate of i, the accumulated value of the annuity at time +( 1)n is $13.776. It is also known that

+ =(1 ) 2.476ni .

Calculate n.

A 4 B 5 C 6 D 7 E 8

Question 25 A bank customer takes out a loan of $500 with a 16% nominal interest rate convertible quarterly. The customer makes payments of $20 at the end of each quarter.

Calculate the amount of principal in the fourth payment.

A 0.0 B 0.9 C 2.7 D 5.2 E There is not enough information to calculate the amount of principal.

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SOA Exam FM / CAS Exam 2

May 2005 Exam Solutions Solution 1

E Answer choice (A) is an expression for the annuity-immediate present value factor and it is correct since:

nn na v s=

Answer choices (B) and (C) are also correct expressions. Answer choice (D) is also correct since the part in the brackets is another way to express the present value of an annuity-due:

− − −+ + + + = = =

−2 1 1 11

1

n nn

nv vv v v av d

We then have:

n na va=

Answer choice (E) is not a correct expression for the annuity-immediate present value factor. It should be:

(1 )

nn n

sa

i=

+

Solution 2 C The total payment for a sinking fund loan for any period is comprised of the service payment

(SP) and the sinking fund payment (SFP). Since Lori pays the interest due on the loan each year, the service payment equals the interest due on the loan:

SP 10,000(0.09) $900.00= =

The accumulated value of the sinking fund payments at the sinking fund interest rate must equal the loan value at the maturity of the loan. The annuity-immediate accumulated value factor is:

( )10

10 8%1.08 1

14.486560.08

s−

= =

The annual sinking fund payment is then:

10 8%

10,000SFP $690.29489s

= =

The total annual payment on the loan is:

+ = + =SP SFP 900.00 690.29489 $1, 590.29489

The total payments made over the 10-year period is:

1, 590.29489 10 $15,902.95X = × =

Exam FM May 2005 Exam – Solutions

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Solution 3 B Macaulay duration can be calculated directly:

2 3

2 3

100(1) 100(2) 1,100(3)2,131.944441.2 1.2 1.2 2.70100 100 1,100 789.35185

1.2 1.2 1.2

MacD+ +

= = =+ +

Solution 4 A Seth receives n annual payments of $X from time 1 year to time n years. The present value of

Seth’s annuity-immediate at time 0 is just:

nXa

Susan receives m annual payments of $X from time 1n + years to time n m+ years. Since the annuity-immediate present value factor is valued one period before the first cash flow, the present value at time 0 of Susan’s payments is:

nmXv a

The difference between the present value of Seth’s and Susan’s payments is:

nn mX a v a −

Solution 5 B We determine the annual effective interest rate from the price of the zero-coupon bond:

12

12

1,000624.60(1 )

1,000(1 ) 1.60102624.60

1 1.04000

Pi

i

i

= =+

+ = =

+ =

The annual effective interest rate is 4.0%. The semi-annual effective interest rate is then:

(2)

1/2(1.04) 1 0.019802

i= − =

The 6% coupon bond pays semi-annual coupons of:

0.06 1,000 $302

=

The annuity-immediate present value factor for 20 semi-annual periods at 1.980% semi-annual effective interest is:

20

20 1.980%1 (1.01980) 16.38242

0.01980a

−−= =

The price of the 6% coupon bond is:

2020 1.980%30 1,000(1.01980)

30(16.38242) 1,000(0.67556)1,167.04

P a −= +

= +=

May 2005 Exam – Solutions Exam FM

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Solution 6 D This is just a straightforward weighted average question. The total value of the portfolio is:

980 1,015 1,000 2,995+ + =

The duration of the portfolio is then:

× + × + × =980 1,015 1,00021.46 12.35 16.67 16.77

2,995 2,995 2,995

Solution 7 A The equation of value for the present value of the cash flows is:

2

2

364.46 100 200 100

100 200 264.46 0

v v

v v

= + +

+ − =

We can solve this equation for v using the quadratic equation:

2200 200 4 100 ( 264.46)

2 1000.90908 or 2.90908

v− ± − × × −

=×

= −

We discard the negative solution since it doesn’t make sense when determining an interest rate. We use v to solve for i:

1(1 ) 0.909081 1.10

0.10

v iii

−= + =+ =

=

Solution 8 C The initial loan payment is determined by the equation:

n i

LPa

=

The annuity-immediate present value factor is:

25

25 8%1 (1.08) 10.67478

0.08a

−−= =

The initial loan amount is:

30010.674783,202.43286

L

L

=

=

The outstanding loan balance immediately after the 10th loan payment by the retrospective method is:

= −

−= −

= −=

1010 10 8%

10

3, 202.43286(1.08) 300

(1.08) 13, 202.43286(2.15893) 3000.08

6,913.81235 4,345.968742, 567.84361

B s


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An extra $1,000 is paid at this time, so the outstanding balance becomes:

− =2,567.84361 1,000 1,567.84361

This new balance is repaid over 10 years with a revised annual payment. The new payment is:

−

= = = −

1010 8%

1,567.84361 1,567.84361 233.651 (1.08)

0.08

Pa

Solution 9 D The first payment is $100 at time 1, and each subsequent annual payment increases by $1. There

are 50 payments, so the last payment of $149 occurs at time 50 years. This series of payments can be split into two parts consisting of a level annuity-immediate and an increasing annuity-immediate.

The first part is a level annuity-immediate in which payments of $99 are made at the end of each year for 50 years. The second part is a 50-year increasing annuity-immediate, where the first payment is $1 at time 1, and then each subsequent annual payment increases by $1.

The present value of both parts of this payment series is:

50 9% 50 9%99 ( )a Ia+

Determining the required values, we have:

50

50 9%

50 9%50

50 9%

1 (1.09) 10.961680.09

(1.09)(10.96168) 11.94823

11.94823 50(1.09)( ) 125.286750.09

a

a

Ia

−

−

−= =

= =

−= =

So the present value of the payment series is:

99(10.96168) 125.28675 1, 210.49X = + =

Solution 10 C or E The SOA originally scored answer choice (C) as the correct answer, but then it also scored answer

choice (E) as correct due to ambiguity regarding which forward rate is to be determined.

Answer choice (C): The one-year forward rate for year two can be interpreted as 1f , the one-year forward rate in effect from time 1 year to time 2 years, since the second year is from time 1 year to time 2 years. Forward rates can be determined from the yields of the zero-coupon bonds. In this case, we use the yield from the two-year zero-coupon bond since the yield on a two-year zero-coupon bond is the same as the two-year spot rate. We have:

+ = + +

= +

+ =

=

22 0 1

21

21

1

(1 ) (1 )(1 )

1.095 (1.085)(1 )

1.09511.085

10.51%

s f f

f

f

f


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Answer choice (E): The one-year forward rate for year two can also be interpreted as the one-year forward rate that starts at time two years. This forward rate is 2f , the one-year forward rate in effect from time 2 years to time 3 years. In this case, we use the yield from the 3-year zero-coupon bond since the yield on the 3-year zero-coupon bond is the same as the 3-year spot rate. Using the above result for 1f , we have:

+ = + + +

= +

+ =

=

33 0 1 2

32

32

2

(1 ) (1 )(1 )(1 )

1.105 (1.0850)(1.1051)(1 )

1.1051(1.0850)(1.1051)12.53%

s f f f

f

f

f

Solution 11 C We are given two potential bond purchase prices, either one of which can be used to determine

the desired yield rate. Assuming the bond is called after 10 years for the call price of $1,050, the equation of value for the bond price is:

1010957 80 1,050(1 )iP a i −= = + +

Using the TI BA-35 calculator, we can easily determine the yield. Press 957 [PV], 80 [PMT], 1,050 [FV], 10 [N], and then [CPT] [%i]. The desired yield is 9.0%.

Assuming the bond is held until maturity, the equation of value for the bond price is:

3030897 80 1,000(1 )iP a i −= = + +

Using the TI BA-35, press 897 [PV], 80 [PMT], 1,000 [FV], 30 [N], and then [CPT] [%i]. The desired yield is again 9.0%.

This investor buys the bond at the highest price that guarantees she will receive at least her desired yield of 9.0% regardless of when the bond is called. The bond is called after 20 years. The investor either purchased this bond for a price of $957 or $897. The price that provides a higher yield is the lower price, so she purchased the bond for $897.

Assuming she paid $897 for the bond, we can determine her actual yield. The equation of value assuming the bond is called after 20 years for the call price of $1,050 is:

2020897 80 1,050(1 )iP a i −= = + +

Using the TI BA-35, press 897 [PV], 80 [PMT], 1,050 [FV], 20 [N], and then [CPT] [%i]. Her actual yield is 9.24%.

If this investor had paid $957 for the bond, her yield would have been 8.56%, which is lower than her desired yield. Since she bought the bond at a price to guarantee at least a 9% yield, she must have paid $987 for the bond.

Solution 12 B Statement I is false. Statement I is not true for level perpetuities-due or variable perpetuities.

Statement II is true by definition.

Statement III is false. Statement III is not true for variable perpetuities.


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Solution 13 D The $1,000 deposit accumulates from time 0 to time 2 years and the $1,500 deposit accumulates

from time 1 year to time 2 years. Since the nominal interest rate i is convertible semiannually, let’s work in semiannual periods. So the first deposit accumulates for 4 semiannual periods and the second deposit accumulates for 2 semiannual periods. The equation of value is:

+ + + =

4 21,000 1 1,500 1 2,600

2 2i i

Let = +

21

2ix , and we can rewrite the equation of value so that we can solve for x using the

quadratic equation:

+ − =

− ± − × × −=

×= −

2

2

1,000 1,500 2,600 0

1,500 1,500 4 1,000 ( 2,600)2 1,000

1.02834 or 2.52834

x x

x

x

We discard the negative solution since it doesn’t make sense with interest rates. We can now solve for i:

= + =

+ =

=

21 1.02834

2

1 1.014072

2.81%

ix

i

i

Solution 14 E Payments of $20 occur at the end of each year from time 1 year to time 10 years, and then the

payments decrease by $1 each year so that a payment of $19 occurs at time 11 years, $18 at time 12 years, and so on, down to $1 at time 29 years.

If we split the payments into two parts, it will be easier to fit them to standard annuity factors. The first part is a 9-year annuity-immediate of $20, and the second part is a 20-year decreasing annuity-immediate in which the first payment occurs at time 10 years. The present value factor of the decreasing annuity-immediate is valued one year before the first cash flow, which is at time 9 years in this case, so this present value needs to be discounted for 9 years to bring it to time 0. The present value of both parts of the payment series is then:

+ 99 2020 ( )a v Da

Calculating the required values, we have:

−

−

−= =

−= =

−= =

9

9

20

20

20

1 (1.06) 6.801690.06

1 (1.06) 11.469920.0620 11.46992( ) 142.16798

0.06

a

a

Da

The present value of the payment series is:

−+ =920(6.80169) (1.06) (142.16798) 220.18


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Solution 15 D With the dedication investment strategy, we start matching liability and asset cash flows at the

end and work backwards to the beginning. At time 2 years, the liability obligation is $10,000. This amount needs to be matched at time 2 with $10,000 of asset cash flows.

Assuming a par value of $100, the 2-year bond pays $6 at time 1 year and $106 at time 2 years. To match the liability cash flow at time 2, we need to buy 94.33962 units of the 2-year bond:

=10,000 94.33962

106

Since we purchase 94.33962 units of the 2-year bond, it pays × =94.33962 6 566.03774 at time 1. So the net liability cash flow that needs to be matched at time 1 by the 1-year bond is

− =10,000 566.03774 9, 433.96226 .

Assuming a par value of $100, the 1-year bond pays $104 at time 1 year. To match this net liability cash flow at time 1, we need to buy 90.71118 units of the 1-year bond:

=9, 433.96226 90.71118

104

The combination of 94.33962 units of the 2-year bond and 90.71118 units of the 1-year bond exactly matches the liability cash flows at times 1 and 2. The prices of the 1-year and 2-year bonds are:

−

−

= =

= + =

1

2 2

104 99.047621.05

6 106 101.859411.05 1.05

yr

yr

P

P

The cost to buy 94.33962 units of the 2-year bond and 90.71118 units of the 1-year bond is:

× + × =94.33962 101.85941 90.71118 99.04762 18, 594.10

There is a shortcut to working this problem if we recognize that the yield on both bonds is 5%, so all asset cash flows are discounted at 5%. To match the liability cash flows, the asset portfolio must have cash flows of $10,000 at time 1 and $10,000 at time 2. The price of this portfolio is:

+ =210,000 10,000 18, 594.10

1.05 1.05


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Solution 16 A The dollar-weighted yield is just the IRR of the cash flows. In this case, the answer choices are

given as annual effective rates, so let’s work in annual periods.

The $1,000 deposit at time 0 accumulates for 1 year. The $1,000 deposit at time 4 months accumulates for − = =(12 4)/12 8/12 2 /3 years. The $200 withdrawal at time 6 months accumulates for − = =(12 6)/12 6 /12 1/2 years. The $500 withdrawal at time 8 months accumulates for − = =(12 8)/12 4/12 1/3 years. Let i be the annual effective interest rate, and we have:

+ + + − + − + =1 2 /3 1/2 1/31,000(1 ) 1,000(1 ) 200(1 ) 500(1 ) 1, 560i i i i

There are too many terms in this equation to solve using the quadratic equation. One approach to determine i is to use trial and error at this point on the left-hand side and see which interest rate results in an answer of $1,560. Starting with a guess of 22.61%, we get $1,615.05, which is a good bit too high, so our next guess should be a lower interest rate. If our second guess is 18.57%, we get $1,558.96, which is very close and is the correct solution. With trial and error, we can usually answer the question after 2 or 3 guesses.

Another approach is to use the simple interest approximation since all of these cash flows are within a year of each other. Using the simple interest approximation, it is fairly easy to solve the equation:

+ + + − + − + ==

=

1,000(1 ) 1,000(1 0.66667 ) 200(1 0.5 ) 500(1 0.33333 ) 1,5601, 400 260

0.1857

i i i ii

i

Solution 17 B The present value of the cash flows is:

+ + + +2 3 4200 250 300 350v v v v

The pattern of cash flows does not exactly match the pattern expected by any of the standard perpetuity factors, so we need to split the cash flows until they do.

When we subtract out a level perpetuity-immediate of $200, we are left with $50 at time 2 years, $100 at time 3 years, $150 at time 4 years, and so on. The remaining cash flows fit the pattern of an increasing perpetuity-immediate in which the first cash flow of $50 occurs at time 2 years and each subsequent annual cash flow increases by $50. Since the present value factor for a perpetuity-immediate is valued one year before the first cash flow, this value must be discounted for one more year to bring it to time zero in this case. So the present value of both parts is:

∞ ∞+ =200 50 ( ) 46,530a v Ia

If we don’t recall the formula for the present value factor for an increasing perpetuity-immediate, it is easy to derive from the formula for the present value factor for an increasing annuity-immediate:

∞ →∞ →∞ →∞

−−

− + += = = = 2

1/(1 ) 1( ) lim ( ) lim lim

nn

nn

nn n n

v nva nv i i iIa Iai i i


9

So the present value of the payment series becomes:

++ × =

+

+ =

2

2

200 50 1 46,5301

200 50 46,530

ii i i

i i

Now we can solve for i using the quadratic equation:

− − =

± − − × × −=

×=

2

2

46,530 200 50 0

200 ( 200) 4 46, 530 ( 50)2 46, 530

0.035

i i

i

i

Solution 18 E Since the answers are expressed as accumulated values at time 2 months, we let the valuation

date be at time 2 months. The option to pay 90% of the purchase price two months after the date of sale is valued at 0.90 times the purchase price at time 2 months.

The option to deduct X% off the purchase price at the date of sale (time 0) implies that customers

pay −

1100X times the purchase price at time 0. Since this amount is paid at time 0, it needs to

be accumulated to time 2 months so that we can equate it to the value of the second option. The two-month accumulation factor using an annual effective interest rate of 8% is

=2 12 1 6(1.08) (1.08) .

When we equate these two values at time 2 months, the purchase price cancels out of each side and we have:

− =

1 61 (1.08) 0.90100X

Solution 19 C We can relate the nominal discount rate convertible monthly to the nominal interest rate

convertible monthly and solve for the nominal discount rate convertible monthly:

−

−

+ = −

+ = −

= −

=

1(12) (12)

1(12)

(12)

(12)

1 112 12

0.1891 112 12

0.98449 112

0.18612

i d

d

d

d


10

Solution 20 A We are given that the investor makes level deposits into a bank account at the beginning of each

year for 10 years and that it earns interest at the end of every year at an annual effective interest rate of 12%. The interest earns interest at an annual effective rate of 8%. We’ll let the amount of each annual level deposit be X.

At the end of the first year, the investment of X has earned 0.12X in interest, which is reinvested at 8% for 9 years to time 10 years. At the end of the second year, two payments of X have earned 0.12(2 )X in interest, which is reinvested at 8% for 8 years until time 10 years. At the end of the third year, three payments of X have earned 0.12(3 )X in interest, which is reinvested at 8% for 7 years until time 10 years. That pattern continues, until at the end of ten years, ten payments of X have earned 0.12(10 )X in interest, but since this occurs at time 10, there is no time for this to be reinvested at 8% until time 10 years.

In total, there are 10 annual payments of X into the fund. The total investment over 10 years is 10X . We can set up the equation for this series of investments valued at time 10 years as:

( ) ( ) ( ) ( )+ + × + × + ⋅⋅⋅ + ×9 8 7 010 0.12 1.08 2 0.12 1.08 3 0.12 1.08 10 0.12 1.08X X X X X

This can be reduced to:

( ) ( ) ( ) ( ) + + + + ⋅⋅⋅ +

9 8 7 010 0.12 1 1.08 2 1.08 3 1.08 10 1.08X X

The part in the brackets is the accumulated value of an increasing annuity-immediate, or ( )10 8%Is . The accumulated value at time 10 is $10,000, so we can set up the equation of value:

( )= + 10 8%10,000 10 0.12X X Is

Determining the required values, we have:

( ) −= =

10

10 8%1.08 1

15.645490.08/1.08

s

( ) −= =10 8%

15.64549 10 70.568590.08

Is

Plugging these values into the equation of value, we have:

= +

=

=

10,000 10 0.12 (70.56859)10,000

18.46823541.47

X X

X

X


11

Solution 21 D Let the purchase price be X. This purchase price is divided into 12 monthly payments of /10X

that begin today. Working in months with the monthly effective interest rate, the equation of value is:

=

=

(12)12

(12)12

12

12

1011

10

i

i

XX a

a

Using the TI BA-35 calculator, we can solve for the monthly effective interest rate. Press [2nd] [BGN] for beginning of month payments, 1 [PV], 0.1 [PMT], 10 [N], and [CPT] [%i]. The monthly effective interest rate is 3.50315%. The annual effective interest rate is then:

= − =12(1.0350315) 1 51.16%i

Solution 22 B Karen sells the stock short for $50 and receives margin interest of $4 at the end of the year. The

stock pays a dividend of $2 at the end of the year. Her short sale yield is 20%. Plugging these values into the short sale yield formula, we can solve for the buyback price, B:

− + −=

−=

− ==

(50 ) 4 20.2050(0.8)

520.2040

52 840

B

B

BB

Solution 23 D The price of the stock is $75. The first dividend one year from now is $6, and each subsequent

dividend grows by 6% per year. This matches the pattern of payments expected by the constant dividend growth stock valuation formula, so we use this formula to solve for the annual effective interest rate i:

=−

− =

=

6750.03

60.03750.11

i

i

i


12

Solution 24 E We use the accumulated value of the annuity at time +( 1)n to solve for the annual effective

interest rate:

+ = = +

=+

+ −=

+−

=+

+ ===

1 13.776 (1 )13.776

1(1 ) 1 13.776

12.476 1 13.776

11.476(1 ) 13.776

12.3 1.4760.12

n n

n

n

AV s i

si

ii i

i ii iii

We use the annual effective interest rate to solve for n:

+ ==

=

=

(1 ) 2.476ln(1.12) ln(2.476)

ln(2.476)ln(1.12)

8

nin

n

n

Solution 25 A The amount of interest in the fourth payment is the periodic effective interest rate times the loan

balance at time 3 quarters. The amount of principal in the fourth payment is the total payment minus the amount of interest in the fourth payment.

The quarterly effective interest rate is = =(4) 0.16 0.044 4

i .

Working in quarters, we calculate the loan balance at time 3 quarters using the retrospective method as the accumulated value of the loan minus the accumulated value of the payments:

= −

−= −

=

33 3 4%

3

500(1.04) 20

(1.04) 1500(1.12486) 200.04

500

B s

The loan payment exactly matches the amount of interest due on the loan each quarter, so the loan amount stays level. Since all of the loan payment is used to pay interest, there is nothing left over to pay down the principal, so the amount of principal in the fourth payment is zero. Mathematically, we can determine the amount of interest in the fourth payment:

= = × =(4)

4 3 0.04 500 20.04

iI B

So the amount of principal in the fourth payment is:

= − = − =4 4 20.0 20.0 0.0P P I

Course FM May 2005 Exam Index

© BPP Professional Education: 2005 exams Page 1

May 2005 Exam Index

This index provides details of how the sections in the BPP Course FM course notes correspond to the questions in the May 2005 SOA Exam FM / CAS Exam 2.

Question & Answer

Corresponding Section(s) in the BPP Course FM Course Notes Comments

1: E 2.1 Level annuity-immediate PV factor 2: C 5.6 Sinking fund loan 3: B 7.3 Macaulay duration 4: A 2.1 Level annuity-immediate 5: B 6.2 Bond valuation 6: D 7.7 Weighted average duration 7: A 2.6 Determining i from PV cash flows 8: C 5.5 Annual payment loan; refinance loan 9: D 3.1 Increasing annuity-immediate

10: C or E 8.1, 8.3 Forward rates and yields 11: C 6.2 Callable bond 12: B 2.5 Perpetuities 13: D 4.2, 4.4 Non-annual interest rates 14: E 2.1, 3.3 Level & decreasing annuity-immediate 15: D 7.9 Dedication 16: A 5.3 Dollar-weighted interest rate 17: B 3.1 Increasing perpetuity-immediate 18: E 2.6 Equation of value 19: C 4.2, 4.3 Non-annual interest & discount rates 20: A 5.1 Interest reinvested at different rate 21: D 4.5 Non-annual annuity-due 22: B 6.4 Short sale 23: D 6.3 Stock valuation; dividend growth 24: E 2.1 Annuity-immediate AV factor 25: A 5.5 Principal amount in loan payment

FM SOA Exam May 05 questions fall 05 - bpptraining.com SOA Exam May 05.pdfExam FM May 2005 Exam - Questions 2 Question 4 An estate provides a perpetuity with payments of X at the end

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