Integrated Case

First National Bank

Feb-42 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 3

100 100 100

Uneven cash flow stream

0 1 2 3

-50 100 75 50

0 10% 1 2 3

100 FV=?PV= 100 N= 3

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

Time Value of Money Analysis 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?

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 3

PV=? 100

PV = FVn/(1+I)^N FV= 100 I= 10%PV = 100/(1+.01)^3 N= 3PV = 75.13 PV= $75.13

0 10% 1 2 3

100 125.97

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 3.8

1 2

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

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

0 100 100 100

AnswerAn 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

100 100 100 0110 1121 2331

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

100 100 100

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?

110 1121 2

133.1 3364.1

0 10% 1 2 3

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 7

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

614.4567

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?

I= 10%

0 1 2 3 4 5 6 7

100 100 100 100 100 100 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

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

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

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

PMT= 1095N= 45I= 12%

FV= ?

FV= ###

PMT= 1095N= 25I= 12%

FV= ?

FV= ###

PMT= ?N= 25I= 12%

FV= 1487262

PMT= ###

0 10% 1 2 3

100 300 3001 90.909092 247.9339

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?

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%.

3 225.39444 -34.15067

530.0867

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

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))^4-10.103812910.381289 %

10% Compounded daily

EAR = (1+(0.10/360))^360-10.105155610.515557

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

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?

Answer 10% semiannual compoundingPV= 100

N= 3I= 10%

M= 2

FV= 100(1+(.01/2))^(2*3)FV= 134.00956

10% Quarterly compoundingPV= 100

N= 3I= 10%

M= 4

FV= 100(1+(.01/4))^(4*3)FV= 134.48888

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

0 2 4 6

0 100 100 100Answer

0 5% 2 4 6

0 100 100 100 0110.25 2

121.5506 4331.8006

0 5% 2 4 6

100 100 1002 90.70295

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

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?

4 82.270256 74.62154

247.5947

rather than the EAR or the periodic rate, INOM/2 = 10%/2 = 5% to solve them?

Loan Amount= 1000I= 10%

N= 3PMT= ?

PMT= 402.1148

Period Beginning Balance Payment Interest Principa Ending Balance1 1000 402.1148 100 302.1148 697.8851963746232 697.885196374623 402.1148 69.78852 332.3263 365.5589123867083 365.558912386708 402.1148 36.55589 365.5589 0

L. (3) What would be wrong with your answer to parts L(1) and L(2) if you used the nominal rate, 10%,

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?

part of its evaluation process, you must take an examination on time value of money analysis

an ordinary annuity

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

4

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

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

4 5

100 100

8 9 10

100 100 100

8 9 10 11 12 13 14 15 16 17

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

4

-50

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

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%.

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

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

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

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

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

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

Aproximately = 0

What would be wrong with your answer to parts L(1) and L(2) if you used the nominal rate, 10%,

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?

18 19 20 21 22 23 24 25

100 100 100 100 100 100 100 100