Scan 0008

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II-30

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PRACTICE MULTIPLE CHOICE TEST 8

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1.

You are given two loans, with each loan to be repaid by a single payment in the future . Each payment includes both principal and interest.

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The first loan is repaid by a 3000 payment at the end of four years. The interest is accrued at 10% per annum compounded semiannuaIly .

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The second 10an is repaid by a 4000 payment at the end of five years. The interest is accrued at 8% per annum compounded semiannuaIIy .

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The two loans are to be consolidated. The consolidated loan is to be repaid by two equal instaIlments of X, with interest at 12% per annum compounded semiannuaIly. The first payment is due immediately and the second payment is due one year from now .

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Calculate X .

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(B) 2485

(A) 2459

2.

re}

Fund X starts with 1000 and accumulates 0< t < IS.

2504

CD)

2521

with force of interest

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(E) 2537

Ot

=

15 ~

t'

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••

•.

for

,

Fund Y starts with 1000 and accumulates with an interest rate of 8% per annum compounded semiannuaIly for the first three years and an effective interest rate of per annum thereafter.

i

Fund X equals Fund Y at the end of four years. Calculate i. r'"

(A) .0750

3.

(,~) .0775

(C) .0800

CD)

.0825

(E) .0850

Jeff puts 100 into a fund that pays an effective annual rate of discount of 20% for the first two years and a force of interest of Ot = t2 ~ 8' for 2 < t < 4, for the next two years. At the end of four years, the amount in Jeft's account is the same as what it would have been if he had put 100 into an account paying interest at the nominal rate of per annum compounded quarterly for four years.

i

Calculate i.

(A) .200

4.

(B) .219

(C) .240

(D) .285

The present value of a series of payments of 2 at the end of every eight years, forever, is equal to 5. Calculate the effective rate of interest. (A) .023

(B) .033

(C) .040

@?) .043

(E) .052

l

II-31

On January 1, 1990, Jack deposited 1000 into Bank X to earn interest at rate j per annum compounded semiannually. On January 1, 1995, he transferred his account to Bank Y to earn interest at rate k per annum compounded quarterly. On January 1, 1998, the balance at Bank Y is 1990.76.

5.

If Jack could have earned interest at rate k per annum compounded quarterly from January 1, 1990 through January 1, 1998, his balance would have been 2203.76. Calculate the ratio ~ 6.

1. CB) 1.30

1.25

(D) lAO

(C) 1.35

(E) 1A5

You are given an annuity-immediate with 11 annual payments of 100 and a final larger payment at the end of 12 years. At an annual effective interest rate of 3.5%, the present value at time 0 of all payments is 1000. Using an annual effective interest rate of 1%, calculate the present value at the beginning of the ninth year of all remaining payments. (A) 412

7.

(B) 419

Using an annual effective interest rate j

@ 439

(C) 432

(E) 446

> 0, you are given:

(i) The present value of 2 at the end of each year for 2n years, plus an additional 1 at the end of each of the first n years, is 36. (ii) The present value of an n-year deferred annuity-immediate years is 6. Calculate (A) .03

n

j. (B) .04

(C) .05

CD)

\;. '\

8.

paying 2 per year for

\.

H

\)

.06

1--

~2·07

c:

An l1-year annuity has a series of payments 1, 2, 3,' 4, 5, 9, 5, 4, 3, 2, 1, with the first payment made at the end of the second year. The present value of this annuity is 25 at interest rate i. A 12-year annuity has a series of payments 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, with the first payment made at the end of the first year. Calculate the present value of the 12-year annuity at interest (A) 29.5

(C) 30.5

i. (D) 31.0

(E) 31.5 " I.

r I I

I :1:) I

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9.

Joan has won a lottery that pays 1000 per month in the first year, 1100 per month in the second year, 1200 per month in the third year, and so on. Payinents are made at the end of each month for 10 years. Using an effective interest rate of3% per annum, calculate the present value of this prize. (B) 114,000

(A) 107,000

10.

(C) 123,000

(D) 135,000

\(E) 148,000

A 5% lO-year loan of 10,000 is to be repaid by the sinking fund method, with interest and sinking fund ~ayments made at the end of each year. The effective rate of interest earned in the sinking fund is 3% per annum. Immediately before the fifth year's payment would have fallen due, the lender requests that the outstanding principal be repaid in one lump sum. Calculate the amount that must be paid, including interest, to extinguish the debt. (A) 6350

11.

(B) 6460

6740

((l)

(E) 7000 6850

A loan of 1000 is being repaid in ten years by semiannual installments of 50, plus interest on the unpaid balance at 4% per annum compounded semiannually. The installments and interest payments are reinvested at 5% per annum compounded semiannually. Calculate the annual effective yield rate of the loan. (B) .048

~).046

12.

@

(C) .050

(D) .052

(E) .054

A company agrees to repay a loan over five years. Interest payments are made annually and a sinking fund is built up with five equal annual payments made at the end of each year. Interest on the sinking fund is compounded annually. You are given: (i) The.amount in the sinking fund immediately after the first payment is X. (ii) The amount in the sinking fund immediately after the second payment is Y. (iii) The ratio Y IX = 2.09. (iv) The net amount of the loan immediately after the fourth payment is 3007.87. Calculate the amount of the sinking fund payment. (A) 1931

13.

(B) 2031

~2131

(D) 2231

(E) 2431

An n-year 1000 par value bond with 8% annual coupons has an annual effective yield of i, i> O. The book value of the bond at the end of year 3 is 1099.84 and the book value at the end of year 5 is 1082.27. Calculate the purchase price of the bond. re--

(A) 1112

(C) 1132 ~

1122

(D) 1142

(E) 1152

ll-33

14.

A 1000 par value 3-year bond with annual coupons of 50 for the first year, 70 for the second year, and 90 for the third year is bought to yield a force of interest for t

> O.

=

~t - 1

2(t -t+1)

Calculate the price of this bond. (B) 550

15.

Ot

(C) 600

(D) 650

(E) 700

The proceeds of a 10,000 death benefit are left on deposit with an insurance company for seven years at an effective annual interest rate of 5%. The balance at the end of seven years is paid to the beneficiary in 120 equal monthly payments of X, with the first payment made immediately. During the payout period, interest is credited at an annual effective rate of3%. Calculate X. (A) 117

16.

(B) 118

~135

(E) 158

Roward wishes to borrow 1000. Lynn offers a loan at a 10.65% annual effective rate in which Roward would repay the loan with eight equal annual payments made at the end of each year by the amortization method. Ann offers a loan in which the principal is to be repaid at the end of eight years. In the meantime, 9% annual effective is to be paid on the loan, and Howard is to accumulate the amount necessary to repay the loan by depositing eight annual payments at the end of each year into a sinking fund. Calculate the interest rate the sinking fund must earn so that Howard is indifferent between the two offers. (B) 7.8%

17.

(C) 129

(C) 8.3%

(D) 10.3%

(E) 12.3%

A common stock is purchased at a price equal to ten times current earnings. During the next eight years the stock pays no dividends, but earnings increase 50%. At the end of eight years the stock is sold at a price equal to 16.5 times earnings. Calculate the effective annual yield rate. (A) .052

(B) .065

(C) .083

(D) .102

@.120

;: '

., \1

ID-39

SOLUTIONS TO PRACTICE MULTIPLE CHOICE TEST 8

1.

We seek

X

such that the present value of the consolidated loan equals the sum of the present

values of the two separate loans. We have

=

PVj

Then X

2.

3000(1.05)-8

=

=

=

2030.51 + 2702.26 1 + (1.06)-2

2504.12

ANSWER C. '

After four years we have

X ~ 1000· eXP[1\5d~ and Y

i= 3.

= 4000(1.04)-10 = 2702.26.

2030.51 and PV2

t ]

=

1000· exp[-ln(15

-

i

1000(1.04)6(1 + i) = 1000(1.26532)(1 + i). Equating and solving for we find ~ ,,.,~I.-,,,., - 1 = .07770, ANSWER B.

The accumulated value after four years is

=

100(.80)-2. exp [In (t2+8)1;] ~ 100(.80)-2( et"')

This same accumulated value is reached by 100(1+!i) i

=

4 [(3.1250)'0625 - 1]

=

A1'\'SWER E.

The present value of this inunediate perpetuity is J' where;'

The 1/1/98 balance is 1000 (l+!;')

=

(1 +i)8 - 1 is the effective rate

= ~, so i = (1.40)·125 -

10(l+!k)

12

could have been earned throughout is 1000(1+!k us

= 312.50.

16,and we seek the value of i. \Ve have

.29524,

per eight-year period. Then we easily find;' 5.

1000(1),

=

100(1-.20)-2 . exp [140. dt]

4.

i)I:] ~

= )32

1

= .04296,

A.~SWER D.

1990.76, and the 1/1/98 balance if rate

=

2203.76.

k

The second equa:ior: gives

k = 4[(2.20376)·03125 - 1] = .10, so that (l+!k Y2 = 1.34489 .

Then J. _- 2 [(1.99076)·10 1.34489

- 1] _-

.08. Then finally Jk -_

lJ8 .10 _- 1.25,

A.'\S\\'ER A. ~

III-40

6.

We are given

i

PV = 100aTIi + B . V12 = 1000 at = .035, so that

B = (l000-100aTIi)(1.035)12

=

have pvg

100a31.ol +

(Calculator comments:

= 150.87. Then at t

= 8, the beginning

of the ninth year, we

= 439.07,

150.87v~1

ANSWER D.

To compute the balloon payment at time 12, hit IAC/ONI, enter 12, hit

!El, enter 3.5, hit I%il, enter 100, hit IPMTI, enter 1000, hit IPVI, and then hit ICPTIIFVI to obtain

=

Note that the balloon at t

50.87.

12 is P MT

+ FV =

150.87, due to the calculator

conventions regarding the annuity keys. To compute the balance at time 8 using hit[tl], enter 1, hit~,

7.

=

a-2n I

n a-I

subtracting

+

and then hitICPTIIPV! to obtain 439.08.)

=

We are given PVi

i = 1%, enter 4,

2a2nj

+ a;j =

=

36 and PVii

2vn•

anj

=

6, all at rate j. Recall that

n so the first equation can be written as 2vn . a,ni vn . a-I'

PVi - PVii

we

find

3a;j

=

30,

so

anj

=

1

j vn

=

+

=

3a,n;

10, and

".n

36. Then

=

1 - 10j.

Substituting these values into the second equation we have 2(1 - 10j)(10) 8.

=

6, so j

=

We are given PVj and

we

Clearly P"Vl

seek

=

=

1~ (1 - 260)

=

.07,

A.l'\SWER E.

+ 2v3 + 3v4 + 4v5 + 5v6 + 6v7 + 5v8 + 4v9 + 3vlO + 2,-,ll ~ ".l2 = 25, P"Vl = v+2v2 + 3v3 + .., + 6v6 + 6v7 + 5v8 +4v9 + 31.·]0 + 2vl1 + v12• v2

+ (v+v2+

PVj

...

+v6)

=

25

+ a6ji'

To find a6ji' note that the I I-year

t = 2 to t = 7

annuity is the classic "rainbow" annuity, where a layer of six unit payments from has present value a6j at

t

= 1.

A similar layer from t

Continue in this way to the fi.nal layer from The present

val~e

of the

= 3 to t = 8 has present

value 0;- at

= 2.

t = 7 to t = 12, with the present value Q6' at t = 6.

six a6j values

is (a6j?

= 25,

so a6j = 5.

Ther:. finally

Pili = 25 + 5 = 30, 9.

t

-

A..... \SWERB.

Consider the annuity to be level at 900 per month, plus an increasing monthly annuity of 100 per month the first year, 200 per month the second year, ... , 1000 per month the tenth year. The ]01 present value of these two annuities is (12)(900)a(l2)

+

10 (12)(100)(Ia)C12,),

evaluated at

i = .03.

The monthly annuities are evaluated using text equations (4.9) and (4.41), producing

PV

=

(12)(100) CcL») [9aT61 + (I a)lOi]

_ -

(12)(100)

=

(12)(100)(1.013677)(76.77183

[

.03 12 [(1.03) 1/12 -

_ 1]

] ( 9alOl

+

alOi v . 0310 la) ..

+ 44.83899) =

147,928.88,

ANSWERE.

of:!

10.

The a:mual sinking fund deposit is D payment then due, is

SFB

=

=

1~~000 101·03

=

872.30. The balance at

= 3758.86.

872.30s4j.03(1.03)

+

extinguish the debt at that time is (10,000-3758.86)

11.

t

= 5, just

tr..e

Then the amount needed to

= 6741.14,

.05(10,000)

ANSWER C.

The semiannual returns on the loan are 50 each half-year in principal, plus interest payments of 20, 19, 18, ... , 1.

All such returns are invested at .025 effective per half-year,

accumulated value after 10 years is _ 50s201.025 + (Ds)20/.025 = 50(2).5448)

+

20(1.025)20 - 25.5448 .025

=

leading to i

(1.56634)'10 - 1

so the

= 1566.34.

=

This value represents the final return on the original loan, so we have 1000(l+i)IO

12.

before

1566.34,

ANSWERA.

= .0459,

Item (i) tells us that X is the sinking fund deposit whose value we seek. Item (ii) teIls us that

= Y, so the

X + X(l+j)

= Y XX

sinking fund rate of interest is j

- 1

=

*' - 2

=

.09,

using the information in Item (iii). Then after the fourth payment we have a net amount of loan of L - X·

=

X

=

s4j.09

3007.87. But we also know that L

=

_3007.82 s51.09 -

=

X·

sSi.09'

so we finally have

2130.89,

A..1\JS\\'"ERC.

84/.09

I, f:

13.

Using the book values at times 3 and 5 leads to a quadratic equation for the yield rate. \\"e tave

= 1000(.08) = 80.

FT

=

1082.27

=

BVS

r

Then

(B1t4) (1 +i) - 80

=

[(BV3)(1 +i) - 80](1 +i) -

80

= 1099.84(1 +i)2 - 80(1 +i) - 80. This quadratic

n-

3

= 9:

14.

=

The price is P

=

v2

= (3)-1/2,

Using the calculator

= 9. Now

ICPTIlli] to obtain n - 3

+ 1000(1.065)-12 =

80al2j.065

vn

i = 6.5%.

annuity keys we can ca~c:.:late

enter 6.5, hit I%il, enter 80, hit IPMTI, enter 1099.84, hit IPVI, enter 1000, r.::FVI,

and then hit

P

solves for

=

50v + 70v2

=

exp [-lnOtdt] and

v3

=

1122.38,

+ 1090v3,

A."\,'SV.'"ER B

where

exp [-!In(t2-t+l)I:J

(7)-1/2. Then P

,

reset!Rl to 12 and hit ICPT! !PVI to fi:-:c

=

=

(n2_n+1)-1/2.

50 + 70(3)-1/2 + 1090(7)-1/2

This gives us v

=

= 1,

502.40, .t>u"l"SWER A.

15.

e:Tective monthly rate

14,071

Then X

=

=

X·

aT20jj

IJ:8~i~2

(Calculator comment:

j

=

(1.03)1/12

=

X [(1+j).

= =

X [(1.03)1/12.

1. Then the equation of value is

-

1 - (1~j)-120] ., = X(10386242)

(1.03)1/12 1 1 - (1.03)-10]

To compute

ANS\\ =.:.

a120ij, 11

-7

I,

a120lj

Under Lynn's offer, the annual payment is

+ 1s°.!?0, where 81i

=

104.02, so that X

a~OOO

8/.1065

i is the

= II%il (to compute

=

191.90.

17.

= 9.81333,

and store j), ente~ ~

= ~64~61 =

which solves (by financial calculator) for

135.27.)

Under Ann's offer, the annua:

sinking fund interest rate.

between the two offers, these annual payments must be equal. s8ji

==

hit IAC/ONI12ndIIBGNI, enter 120, hit [H], en:e~ ::

enter 12, hit I

hitlPMTI, and then hit/CPTIIPV! to obtain

payment is .09(1000)

14,071. The payout period is 1:= =:':.n::_ '-

135.48,

hit 12ndll t> APRI, enter 12, hit I =

16.

=

l::.e balance after seven years is 10,000(1.05)7

If 191.90

i = .05762,

To be indifferent

= 90 +

then

•

ANSWER A

•

ls°.!?O, 81i

If current earnings is E, the stock is purchased for lOE. After eight years the current earnings level is (1.5)E, and the selling price is (16.5)(1.5)E. The effective annual compound yield is i, where 10E = (16.5)(1.5)E vB, so that v8 = .40404, which solves for = .11994, A..:."lSWER E

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