Student Name:_________________

PUID:_________________

MATH 373

Test 1

Fall 2021 September 23, 2021

1. Grace borrows 1000 to buy new ski equipment. She will repay the loan with level monthly

payments over the next 12 months. The loan has an annual effective interest rate of 8.0%.

Calculate the amount of Grace’s monthly payment.

Solution:

(12)

(12) (12) (12)

(12) (12) (12)

(12) 1

12

1

12

1 2 12

12 12 12

1 2 12

12 12 12

1,000

1 12 12

0.08

1.08 1 0.0064340312

1.00643403

Equation of Value : 1,000 ...

1,000 ...

1,00

i

Monthly

i i i

i i i

PV

T

i

i

v

PMT P

Pv Pv Pv

P v v v

(12) (12)

(12)

(12)

(12) (12)

1 13

12 12

1

12

1

12

1 13

12 12

1

1 13

0 ... by Geometric Sum Formula1

1

1,000

1 1.006434031,000 86.859406

1.00643403 1.00643403

i i

i

i

i i

v v

Pv

v

Pv v

P

2. Rura Incorporated manufactures farm equipment. Rura is going to build a new factory. Based

on projections, it is expected that Rura will need to invest 5,700,000 at the beginning of this

project. Additional cash flows over the next five years will be as follows:

End of Year Cash Flow

1 -1,000,000

2 X 3 2,000,000

4 4,000,000

5 5,000,000

After five years, the factory will be obsolete and not generate any additional cash flows.

This factory, based on expected cash flows, will generate an internal rate of return of 10%.

Calculate the Net Present Value of the expected cash flows at an annual effective rate of 11%.

Solution:

1

10%

2 3 4 5

10%

2 3 4 5

3 4

10% 1.1

0 ... by IRR definition

5.7 2 4 5 ... by NPV definition

0 5.7 2 4 5

5.7 2 4 5

IRR IRR

i IRR

i IRR IRR IRR IRR IRR

IRR IRR IRR IRR IRR

IRR IRR IRR IRR

i v

NPV NPV i i

NPV v Xv v v v

v Xv v v v

v v v vX

5

2

1 3 4 5

2

1

11%

2 3 4 5

11% 11% 11% 11% 11% 11%

11%

11%

5.7 1.1 2 1.1 4 1.1 5 1.10.883540947 million

1.1

11% 1.11

5.7 0.883540947 2 4 5

0.253439255 million

$253,439.

IRR

i

i i i i i i

i

i

v

X

i v

NPV v v v v v

NPV

NPV

26

3. You are given that 2

1( ) .

1 0.05v t

t

Calculate 10 10i . Your answer needs to be accurate to 5 decimal places.

Solution:

2

2

2 2

10 2

10

2

2 2

10 2

10 10

11 0.05

1 0.05

1 0.05 10 1 0.05 910 9

9 1 0.05 9

0.188118812

1 0.050.1

1 0.05 1 0.05

0.1 100.166666667

1 0.05 10

0.188118812 0.166666667 0.0214

t

v t a t tt

a ai

a

i

d da t t

tdt dt

a t t t

i

52

4. Ethan, Devesh, and Autumn enter into a financial agreement. Under the agreement, Ethan will

pay Devesh 1000 today. Additionally, Ethan will pay Autumn 2000 at the end of 3 years.

Devesh will pay Autumn X at the end of 2 years.

Autumn will pay Ethan 4106.30 at the end of 6 years.

Under this arrangement, Ethan, Devesh, and Autumn all have the same annual effective yield

rate.

Determine X.

Solution:

6 3

3

2

2

Using Ethan's Cash Flows:

Cash Inflow = Cash Outflow (present valued)

4,106.30 1,000 2,000

Let

4,106.30 2,000 1,000 0

2,000 2,000 4 4,106.3 1,000 ... by Quadratic Formula

2 4,106.3

0.7938

v v

y v

y y

y

y

3

1 3 33

2

2

32007 ... since 0 0 0

0.793832007 0.793832007 1

0.08

Using Devesh's Cash Flows:

Cash Outflow = Cash Inflow (present valued)

1.08 1,000

1,0001,166.40

1.08

v v y

v v i

i

X

X

5. David has the choice of two investments:

a. With Investment A, David will invest 5,000 today. The investment will earn an annual effective interest rate of i . Using the Rule of 72, David believes that he will have 10,000 at the end of 8 years.

b. Investment B is a US Treasury Bill, which has a price of 7,000 today and a maturity value of 7,142.86. The quoted rate on the Treasury Bill is 0.5i .

Calculate the annual effective interest rate earned by Investment B.

Solution:

%

%

%

Investment A :

72Under the Rule of 72, money doubles in years.

Since it takes 8 years for David to double his 5,000 to 10,000;

Thus,

728 9%

Investment B :

360 AmountQuoted Rate

Number of DaysUS

i

ii

160

365

of Interest

Maturity Value

360 7,142.86 7,0000.5 0.09

Number of Days 7,142.86

Number of Days 160

7,000 1 7,142.86

0.047167

i

i

6. Jaxon wants to have 1,000,000 on his 40th birthday. Jaxon’s 20th birthday is today.

Jaxon wants to invest K in an account which has a force of interest of 20.04 0.0003t t so

that he will exactly achieve his goal on his 40th birthday.

Determine .K

Solution:

202

0

203

0

3

0.04 0.0003

0.04 0.0001

0.04 20 0.0001 20 0

1.6

Equation of Value:

1,000,000

1,000,000

1,000,000

1,000,000

201,896.52

t dt

t t

Ke

Ke

Ke

Ke

K

7. Netzel Bank makes 5-year loans to college students. Netzel wants to receive an annual rate of

2.3% compounded continuously to compensate for deferred consumption. Additionally, Netzel

expects that inflation will occur at an annual rate of 3.3% compounded continuously over the

next five years. However, since the inflation rate could be higher, Netzel would like to receive

an annual rate of 0.2% compounded continuously as compensation for the inflation risk.

Additionally, Netzel expects 5% of the loans to default and plans to charge an additional 0.5% to

account for the default risk.

Determine the expected loan recovery rate for those that default.

Solution:

Amount they want to earn if there are no defaults:

0.023 0.033 0.002 0.058

Amount they charge factoring in defaults:

0.058 0.005 0.063

Let the Loan Recovery Rate be . Thus, it follows that:

Amount Ear

x

5 0.058 5 0.063 5 0.063

5 0.058 5 0.063

5 0.063

ned if no defaults Amount Earned factoring if defaults

1 0.05 0.05

0.950.506198

0.05

e e x e

e ex

e

8. Yegor borrows 25,000 from Kunyang. The loan will be repaid with three annual payments of

9619.95.

Kunyang reinvests the payments at an annual effective rate of r .

After reinvestment, Kunyang realizes an annual yield rate of 7% on the loan.

Determine r .

Solution:

3

2

Account Value after Reinvestment at realized yield rate:

3 25,000 3 25,000 1.07 30,626.08

Account Value from Reinvesting Payments at reinvestment rate:

3 9,619.95 1 9,619.95 1 9,619.95

30,626.08 9,6

A a

A r r

2

2

2

2

2

19.95 1 9,619.95 1 9,619.95

0 9,619.95 1 9,619.95 1 9,619.95 30,626.08

0 9,619.95 1 9,619.95 1 21,006.13

Let 1

0 9,619.95 9,619.95 21,006.13

9,619.95 9,619.95 4 9,619.95 21,006.13

2 9,61

r r

r r

r r

x r

x x

x

9.95

1.06 1 1.06

0.06

x r

r

9. Eric has 100,000 in his brokerage account on January 1, 2021.

On March 1, 2021, he has an account balance of 105,000 prior to withdrawing 40,000.

On July 1, 2022, Eric deposited 52,000. Prior to that deposit, he had a balance of 70,000.

On December 31, 2022, Eric had a balance of 130,000 in his account.

Using the simple interest rate approximation, estimate the annual dollar weighted return Eric

earns.

Solution:

1

1

2

100,000

130,000

40,000 52,000 12,000

130,000 100,000 12,000 18,000

1

18,000

2 18100,000 40,000 1 52,000 1

24 24

540.235808

229

1 1

1.235808 1

0.11166

t

T

A C I B I B A C

A

B

C

I

Ij

A C t

j

j

i j

i

i

9

10. Lexy deposited 55,000 into a fund three years ago.

Ten months later, the fund was worth 60,000 and Lexy withdrew 20,000 to pay for a vacation.

Two years ago, after a two month vacation, Lexy had 36,000 in her fund. She also had 6,000

leftover from her vacation money and she deposited this money back into her fund.

One year ago, Lexy had 44,000 in her fund. She also earned a bonus of 18,000 at work. She

decided to deposit this into her fund.

Today she has a fund that is worth 66,000.

Calculate her annual time weighted return over this three year period.

Solution:

1

2

3

4

1 1

3

60,000 121

55,000 11

36,000 91

60,000 20,000 10

44,000 221

36,000 6,000 21

66,000 331

44,000 18,000 31

12 9 22 331 1.094931

11 10 21 31

1 1 1.094931 1

0.030692

TW

TTW

j

j

j

j

j

i j i

i

T (MONTHS) VALUE CONTRIBUTION

0 55,000 0

10 60,000 -20,000

12 36,000 6,000

24 44,000 18,000

36 66,000 -

11. Catherine invests 1100 in an account earning compound interest. At the end of 10 years,

Catherine has 2200.

Kate invests 4000 in an account earing simple interest. At end of 8 years, Kate has 8000.

Let 5

Compoundd be the effective discount rate earned by Catherine in the 5th year.

Let 5

Simplei be the effective interest rate earned by Kate in the 5th year.

Calculate 5 5

Simple Compoundi d . Your answer must be accurate to five decimal places.

Solution:

10

1

10

5 4

5

5

Catherine's Equation of Value:

10 1,100 10

2, 200 1,100 1

2, 2001 .07177346

1,100

(5) (4) (1.07177346) (1.07177346)

(5) (1.07177346)

0.066967

Kate's

Compound

Compound Compound

A a

i

i

a ad

a

d d

5

5

5 5

Equation of Value:

8 1,100 8

8,000 4,000 1 8

0.125

1 5 0.125 1 4 0.1255 4

4 1 4 0.125

0.083333

0.083333 0.066967 0.01637

Simple

Simple

Simple

Simple

Simple Compound

A a

i

i

a ai

a

i

i d

12. Zhouyuan invests 11,000 in an account for eight years. The account earns:

a. An interest rate equivalent to annual effective discount rate of 6% for the first two

years;

b. A force of interest of 8% for the next three years; and

c. An interest rate of 4% compounded quarterly for the last three years.

Determine the amount in Zhouyuan’s account at the end of eight years.

Solution:

4 32 0.08 3 0.04

8 11,000 1 0.06 14

8 17,832.99

A e

A