Scan 0009

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II-34

PRACTICE MULTIPLE CHOICE TEST 9 1.

A fund earns interest at a force of interest 8t = kt. grow to 250 at the end of 5 years. Determine k.

A deposit of 100 at time

(D) .08

(/i~ 2.

.08(1n 1.5)

(B) .06

At an annual effective interest rate of

t = 0 will (E) In 2.5

~C)\.08(ln 2.5)

i, i > 0, the

••

Calculate the present value of a payment of 8000 at the end of year annual effective interest rate .

3.

r(BJi 1415

\J

(C) 1600

(D) 1775

t + 3 using the same (E) 2000

Barbara purchases an increasing perpetuity with payments occurring at the end of every 2 years. The first payment is 1, the second one is 2, the third one is 3, and so on. The price of the perpe-tuity is 110. Calculate the annual effective interest rate. (A) 4.50%

(B) 4.62%

(C) 4.75%

(E) 5.00% ~4.88%

4.

Eric receives 12000 from a life insurance policy. He uses the fund to purchase two different annuities, each costing 6000. The first annuity is a 24-year annuity-immediate paying K per year to himself. The second is an 8-year annuity-immediate paying 2K per year to his son. Both annuities are based on an effective annual interest rate of i, i > O. Determine i. (A) 6.0%

(C) 6.4% ®6.2%

5.

(D) 6.6%

(E) 6.8%

len'\

Victor wants to purchase a perpetuity paying 100 per year with the first payment due at the end of year 11. He can purchase it in either of two ways: (i) He can pay 90 per year at the end of each year for 10 years. (ii) He can pay K per year at the end of each year for the first 5 years, and nothing for the next 5 years. Calculate K . •.(A) 150

\,-.",

(B) 160

(C) 170

CD) 175

••• ••

following are all equal:

(i) The present value of 10,000 at the end of 6 years (ii) The sum of the present values of 6000 at the end of year t and 56,000 at the end of year 2t (iii) 5000 immediately

(A) 1330

•• ••• •••

(E) ] 80

•• •• •••

•• ••

••

till

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ll-34

PRACTICE MULTIPLE CHOICE TEST 9 1.

A fund earns interest at a force of interest 6t = kt. grow to 250 at the end of 5 years. Determine k.

;lf~ 2.

A deposit of 100 at time t

(D) .08 .08(1n 1.5)

(B) .06

~f:?08(1n

At an annual effective interest rate of i, i

=0

will

(B) In 2.5

2.5)

> 0, the following are all equal:

(i) The present value of 10,000 at the end of6 years (ii) The sum of the present values of 6000 at the end of year t and 56,000 at the end of year 2t (iii) 5000 inunediately Calculate the present value of a payment of 8000 at the end of year annual effective interest rate . (A) 1330

3.

(B) 2000

1415

(B) 4.62%

(C) 4.75%

'0

03)\4.88%

(B) 5.00%

Eric receives 12000 from a life insurance policy. He uses the fund to purchase two different annuities, each costing 6000. The first annuity is a 24-year annuity-immediate paying K per year to himself. The second is an 8-year annuity-inunediate paying 2K per year to his son. Both annuities are based on an effective annual interest rate of i, i > O. Determine i. (A) 6.0%

(C) 6.4%

®6.2% 5.

(D) 1775

Barbara purchases an increasing perpetuity with payments occurring at the end of every 2 years. The first payment is 1, the second one is 2, the third one is 3, and so on. The price of the perpe-tuity is 110. Calculate the annual effective interest rate.

(A) 4.50%

4.

V

(C) 1600

t + 3 using the same

(D) 6.6%

(B) 6.8%

lCn'\

Victor wants to purchase a perpetuity paying 100 per year with the first payment due at the end of year 11. He can purchase it in either of two ways: (i) He can pay 90 per year at the end of each year for 10 years. (ii) He can pay K per year at the end of each year for the first 5 years, and nothing for the next 5 years. Calculate \ (A:) 150 \

K. (B) 160

(C) 170

(D) 175

(B) 180

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•.,

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II-35

6.

Esther invests 100 at the end of each year for 12 years at an annual effective interest rate of i. The interest payments are reinvested at an annual effective rate of 5%. The accumulated value at the end of 12 years is 1748.40. Calculate i.

\5

(A) 6%

7.

At time t

= 0, Billy

(C) 8%

(D) 9%

(E) 10%

7%

puts 625 into an account paying 6% simple interest. At the end of year

2, George puts 400 into an account paying interest at a force of interest Ot = 6~t' for t 2: 2. If both accounts continue to earn interest indefinitely at the levels given above, the amounts in the two accounts will be equal at the end of year n. Calculate n. (A) 23

(B) 24

&> 26

(E) 27

~ 25 r<4) 8.

A bond with a par value of 1000 and 6% semiannual coupons is redeemable for 1100. You are glVen: (i) The bond is purchased at P to yield 8%, convertible semiannually. (ii) The amount of principal adjustment for the 16th semiannual period is 5. Calculate P. \ EA)

9.

760

(C) 790

(D) 800

(E) 820

A perpetuity with annual payments is payable beginning 10 years from now. The first payment is 50. Each annual payment thereafter is increased by 10 until a payment of 150 is reached. Subsequent payments remain level at 150. This perpetuity is purchased by means of 10 annual premiums, with the first premium of P due immediately. Each premium after the first is 105% of the preceding one. The annual effective interest rates are 5% during the first 9 y~ars and 3% thereafter. Calculate P. (A) 281

10.

(B) 770

(B) 286

(9291

(D) 296

(E) 301

A deposit of 1 is made at the end of each year for 30 years into a bank account that pays interest at the end of each year at j per annum. Each interest payment is reinvested to earn an annual effective interest rate of j12. The accumulated value of these interest payments at the end of30 years is 72.88. Determine j. (A) 8.0%

(B) 8.5%

(C) 9.0%

(D) 9.5%

(~10.0%

II-36

11. An investment fund is estaplished at time 0 with a deposit of 5000. 1000 is added at the end of 4 months, and an additional 4000 is added at the end of 8 months. No withdrawals are made. The fund value, including interest, is 10,560 at the end of 1 year. The force of interest at time t is , ,

I~

.•.\1_,

for 0 ::; t::; 1. Determine k. f

(A) .072

(B) .076

r",'

'8

.080

(D) .084

(E) .088

12. Bart buys a 28-year bond with a par value of 1200 and annual coupons. The bond is redeemable at par. Bart pays 1968 for the bond, assuming an annual effective yield rate of i. The coupon rate on the bond is twice the yield rate. At the end of 7 years, Bart sells the bond for P, which produced the same annual effective yield rate of to the new buyer. Calculate P.

i

(A) 1470

(B) 1620

(C) 1680

(15)

1840

(E) 1880

13. An investment will triple in 87.88 years at a constant force of interest O. Another investment will quadruple in t years at a nominal rate of interest numerically equal to 0 and convertible once every 4 years. Calculate t. (A) 101 years

(B) 106 years

(C) 109 years

(E) 125 years

14. John borrows 10,000 for 10 years and uses a sinking fund to repay the principal. The sinking fund deposits earn an annual effective interest rate of 5%. The total required payment for both the interest and the sinking fund deposit made at the end of each year is 1445.04. Calculate the annual effective interest rate charged on the loan. (A) 5.5%

(B) 6.0%

®J6.5% ~'

(D) 7.0%

(E) 7.5%

15. You are given: (i)

The present value of an annuity-due that pays 300 every 6 months during the first 15 years and 200 every 6 months during the second 15 years is 6000.

(ii)

The present value of a 15-year deferred annuity-due that pays 350 every 6 months for 15 years is 4000.

(iii) The present value of an annuity-due that pays 100 every 6 months during the first 15 years and 200 every 6 months during the next 15 years is X. The same interest rate is used in all calculations. (A) 3220

(B) 3320

Determine X.

(C) 3420

/" \ (Dy 3520 "'-*/

(E) 3620

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II-37

16.

An n-year 1000 par value bond with 4.20% annual coupons is purchased at a price to yield an annual effective rate of i. You are given: i)

If the annual coupon rate had been 5.25% instead of 4.20%, the price of the bond would have increased by 100.

ii)

At the time of purchase, the present value of all the coupon payments is equal to the present value of the bond's redemption value of 1000.

Calculate i. (A) 5.0%

17.

(B) 5.5%

(C) 5.9%

(E) 6.5%

A twenty-six week Treasury bill maturing for 10,000 is bought at a discount to yield 3.51 % annually. For the same purchase price, a zero-coupon bond maturing for 50,000 at the end of 20 years is available. The nominal yield rate convertible semiannually on this bond is i. Calculate i. (A) 4.15%

(B) 4.25%

·~.30%

(D) 8.50%

(E) 18.10%

ii

I; ill ili

I

I

I

ID-43

SOLUTIONS TO PRACTICE MULTIPLE CHOICE TEST 9 1.

= ~,

ktdt

15

=

so a(5)

e25k/2

and 250

=

100a(5). Taking logs, k

= f5 ·In(2.5), At~SWERC.

2.

=

We are given 10,000v6

6000vt + 56,000v2t

and the first equation implies

=

Then 8000vt+3

v6

8000vt. v3

=

5000. The second equation implies vt

= .50, so that v3 = J50. = (8000)(.25)· J50 = 1414.21,

=

.25

ANSWER B

.. a- nvn 3.

=

(1 a)-In

that

nl.

(1.10)1/2 - 1

=

6000

5.

=

=

K· = 2K.

a241 aSj

.618 and i

=

1ii

and cross multiply to obtain 110i2 - i-I

=

0, so

=

0,

ANSWER D

=

K·

=

=

1 (1_V24) 2" 1-v8

a241

2K· a8j' Then we have

=

•.. 1, whIch ImplIes 2

=

1 + v8 + v16, from which we find

= .062,

ANSWER B

Item (i) tells us that 90slOfi

=

1?0

=

100/i, from which we find (1+i)lO

(K,sSli)(1+i)5

imp1iesK

=

190/90, and i

= 151.93,

= .0776.

ANSWER A

The series of interest payments is 100i, 200i, ... , 1100i, made and immediately reinvested at times

t=

2, 3, ... , 12. The accumulation of the interest payments at time 12 is thus

100i . (1S >m.05 1748.40

=

=

100i

[:5-11~05 - 11]

=

7834.25i.

Then at the end of 12 years we have

1200 + ~ 7834.25i, which solves for i '--v---' deposits

7.

d

-d1and n-+cc hm nvn

.10 as an effective biannual rate. Then the effective annual rate is

Then,fromitem(ii), 6.

Substitute

=

.0488,

We are given 6000

v8

= 110.

(1a)ool = (~)

which implies i

4.

Taking the limit as n -t 00, we find n--+oo lim a-In

Z

= .07,

After n years, the amount in Billy's account is A(n) accumulates from

t=

400 . exp

i

(in

.625(l+.06n)

6

=

ANSWER B

ace. int.

2 to

t dt)

400( ~),

=

625(1+.06n).

George's account

t = n, producing

=

400·

e1n(6+n)-ln

(6+2)

which solves for n

= 400 ( ~

= 26,

). Then at time t

= n we have ANSWER D

(

•

ill-44

8.

•

•

The semiannual coupon is 30 and the semiannual yield rate is .04. Item (ii) tells us that 5

= BVl6

spectively,

= (1.04)BV1S-30 - BVjs, from which we find BVjs = 875. = p(1.04)IS - 30SiSj.04' and P = 30aiSj.04 + 875(1.04)-IS =

BVjs

-

BVjs

Then, retro819.41, ANSWERE

9. 3%

... ... I

50

••

The accumulated value of the deposits at p(1.05)9

+P(1.05)(1.05)8+

drawals, also at t

...

10.

=

•••

t = 9 is

+ p(1.05)9 =

10P(1.05)9, and the present value of the with-

= 9, is

40aoo/.03 + 1O(Ia)cci.o3 - 1O(1.03)-11(Ia)ool.03 Equating att

9 we find P

=

4510.07 10(1.05)9

=

= .b~+ lOC~) 290.72

find

8291J!2

value of the interest is 72.88

= 65.44.

= j.

(I

[1-(1.03)-11]

=

4510.07 . ANSWER C

'

= 2, 3, ...

, 30, respectively.

= j [8-29U;/2- 29] ' from finds j/2 = .05, so j = .10,

s)29/i/2

Then the BA-35 calculator directly

The

which we

ANSWERE .

11.

The accumulated fund at

5000 . exp

=

Notethatl10tdt 10,560

12.

=

Here g

= r = 2i,

1968

=

Po

which we find P7

=

1200V21

=

[J,;, 8, dt]

=

-In[l+(l-t)kJI;

+

5000(1+k)

1000(l+~k)

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till

•••

••

fill ••• ~

t = 1 is

[1' 0, dt] + 1000 . exp

••

••

The sequence of interest payments is j, 2j, ... , 29j at times t accumulated

•

• •

v'

2150 20 21 19 Out ... 9P(l.05) 1I140 I P(1.05)2 ,60 I.11 .. 10 P(1~05)9 5%

• • •

+ 4000

. exp [/';,

In[l+(1-r)kJ.

+ 4000(1+1k),

8, dt] .

Then the fund value is

which solves for k

= .08,

ANSWER C

so K

+ (g/i)(C-K)

i = .03716.

=

K

+ 2(l200-K),

so that K

= 432 =

l200vf8,

from

Then at time 7,

+ '----v-" (g/i)(1200-1200v21)

= 1842.29,

ANSWER D

still 2

1

I

1

ill-45

13.

14.

We are given 3 4

=

If

i' is the rate

(l +46)t/4

1445.04

15.

1O,000i'

+

~O,OOO , from which we find 1Oj.05

=

Th en

an d"a301

30 I" a301

=

ANSWER D

charged on the loan, then we have

We are given PV;

PViii

16.

=

= e87.886,s06=.0125l7. We are also given = (1.050068i/4, from which we find t = 113.64,

= /80

100i:i301

300i:i301

+ 200

+ 200· 260 = 2T'

. 301i:i301

=

301i:i301

=

i' = .065,

ANSWER C

=

6000, and PV;i

350·

301i:i30j

=

4000.

W e see k

100 (22610)+ 200 (~)

=

3523.81,

ANSWER D

We are given P

= 42anji + 1000vf

and

+ 100 = 52.5anji + 1000vf,

P

= 1OOOvf. Subtracting the two price equations = l8.~and 42 ( l8.~)= 1OOOvi, from which we find vi = .40.

where anji

42anji

1-

anji

which solves for

17.

= -z-'

gives

100

= 10.5anli,

Then

vi _ 1 - .40 _ 100 - -

i-m,

i = .063,

ANSWER D

The purchase price of the 26-week (l82-day) T-bill is calculated as

From the zero-coupon bond data we find 9822.55 from which we find

i(2)

=

so

.083,

=

50,000

(

1+

T '(2») -40 ' ANSWERC