Fm Case - First National Bank

Integrated Case

First National Bank

part of its evaluation process, you must take an examination on time value of money analysis covering the following questions.

at the end of Years 0 through 3.

Answer Lump sum

0 1 2

100Annuity

0 1 2

100 100

Uneven cash flow stream

0 1 2

-50 100 75

0 10% 1 2

100PV= 100 N=

FVN = PV(1 + I)^N So FV3 = 100(1.10)^3 = 100(1.3310) = 133.10. FV=

B. (2) What’s the present value of 100 to be received in 3 years if the interest rate is 10%,annual compounding?

0 10% 1 2

You have applied for a job with a local bank. As

A Draw time lines for (1) a 100 lump sum cash flow at the end of Year 2, (2) an ordinary annuity of 100 per year for 3 years, and (3) an uneven cash flow stream of -50, 100, 75, and 50

B. (1) What’s the future value of 100 after 3 years if it earns 10%, annual compounding?

PV=?

PV = FVn/(1+I)^N FV= 100 I=PV = 100/(1+.01)^3 N= 3PV = 75.13 PV=

0 10% 1 2

100

100(1 + I)^3 = $125.97. PV= 100(1+I)^3=125.97/100 FV= 125.97(1+I)^3=1.2597 N= 3(1+I)^(3*1/3)=1.2597^1*3 I= ?

41.99 (1+I)=1.07990.4199 I=1.0799-1 I= 8%

I=.0799I=8%

2 = 1(1 + I)^N Pv= 12 = 1(1.20)^N. FV= 22/1=(1.20)^N I= 20%2=(1.20)^N N= ?ln2=N ln(1.20)N= ln2/ln(1.20) N= -3.80178N=3.8

0 20% 1 2 3

1

is shown here? How would you change it to the other type of annuity?0 1 2 3

0 100 100 100

Answer

C. What annual interest rate would cause 100 to grow to 125.97 in 3 years?

D. If a company’s sales are growing at a rate of 20% annually, how long will it take sales to double?

E. What’s the difference between an ordinary annuity and an annuity due? What type of annuity

An ordinary annuity has end-of-period payments,while an annuity due has beginning-of-period payments.

The annuity shown above is an ordinary annuity. To convert it to an annuity due,shift each payment to the left.

0 1 2 3

100 100 100 0

Answer0 10% 1 2 3

100 100 100110121331

FVAn = 100(1) + 100(1.10) + 100(1.10)^2 = 100[1 + (1.10) + (1.10)2] = 100(3.3100) = 331.00.

0 10% 1 2 3

100 100 1001 90.909092 82.644633 75.13148

248.6852

0 10% 1 2 3

100 100 100110121

133.1364.1

0 10% 1 2 3

F. (1) What is the future value of a 3-year, 100 ordinary annuity if the annual interest rate is 10%?

F. (2) What is its present value?

F. (3) What would the future and present values be if it were an annuity due?

0 100 100 1001 90.909092 82.64463

173.5537

0 10% 1 2 3

100 100 1001 90.909092 82.644633 75.131484 68.301355 62.09213

379.0787

I= 10%

0 1 2 3 4 5 6

100 100 100 100 100 1001 90.909092 82.644633 75.131484 68.301355 62.092136 56.447397 51.315818 46.650749 42.4097610 38.55433

614.4567

I= 10%

0 1 2 3 4 5 6

G. A 5-year $100 ordinary annuity has an annual interest rate of 10%. (1) What is its present value?

G. (2) What would the present value be if it was a 10-year annuity?

G. (3) What would the present value be if it was a 25-year annuity?

100 100 100 100 100 1001 90.909092 82.644633 75.131484 68.301355 62.092136 56.447397 51.315818 46.650749 42.4097610 38.5543311 35.0493912 31.8630813 28.9664414 26.3331315 23.939216 21.7629117 19.7844718 17.9858819 16.350820 14.8643621 13.5130622 12.284623 11.1678224 10.1525625 9.2296

907.704

FV= 100I= 10%

PV= 1000

At the end of each year, she invests the accumulated savings ($1,095) in a brokerage accountwith an expected annual return of 12%.

G. (4) What would the present value be if this was a perpetuity?

H. A 20-year-old student wants to save $3 a day for her retirement. Every day she places $3 in a drawer.

(1) If she keeps saving in this manner, how much will she have accumulated at age 65?

PMT= 1095N= 45I= 12%

FV= ?

FV= $1,487,261.89

PMT= 1095N= 25I= 12%

FV= ?

FV= $146,000.59

PMT= ?N= 25I= 12%

FV= 1487261.89

PMT= $11,154.42

0 10% 1 2

100 3001 90.909092 247.93393 225.39444 -34.15067

530.0867

for example, semiannually, holding the stated (nominal) rate constant? Why?

H. (2) If a 40-year-old investor began saving in this manner, how much would he have at age 65?

H. (3) How much would the 40-year-old investor have to save each year to accumulate the same amount at 65 as the 20-year-old investor?

I. What is the present value of the following uneven cash flow stream? The annual interest rate is 10%.

J. (1) Will the future value be larger or smaller if we compound an initial amount more often than annually,

AnswerAccounts that pay interest more frequently than once a year,for example, semiannually, quarterly, or daily, have future values that are higherbecause interest is earned on interest more often.

Answer The quoted, or nominal, rate is merely the quoted percentage rate of return.

Answer The periodic rate is the rate charged by a lender or paid by a borrower each period.

Answer The effective annual rate (EAR) is the rate of interest that would provide an identical future dollar value under annual compounding.

Answer10% compounded semiannually

EAR = (1+(0.10/2))^2-10.102510.25 %

10% Compounded quarterly

EAR = (1+(0.10/4))^4-10.103812890625

10.3812890625 %

10% Compounded daily

EAR = (1+(0.10/360))^360-10.105155571428058

10.5155571428058

Answer 10% semiannual compoundingPV= 100

N= 3I= 10%

J. (2) Define (a) the stated, or quoted, or nominal, rate,

(b) the periodic rate,

(c) the effective annual rate (EAR or EFF%).

J. (3) What is the EAR corresponding to a nominal rate of 10% compounded semiannually? Compounded quarterly? Compounded daily?

J. (4) What is the future value of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?

M= 2

FV= 100(1+(.01/2))^(2*3)FV= 134.0095640625

10% Quarterly compoundingPV= 100

N= 3I= 10%

M= 4

FV= 100(1+(.01/4))^(4*3)FV= 134.48888242463

Answer If annual compounding is used, then the nominal rate will be equal to the effective annual rate.

0 2 4

0 100 100Answer

0 5% 2 4 6

0 100 100 100110.25

121.5506331.8006

0 5% 2 4 6

100 100 1002 90.702954 82.270256 74.62154

247.5947

K. When will the EAR equal the nominal (quoted) rate?

L. (1) What is the value at the end of Year 3 of the following cash flow stream if interest is 10%, compounded semiannually?

L. (2) What is the PV?

M. (1) Construct an amortization schedule for a $1,000, 10% annual interest loan with 3 equal installments. (2) What is the annual interest expense for the borrower, and the annual interest income for the lender, during Year 2?

Loan Amount= 1000I= 10%

N= 3PMT= ?

PMT= 402.1148

Period Beginning Balance Payment Interest Principa Ending Balance1 1000 402.1148 100 302.1148 697.8851963746222 697.885196374622 402.1148 69.78852 332.3263 365.5589123867073 365.558912386707 402.1148 36.55589 365.5589 0

3

100

3

50

3

FV=?3

$133.10

3

an ordinary annuity

100

10%

$75.13

3

125.97

3.8 4

2

If a company’s sales are growing at a rate of 20% annually, how long will it take sales to double?

What’s the difference between an ordinary annuity and an annuity due? What type of annuity

012

123

What is the future value of a 3-year, 100 ordinary annuity if the annual interest rate is 10%?

4 5

100 100

7 8 9 10

100 100 100 100

7 8 9 10 11 12 13 14 15 16

100 100 100 100 100 100 100 100 100 100

At the end of each year, she invests the accumulated savings ($1,095) in a brokerage account A 20-year-old student wants to save $3 a day for her retirement. Every day she places $3 in a drawer.

3 4

300 -50

If a 40-year-old investor began saving in this manner, how much would he have at age 65?

How much would the 40-year-old investor have to save each year to accumulate the same amount at 65 as the 20-year-old investor?

What is the present value of the following uneven cash flow stream? The annual interest rate is 10%.

Will the future value be larger or smaller if we compound an initial amount more often than annually,

The effective annual rate (EAR) is the rate of interest that would provide an identical future dollar value under annual compounding.

What is the EAR corresponding to a nominal rate of 10% compounded semiannually? Compounded quarterly? Compounded daily?

What is the future value of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?

If annual compounding is used, then the nominal rate will be equal to the effective annual rate.

6

100

024

What is the value at the end of Year 3 of the following cash flow stream if interest is 10%, compounded semiannually?

Construct an amortization schedule for a $1,000, 10% annual interest loan with 3 equal installments. What is the annual interest expense for the borrower, and the annual interest income for the lender, during Year 2?

Ending Balance697.885196374622365.558912386707

0 Aproximately = 0

17 18 19 20 21 22 23 24 25

100 100 100 100 100 100 100 100 100