Solutions to EA-1(A) Examination

EA-1 Examination Questions

Spring, 2002

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Question 1 (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.

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

Question 2 (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.

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

Question 3 (3 points)

Given values:

= 180.24943

d^(m) = 0.08

In what range is ?

[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

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 for the first time.

d⁽⁴⁾ = 0.08.

What is N?

[A] 67

[B] 68

[C] 171

[D] 172

[E] 173

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.

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

Question 6 (4 points)

Smith obtained 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 paid in the even-numbered payments.

B = Sum of principal paid in the odd-numbered payments.

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

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 paid in the 100^th payment of the first mortgage.

B = Amount of principal paid in the 100^th payment of the refinanced mortgage.

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

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.

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

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.

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

Question 10 (5 points)

A bank issues a 20-year loan for $100,000 on 1/1/2000. Level monthly payments are calculated based upon 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.

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

Question 11 (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.

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

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.

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

Question 13 (3 points)

From Mortality Table A: l_x = 20,000 – 100x – x²

Mortality Table B has a constant force of mortality equal to m₄₁ from Mortality Table A. In addition, from Mortality Table B, l₄₅ = 100,000.

In what range is l₄₁ 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

Question 14 (3 points)

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

_0.5q_40.4 = 0.025

_0.9p₄₁ = 0.955

m_42.2 = 0.05

l₄₃ = 100,000

In what range is l₄₀?

[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

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%

p₆₅ = 0.9887

p₆₆ = 0.9873

= 10.3316