SOCIETY OF ACTUARIES

########  AMERICAN SOCIETY OF PENSION ACTUARIES

JOINT BOARD FOR THE ENROLLMENT OF ACTUARIES

# ENROLLED ACTUARIES BASIC EXAMINATION

######### MAY 2002 EA-1 EXAMINATION

EA1-10-02 Printed in U.S.A.

## 

[Data For Question 1]{.underline} (3 Points)

Loan repayment period: 5 years.

Beginning loan amount: \$75,000.

Repayment Plan #1: Level annual payments at the beginning of each year.

Repayment Plan #2: Level semi-annual payments at the end of each 6-month
period.

A = Annual payment under Repayment Plan #1.

B = Total payments in a year under Repayment Plan #2.

1000 d^(4)^ = 76.225.

[Question 1]{.underline}

In what range is the absolute value of \[A -- B\]?

\[A\] Less than \$1,000

\[B\] \$1,000 but less than \$1,025

\[C\] \$1,025 but less than \$1,050

\[D\] \$1,050 but less than \$1,075

\[E\] \$1,075 or more

[Data For Question 2]{.underline} (5 Points)

Annual payments into a fund: \$10,000 at the end of year one, increasing
by \$500 per year in the second through the tenth years. After the tenth
year, each payment increases by 3.5% over the prior payment.

Interest rate: 7% compounded annually.

[Question 2]{.underline}

In what range is the accumulated value of the fund at the end of 20
years?

\[A\] Less than \$500,000

\[B\] \$500,000 but less than \$550,000

\[C\] \$550,000 but less than \$600,000

\[D\] \$600,000 but less than \$650,000

\[E\] \$650,000 or more

[Data For Question 3]{.underline} (3 Points)

Given values:

![](media/image1.wmf)

d^(m)^ = 0.08

[Question 3]{.underline}

In what range is ![](media/image2.wmf)

\[A\] Less than 2,930

\[B\] 2,930 but less than 2,970

\[C\] 2,970 but less than 3,010

\[D\] 3,010 but less than 3,050

\[E\] 3,050 or more

# Data For Question 4 (3 Points)

A 20-year immediate annuity certain is payable monthly. Immediately
after the 43^rd^ payment has been made, the present value of the
remaining annuity payments is calculated to be X.

N is the number of the payment after which the present value of the
remaining annuity payments is less than ![](media/image3.wmf) for the
first time.

![](media/image4.wmf)

[Question 4]{.underline}

What is N?

## \[A\] 67

\[B\] 68

\[C\] 171

\[D\] 172

\[E\] 173

# Data For Question 5 (4 Points)

Two \$10,000 loans have the following repayment characteristics:

##### Loan 1: Level quarterly payments at the end of each quarter for five years.

###### Loan 2: Monthly interest payments on the original loan amount at the end of each month for 48 months plus a balloon repayment of principal at the end of the fourth year. The balloon repayment will be made using the accumulated value of a sinking fund created by level annual deposits at the beginning of each of the four years.

Effective annual interest rate on the loan: 8%.

Effective annual interest rate on the sinking fund: 9%.

A = Sum of repayments under Loan 1.

B = Sum of interest payments on Loan 2 plus sum of sinking fund
payments.

[Question 5]{.underline}

In what range is the absolute value of \[A -- B\]?

## \[A\] Less than \$875

\[B\] \$875 but less than \$950

\[C\] \$950 but less than \$1,025

\[D\] \$1,025 but less than \$1,100

\[E\] \$1,100 or more

# Data For Question 6 (4 Points)

Smith obtains a loan for \$10,000 with 40 annual payments at an
effective annual interest rate of 7%. The first payment is due one year
from now.

A = Sum of [interest]{.underline} paid in the even-numbered payments.

B = Sum of [principal]{.underline} paid in the odd-numbered payments.

[Question 6]{.underline}

In what range is \[A + B\]?

## \[A\] Less than \$13,800

\[B\] \$13,800 but less than \$14,200

\[C\] \$14,200 but less than \$14,600

\[D\] \$14,600 but less than \$15,000

\[E\] \$15,000 or more

# Data For Question 7 (4 Points)

Smith purchases a house for \$120,000 and agrees to put 20% down. He
takes out a 30-year mortgage, with monthly payments, with the first
payment one month after the date of the mortgage. The interest rate is
8% compounded monthly.

Immediately following the 180^th^ payment, Smith refinances the
outstanding balance with a new 10-year mortgage, also with monthly
payments, with the first payment one month after the date of the new
mortgage. The new interest rate is 7.5% compounded monthly.

A = Amount of [interest]{.underline} paid in the 100^th^ payment of the
first mortgage.

B = Amount of [principal]{.underline} paid in the 100^th^ payment of the
refinanced mortgage.

##### Question 7

In what range is \[A + B\]?

## \[A\] Less than \$1,300

\[B\] \$1,300 but less than \$1,325

\[C\] \$1,325 but less than \$1,350

\[D\] \$1,350 but less than \$1,375

\[E\] \$1,375 or more

# Data For Question 8 (4 Points)

A serial bond issue bearing 6% annual coupons, payable semiannually, is
to be redeemed at par value at annual intervals over a 20-year period.
The first redemption will occur at the end of year 10 in the amount of
\$20,000. Each subsequent annual redemption will be \$1,000 less than
the preceding one.

[Question 8]{.underline}

In what range is the maximum price an investor would pay for the entire
issue to realize an effective annual yield of 7%?

## \[A\] Less than \$190,500

\[B\] \$190,500 but less than \$191,000

\[C\] \$191,000 but less than \$191,500

\[D\] \$191,500 but less than \$192,000

\[E\] \$192,000 or more

# Data For Question 9 (3 Points)

Face value of a bond: \$1,000.

Redemption value: \$1,050.

Time to maturity: 10 years.

Coupon rate: 9.00% per annum, convertible semi-annually.

Yield rate: 10.25% per annum.

The bond is not callable.

[Question 9]{.underline}

In what range is the increase in the book value of the bond during the
third year?

## \[A\] Less than \$7.00

\[B\] \$7.00 but less than \$7.50

\[C\] \$7.50 but less than \$8.00

\[D\] \$8.00 but less than \$8.50

\[E\] \$8.50 or more

# 

# 

# 

# 

# 

**2002 EA-1**

**Answer Key**

  ----------------- ----------------- ----------------- -----------------
  **Question        **Answer**        **Question        **Answer**
  Number**                            Number**          

  1                 C                 16                D

  2                 C                 17                B

  3                 B                 18                A

  4                 E                 19                B

  5                 D                 20                A

  6                 D                 21                C

  7                 C                 22                B

  8                 E                 23                A

  9                 B                 24                C

  10                B                 25                B

  11                E                 26                C

  12                B                 27                B

  13                C                                   

  14                B                                   

  15                E                                   

                                                        
  ----------------- ----------------- ----------------- -----------------

# Data For Question 10 (5 Points)

A bank issues a 20-year loan for \$100,000 on 1/1/2000. Level monthly
payments are calculated based on a 7% annual interest rate compounded
monthly, with payments due at the end of each month. The borrower can
repay the loan in full without penalty on the first day of any year. On
1/1/2002, the bank sells the loan to an investor for \$90,000.

[Question 10]{.underline}

What is the latest full repayment date for which the investor's yield
exceeds 8%, compounded monthly?

## \[A\] 1/1/2009

\[B\] 1/1/2010

\[C\] 1/1/2011

\[D\] 1/1/2012

\[E\] 1/1/2013

[Data For Question 11]{.underline} (3 Points)

S1 = The accumulated value as of 12/31/2002 of \$500 invested at the end
of each month during 2002 at a nominal interest rate of 8% per year,
convertible quarterly.

A1 = The present value as of 1/1/2002 of S1, at a nominal discount rate
of 6% per year, convertible semiannually.

S2 = The accumulated value as of 12/31/2002 of \$1,500 invested at the
end of each quarter during 2002 at a nominal discount rate of 6% per
year, convertible monthly.

A2 = The present value as of 1/1/2002 of S2, at a nominal interest rate
of P% per year, convertible once every two years.

[Question 11]{.underline}

In what range is P% such that A1 = A2?

\[A\] Less than 4.60%

\[B\] 4.60% but less than 4.70%

\[C\] 4.70% but less than 4.80%

\[D\] 4.80% but less than 4.90%

\[E\] 4.90% or more

# Data For Question 12 (3 Points)

For a group of lives observed over the age interval (x,x+1\], you are
given:

(i) 100 lives entered observation at exact age x.

(ii) 40 of these lives are scheduled to leave observation at age x +
     0.75.

(iii) 23 deaths were observed.

(iv) No other lives entered or left observation.

(v) The underlying survival distribution is Balducci.

[Question 12]{.underline}

In what range is the moment estimate of q~x~?

\[A\] Less than 0.249

\[B\] 0.249 but less than 0.254

\[C\] 0.254 but less than 0.259

\[D\] 0.259 but less than 0.264

\[E\] 0.264 or more

# Data For Question 13 (3 Points)

From Mortality Table A: l~x~ = 20,000 -- 100x -- x^2^.

Mortality Table B has a constant force of mortality equal to µ~41~ from
Mortality Table A. In addition, from Mortality Table B, l~45~ =100,000.

##### Question 13

In what range is l~41~ from Mortality Table B?

\[A\] Less than 100,000

\[B\] 100,000 but less than 105,000

\[C\] 105,000 but less than 110,000

\[D\] 110,000 but less than 115,000

\[E\] 115,000 or more

# Data For Question 14 (3 Points)

Assume a uniform distribution of decrement over each interval \[x,x+1\].

~0.5~q~40.4~ = 0.025

~0.9~p~41~ = 0.955

µ~42.2~ = 0.05

l~43~ = 100,000

##### Question 14

In what range is l~40~?

## \[A\] Less than 116,000

\[B\] 116,000 but less than 116,500

\[C\] 116,500 but less than 117,000

\[D\] 117,000 but less than 117,500

\[E\] 117,500 or more

# Data For Question 15 (5 Points)

Age of retiree on 1/1/2002: 65.

Normal form of payment: Single life annuity of \$20,000 payable at the
beginning of each year.

Optional form of payment: \$X payable at the beginning of each year
while the retiree is alive

and

If the retiree dies during 2003, a ten-year decreasing certain annuity
starting on January 1, 2004. The initial payment on this date is \$X and
subsequent annual payments are each 95% of the prior payment.

Selected values:

  -----------------------------------------------------------------------
  i = 7%

  p~65~ = 0.9887

  p~66~ = 0.9873

  ä~65~ = 10.3316

  ä~67~ = 9.8614
  -----------------------------------------------------------------------

The optional form of payment and the single life annuity are actuarially
equivalent on 1/1/2002.

[Question 15]{.underline}

In what range is \$X?

## \[A\] Less than \$18,055

\[B\] \$18,055 but less than \$18,655

\[C\] \$18,655 but less than \$19,255

\[D\] \$19,255 but less than \$19,855

\[E\] \$19,855 or more

# Data For Question 16 (3 Points)

For a group of lives, the following is given:

  -------------- -------------- -------------- -------------- --------------
                 x              l~x~           d~x~           L~x~

                                                              

                 35             10,000         300            9,851

                 36             9,700                         9,456

                 37                            600            8,913

                                                              

                 ~2~m~35~ =                                   
                 0.0404                                       
  -------------- -------------- -------------- -------------- --------------

There is a constant force of mortality over the interval \[36, 37\].

[Question 16]{.underline}

In what range is l~36.5~?

##### \[A\] Less than 9,449

\[B\] 9,449 but less than 9,452

\[C\] 9,452 but less than 9,455

\[D\] 9,455 but less than 9,458

\[E\] 9,458 or more

# Data For Question 17 (4 Points)

> e~70:5~ = 4.66234
>
> e~70:15~ = 11.45220
>
> e~80~ = 8.26871
>
> e~75:10~ = 7.70883
>
> e~75:5~ = 4.43230
>
> e~80:5~ = 4.08531

[Question 17]{.underline}

In what range is e~70~?

## \[A\] Less than 14.00000

\[B\] 14.00000 but less than 15.00000

\[C\] 15.00000 but less than 16.00000

\[D\] 16.00000 but less than 17.00000

\[E\] 17.00000 or more

# Data For Question 18 (3 Points)

# 

# µ~x~ = 0.1, x\>0.

# 

# N = the average number of years lived between age 60 and age 80 by those who die between age 60 and age 80.

# 

# 

# 

##### Question 18

# 

# In what range is N?

# \[A\] Less than 7.0

# 

# \[B\] 7.0 but less than 7.7

# 

# \[C\] 7.7 but less than 8.4

# 

# \[D\] 8.4 but less than 9.1

# 

# \[E\] 9.1 or more 

# Data For Question 19 (5 Points)

An employee age 65 with a spouse age 65 is retiring under one of three
actuarially equivalent optional forms of payment, all of which are
payable at the beginning of each month.

Option 1: Life annuity of \$X per month.

Option 2: Life annuity of \$Y per month with the first 60 months
guaranteed.

Option 3: Joint and last survivor annuity with the following monthly
payments:

> a\) \$Y during the joint lives of the employee and spouse.
>
> b\) \$X for the remaining lifetime of the employee if the spouse dies
> first.
>
> c\) P% of \$Y for the remaining lifetime of the spouse if the employee
> dies first.

Selected actuarial factors:

  ----------------------------------- -----------------------------------
  ![](media/image5.wmf)               ![](media/image6.wmf)

  ![](media/image7.wmf)               i = 7%
  ----------------------------------- -----------------------------------

##### Question 19

In what range is P%?

## \[A\] Less than 42.00%

\[B\] 42.00% but less than 44.00%

\[C\] 44.00% but less than 46.00%

\[D\] 46.00% but less than 48.00%

\[E\] 48.00% or more

# Data For Question 20 (3 Points)

Selected values from a double decrement table:

> $$\ell_{\text{40}\ }^{\ (T)}\  = \ \text{10},\text{000}$$
>
> $$\ell_{\text{42}\ }^{\ (T)} = \ 7,\text{000}$$
>
> ![](media/image8.wmf)
>
> ![](media/image9.wmf)
>
> ![](media/image10.wmf)
>
> ![](media/image11.wmf)are both linear over the interval \[41, 42\].

[Question 20]{.underline}

In what range is the expected number of decrements due to cause 1
between ages 41 and 42?

##### \[A\] Less than 482

\[B\] 482 but less than 492

\[C\] 492 but less than 502

\[D\] 502 but less than 512

# \[E\] 512 or more 

# Data For Question 21 (5 Points)

Under a pension plan\'s actuarial equivalence definition, the interest
rate is 7% and ![](media/image12.wmf) Under the plan, there are two
actuarially equivalent forms of payment:

Form 1: 10 years certain and payments for life thereafter.

Form 2: · Payments of \$X while the participant and spouse are both
alive.

· Payments of 110% of \$X to participant after the death of spouse.

· Payments of 50% of \$X to spouse after death of participant.

Payments are made annually at the beginning of each year.

For a participant age 72, with a spouse age 75, the benefit amount under
Form 1 is \$100.

[Question 21]{.underline}

In what range is \$X for this participant?

## \[A\] Less than \$86

##### \[B\] \$86 but less than \$92

\[C\] \$92 but less than \$98

\[D\] \$98 but less than \$104

\[E\] \$104 or more

# Data For Question 22 (5 Points)

###### Age at issue for an insured: 25.

Benefit: \$200,000 payable at the end of the year of death if death
occurs before age 55. \$X payable at the end of the year of death if
death occurs on or after age 55 but before age 65.

Premiums: A total of 6 premiums consisting of \$300 at age 25, doubling
each year for the next four premium payments. All premiums are payable
at the beginning of the year and one additional premium of \$4,000 is
payable at age 55.

Selected values:

> 1000A~25~ = 81.6496 i = 6%
>
> 1000A~55~ = 305.1431 ~29~p~26~ = 0.90450
>
> 1000A~65~ = 439.7965 ~40~p~25~ = 0.78766

  ---------------------------------- ------------------------------------
  x                                  p~x~

                                     

  25                                 0.99877

  26                                 0.99873

  27                                 0.99867

  28                                 0.99861

  29                                 0.99854
  ---------------------------------- ------------------------------------

[Question 22]{.underline}

In what range is \$X?

## \[A\] Less than \$110,000

\[B\] \$110,000 but less than \$114,000

\[C\] \$114,000 but less than \$118,000

\[D\] \$118,000 but less than \$122,000

\[E\] \$122,000 or more

# Data For Question 23 (3 Points)

For a study of four automobile engines, you are given:

(i) The engines are subject to a uniform survival distribution over the
    interval \[0,ω\].

(ii) Failures occurred at times 4, 5 and 7; the remaining engine was
     operational at time r.

(iii) The observation period was from time 3 to time r.

(iv) The maximum likelihood estimate of ω is 13.67.

[Question 23]{.underline}

In what range is r?

##### \[A\] Less than 11.3

\[B\] 11.3 but less than 11.8

\[C\] 11.8 but less than 12.3

\[D\] 12.3 but less than 12.8

\[E\] 12.8 or more

# Data For Question 24 (4 Points)

A life insurance policy provides for payment of \$1,000 at the end of
the year of death but provides no payment if death occurs in the first
four years.

For a male age 60, the net single premium for this policy is \$300.

The interest rate is 7%.

The probability of death in each of the first four years is 0.2% (i.e.,
q~60~, q~61~, q~62~ and q~63~ = 0.002).

\$P = the net single premium for a \$1 annual life annuity-due with a
3-year certain period for a male age 60.

[Question 24]{.underline}

In what range is \$P?

## \[A\] Less than \$9.50

\[B\] \$9.50 but less than \$10.10

\[C\] \$10.10 but less than \$10.70

\[D\] \$10.70 but less than \$11.30

\[E\] \$11.30 or more

# Data for Question 25 (3 Points)

![](media/image13.wmf)

i = 5%

[Question 25]{.underline}

In what range is ä~80~?

\[A\] Less than 8.00

\[B\] 8.00 but less than 9.00

\[C\] 9.00 but less than 10.00

\[D\] 10.00 but less than 11.00

\[E\] 11.00 or more

# Data For Question 26 (4 Points)

Interest Rate: 5% per year, compounded annually**.**

A term certain and life annuity issued to a person, age 65, provides
\$500 payable at the end of each month. The load is 8% of the gross
premium. Annuity payments are payable at least until a sum equal to the
gross single premium has been paid.

Selected Values:

> $${}_{\text{13}|}a_{\ \text{65}}^{(\text{12})}\  = \ 3\text{.}\text{31}$$
>
> $${}_{\text{14}|}a_{\ \text{65}}^{(\text{12})}\  = \ 2\text{.}\text{92}$$
>
> $${}_{\text{15}|}a_{\ \text{65}}^{(\text{12})}\  = \ 2\text{.}\text{56}$$

[Question 26]{.underline}

In what range is the gross single premium?

## \[A\] Less than \$83,000

#### \[B\] \$83,000 but less than \$85,000

#### \[C\] \$85,000 but less than \$87,000

#### \[D\] \$87,000 but less than \$89,000

#### \[E\] \$89,000 or more 

# Data For Question 27 (3 Points)

A group of lives is subject to two decrements over the interval
\[0,100\].

Selected values:

> ![](media/image14.wmf)

[Question 27]{.underline}

In what range is ![](media/image15.wmf)

\[A\] Less than 0.093

\[B\] 0.093 but less than 0.096

\[C\] 0.096 but less than 0.099

\[D\] 0.099 but less than 0.102

\[E\] 0.102 or more

**2002 EA-1**

**Answer Key**

  ----------------- ----------------- ----------------- -----------------
  **Question        **Answer**        **Question        **Answer**
  Number**                            Number**          

  1                 C                 16                D

  2                 C                 17                B

  3                 B                 18                A

  4                 E                 19                B

  5                 D                 20                A

  6                 D                 21                C

  7                 C                 22                B

  8                 E                 23                A

  9                 B                 24                C

  10                B                 25                B

  11                E                 26                C

  12                B                 27                B

  13                C                                   

  14                B                                   

  15                E                                   

                                                        
  ----------------- ----------------- ----------------- -----------------