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EA-1(A) Examination Questions

Spring, 1999

Question 1

Market value of a pension fund:

Date Value

1/1/99 $100,000

4/1/99 90,000

7/1/99 95,000

10/1/99 185,000

1/1/2000 180,000

Contribution and Benefit Payments:

Date Contributions Benefit Payments

3/31/99 $0 $20,000

9/30/99 75,000 0

Rates of return on fund:

Time weighted = t

Dollar weighted = d

Dollar weighted assuming cash flows at mid-year = m

Which of the following is true?

[A] t > m > d

[B] t > d > m

[C] m > t > d

[D] d > t > m

[E] None of the above

Question 2

Terms of a 25-year annuity certain:

Date of first payment: 12/31/99.

Frequency of payments: Annual.

Amount of each payment:

First 10 years: $1,000 per year.

Next 10 years: $1,500 per year.

Final 5 years: $2,000 per year.

Interest rate: 8% per year, compounded semiannually.

In what range is the present value of the annuity as of 1/1/99?

[A] Less than $12,500

[B] $12,500 but less than $13,000

[C] $13,000 but less than $13,500

[D] $13,500 but less than $14,000

[E] $14,000 or more

Question 3

Terms of two actuarially equivalent annuities:

Annuity A: $500 at the end of each of the first 3 months, and $1,000 at the end of each of the next 9 months.

Annuity B: $P at the end of each of the first 2 quarters, and $2P at the end of each of the next 2 quarters.

Interest rate: 8% per year, compounded monthly

In what range is $P?

[A] Less than $1,770

[B] $1,770 but less than $1,800

[C] $1,800 but less than $1,830

[D] $1,830 but less than $1,860

[E] $1,860 or more

Question 4

Date of a loan: 1/1/99.

Interest rate: 10.8% per year, compounded monthly.

Date of first repayment: 2/1/99.

Frequency of repayments: Monthly.

Number of repayments: 360.

In what range is the modified duration of the loan?

[A] Less than 95 months

[B] 95 months but less than 96 months

[C] 96 months but less than 97 months

[D] 97 months but less than 98 months

[E] 98 months or more

Question 5

Market value of a pension fund on 12/31/98: $1,750,000.

Benefit payments are made on the first day of each calendar quarter.

Contributions are made on the last day of each calendar quarter and included in the ending balance.

Time-weighted rate of return for 1999: 8.79%.

1999 fund data:

Quarter	Benefit Payments	Contributions	Quarterly Return	Ending Balance
1	$22,000	$30,000	1.02%	$1,775,626
2	$43,000	$85,000	5.19%	$1,907,549
3	$27,994	y	-2.26%	$1,937,033
4	$39,228	$27,617	j%	z

In what range is the dollar-weighted rate of return for 1999?

[A] Less than 8.68%

[B] 8.68% but less than 8.73%

[C] 8.73% but less than 8.78%

[D] 8.78% but less than 8.83%

[E] 8.83% or more

Question 6

Date of a loan: 1/1/99.

Amount of loan: $10,000.

Date of first repayment: 2/1/99.

Frequency of repayments: Monthly.

Term of loan: 10 years.

Interest rate: 7% per year, compounded annually.

Prepayment penalty: 10% of outstanding loan amount.

The borrower establishes a savings account to which annual payments of $P are made. The savings account earns 9% per year, compounded monthly. The first payment to the savings account is made on 12/31/99. Just prior to the 53^rd scheduled loan repayment the savings account balance is sufficient to pay both the outstanding loan amount and the prepayment penalty.

In what range is $P?

[A] Less than $1,475

[B] $1,475 but less than $1,485

[C] $1,485 but less than $1,495

[D] $1,495 but less than $1,505

[E] $1,505 or more

Question 7

Smith takes out a mortgage on 1/1/94 with the following provisions:

Amount of mortgage: $100,000.

Interest rate: 12% per year, compounded monthly.

Number of repayments: 360.

Frequency of repayments: Monthly.

Date of first repayment: 1/31/94.

On 12/31/98, after making the 12/31/98 repayment, Smith repays an additional $3,000 to reduce the outstanding balance of his mortgage. He will continue to make the same monthly repayments he has been making (or the outstanding balance of the mortgage, if less) until the outstanding balance of his mortgage is reduced to zero.

In what range is the total interest paid by Smith?

[A] Less than $225,000

[B] $225,000 but less than $235,000

[C] $235,000 but less than $245,000

[D] $245,000 but less than $255,000

[E] $255,000 or more

Question 8

Term of a perpetuity:

Effective date: 1/1/99.

Frequency of payments: Annual.

Date of first payment: 12/31/99.

Amount of each payment: $2X.

Present value of future payments as of 1/1/2009: $P.

Terms of a 10-year annuity certain:

Effective date: 1/1/99.

Frequency of payments: Annual.

Date of first payment: 12/31/99.

Amount of each payment: $X.

Accumulated value of payments as of 1/1/2009: $P/2.

Interest rate: i%.

In what range is i%?

[A] Less than 7.25%

[B] 7.25% but less than 7.75%

[C] 7.75% but less than 8.25%

[D] 8.25% but less than 8.75%

[E] 8.75% or more

Question 9

Fund balance as of 1/1/99: $12,000.

Deposits for the fund: $100 on the last day of each month for 5 years.

First deposit: 1/31/99.

Withdrawals from the fund: $1,000 on the first day of each quarter.

First withdrawal: 1/1/2006.

No other deposits or withdrawals are made.

Interest rate: 8% per year, compounded monthly.

In what range is the fund balance as of 12/31/2010?

[A] Less than $13,500

[B] $13,500 but less than $15,000

[C] $15,000 but less than $16,500

[D] $16,500 but less than $18,000

[E] $18,000 or more

Question 10

Purchase date of a perpetuity: 1/1/99.

Date of first payment: 3/31/99.

Frequency of payments: Quarterly.

Quarterly payments during each year as follows:

Quarter Amount

1 $100

2 200

3 300

4 400

Interest rate: 10% per year, compounded annually.

In what range is the purchase price of the perpetuity?

[A] Less than $10,125

[B] $10,125 but less than $10,250

[C] $10,250 but less than $10,375

[D] $10,375 but less than $10,500

[E] $10,500 or more

Question 11

Face amount of a bond: $100,000.

Purchase date: 1/1/99.

Coupon rate: 6% per year, payable semiannually on each 6/30 and 12/31.

Redemption date: 1/1/2009.

Redemption amount: $100,000.

Coupons are reinvested at 5% per year, compounded semiannually.

Overall yield rate to purchaser: 5.5% per year, compounded semiannually.

In what range is the purchase price of the bond?

[A] Less than $99,000

[B] $99,000 but less than $100,500

[C] $100,500 but less than $102,000

[D] $102,000 but less than $103,500

[E] $103,500 or more

Question 12

Type of insurance policy: Whole life.

Age of insured at issue: 30.

Death benefit: Payable at the end of the year of death.

Net annual premium (Annual benefit premium): $45 per $1,000 of face amount, payable for life.

Net single premium (Single benefit premium): $Y per $1,000 of face amount.

Interest rate: 6% per year, compounded annually.

In what range is $Y?

[A] Less than $400

[B] $400 but less than $450

[C] $450 but less than $500

[D] $500 but less than $550

[E] $550 or more

Question 13

Value of an inheritance: $140,000.

Number of beneficiaries: 3.

Age of each beneficiary: y.

Terms of a monthly annuity, payable at the beginning of each month, to be purchased with inheritance:

• $400 to each beneficiary while all are alive.
• $450 to each surviving beneficiary after the first death.
• $X to the surviving beneficiary after the second death.

Selected values:

= 9.194 = 7.354 = 6.258

In what range is the value of $X?

[A] Less than $200

[B] $200 but less than $400

[C] $400 but less than $600

[D] $600 but less than $800

[E] $800 or more

Question 14

Under a two decrement table used for a pension valuation the absolute rate of death is 2% and the absolute rate of termination is 4% for all ages.

The pension plan provides for a death benefit after 5 years of service and for a retirement benefit for anyone who terminates after attainment of age 55 with 5 years of service.

An employee was hired at age 58 and is currently age 60.

In what range is the probability that the employee receives either a death or retirement benefit from the plan?

[A] Less than 0.828

[B] 0.828 but less than 0.830

[C] 0.830 but less than 0.832

[D] 0.832 but less than 0.834

[E] 0.834 or more

Question 15

Ages as of 1/1/99:

Smith: 60.

Green: 63.

Brown: 65.

Selected values from a mortality table:

l₆₀ = 1,000 l_60+t = 1,000 - [19t - 1], where t 1

In what range is the probability that Green will not die in 2004 and at least one of Smith and Brown will die in 2004?

[A] Less than .0365

[B] .0365 but less than .0375

[C] .0375 but less than .0385

[D] .0385 but less than .0395

[E] .0395 or more

Question 16

Selected annuity values:

x ä_x

50 12.07872

51 11.92117

52 11.75854

Interest rate: 7% per year, compounded annually.

In what range is the probability an annuitant age 50 would die before age 52?

[A] Less than .005

[B] .005 but less than .010

[C] .010 but less than .015

[D] .015 but less than .020

[E] .020 or more

Question 17

Terms from three annuity contracts:

Contract 1 Contract 2 Contract 3

Age at issue 30 30 30

Net single premium $6,000 $75,000 $P

Age at first annuity payment 31 30 31

Maximum number of payments 15 16 15

Frequency of annuity payments Annual Annual Annual

Amount of first annuity payment $1,000 $5,000 $7,500

Amount of increase (decrease) in

each subsequent payment $0 $1,000 ($500)

The net single premium under each contract is based on the same mortality and interest assumptions.

In what range is $P?

[A] Less than $23,000

[B] $23,000 but less than $25,000

[C] $25,000 but less than $27,000

[D] $27,000 but less than $29,000

[E] $29,000 or more

Question 18

Data for pension plan participant Smith:

Date of birth: 1/1/54.

Date of termination: 1/1/99.

Normal retirement benefit: $7,200 commencing at age 65, payable annually for life.

Benefit Option A: $X commencing at age 55, payable annually for life with 15 years certain.

Option A is actuarially equivalent to the normal retirement benefit.

Selected values:

ä₆₀ = 10.2733 ₁₀E₄₅ = .4808

ä₆₅ = 9.1941 ₁₀E₅₅ = .4413

ä₇₀ = 8.0605 ₂₀E₄₅ = .2122

ä₇₅ = 6.8749 ₂₅E₄₅ = .1317

Interest rate: 7% per year, compounded annually.

In what range is $X?

[A] Less than $2,200

[B] $2,200 but less than $2,300

[C] $2,300 but less than $2,400

[D] $2,400 but less than $2,500

[E] $2,500 or more

Question 19

Terms of a life annuity due:

Date of first payment: 1/1/99.

Age at first payment: 60.

Amount of each payment: $15,000.

Frequency of payments: Annual.

Interest rate: 7% per year, compounded annually.

The premium for the annuity is based upon mortality rates based on a standard table with an improved rate of survivorship at age 60 equal to p₆₀ + 1%, where p₆₀ is from the standard table.

Selected values from the standard table:

1000 = .921

1000A₆₀ = 328

In what range is the net single premium for the annuity as of 1/1/99?

[A] Less than $139,000

[B] $139,000 but less than $147,000

[C] $147,000 but less than $155,000

[D] $155,000 but less than $163,000

[E] $163,000 or more

Question 20

Selected values from a mortality table with a one-year select period:

q_[x] = 0.6q_x

a₄₀ = 16

The force of mortality, µ, during the select period is ln(1.04) for all ages.

In what range is a_[40]?

[A] Less than 16.15

[B] 16.15 but less than 16.30

[C] 16.30 but less than 16.45

[D] 16.45 but less than 16.60

[E] 16.60 or more

Question 21

Selected values:

_np_xx = .25

p_x+n = .5

In what range is _nq_x + _nq_xx - _n|q_xxx?

[A] Less than 1.14

[B] 1.14 but less than 1.16

[C] 1.16 but less than 1.18

[D] 1.18 but less than 1.20

[E] 1.20 or more

Question 22