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a.            a)

b)

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c)

b.

FV=?

PV=100

r=10%p.a.

t=3years

c.

PV=?

FV=100

r=10%p.a.

t=3years

Rearranging and calculating:

d. Using the rule of 72:

 where,

‘t’ is the time taken for the investment to double

‘r’ is the rate at which the investment will double

t=3years

r=?

Thus, a 24% interest rate is required to double an investment in 3 years’ time.

e. Annuity is the name given to the terminating (it will end at some point in time) cash flows of fixed amounts in a pre-defined time period.  There are two types of annuities:

u In an ordinary annuity the payments are given out at the end of each period (e.g. a week, a year).

u In an annuity-due the payments are given out at the beginning of every period.

The annuity shown below is an ordinary annuity. It can be changes to annuity-due by moving the cash flows (payments) to one period earlier i.e. instead of the first payment being given at the end of Year 0, it should be given at the beginning of Year 0.

f.

(1) Using the Future Value formula for annuities:

C=$100

r=10%

t=3years

 (2) Using the Present Value formula for annuities:

C=$100

r=10%

t=3years

(3) Using the Future Value formula for annuities-due:

C=$100

r=10%

t=3years

Using the Present Value formula for annuities-due:

C=$100

r=10%

t=3years

g. The present value can be found by summing up the individual present values of each cash flow:

FV=uneven stream of cash flows

t=variable (from 1 to 4)

r=10%p.a

h.

(1)          a) The stated or quoted nominal rate is the simple annual percent rate on a loan. It does have take into account the compounding period and it is the simple rate to be used in annual compounding.

                b) The periodic rate is the interest rate corresponding to the compounding period. It is found by dividing the nominal rate by the number of compounding periods in a year. In short, it is the rate to be used for calculating interest in a given period within a year.

(2) The Future value will be larger if we compound an initial amount more often than a year i.e. there are more compounding periods in a year than 1. Holding the stated interest rate constant, we will find that the future value will be higher. This is due to the quick compounding effect. The sum compounds more often than before, thus there will be more interest accrued in the successive periods than what would have accrued had those consecutive periods been one. This can be explained by the fact that shorter periods result in compounding more than once a year, and thus there will be more interest than what would have been in compounding just once a year.

(3) Annual Compounding:

Semi-annual compounding:

Quarterly compounding:

Monthly compounding:

Daily compounding:

(4) The Effective Annual Rate is an investment’s annual rate of interest when compounding occurs more often than once a year. Calculated as the following:

EAR = [1+ (Quoted Rate/m)]m – 1

Where m is the no of periods per year

EAR for a 12% nominal rate, compounded semi-annually,

EAR for a 12% nominal rate, compounded quarterly,

EAR for a 12% nominal rate, compounded monthly,

EAR for a 12% nominal rate, compounded daily,

i. No, the effective annual rate will not be equal to the quoted rate except for one case – annual compounding. If the compounding is done in periods other than a year, then the EAR will not be equal to the nominal/quoted rate. It will be greater in cases where the compounding period is less than a year.

j.

(1) Since, there will be equal installments, we determine the amount of one installment using the formula:

PV=$1,000

r=10%

t=3years

Rearranging and calculating the value of ‘C’ (one yearly instalment) is:

Date
Opening Balance
Monthly Payment
Interest Expense
Reduction in Principal
Book Value
Year 0

1,000.00
Year 1
1,000.00
402.11
100
302.11
697.89
Year 2
697.89
402.11
69.79
332.32
365.57
Year 3
365.57
402.13
36.57
365.57
0.00

(2) Annual interest expense in Year 2: $69.79

k. Using the simple formula for Present Value:

Days passed for the investment = 273 days

PV=$100

r=11.33463% p.a. (more appropriately for daily compounding r=0.031054)

Thus, on October 1, I will have $108.85 in my account.

l.

(1) Using the formula for Future Value of annuities:

C=$100

r=10%p.a (EAR=10.25%p.a: using 5% semi-annually)

t=3years

(2) Using the formula for calculating the Present Value of a Future amount of money:

FV=331.8

r=5% semi-annual

t=3years (or more appropriately 6 semi-annual periods)

Rearranging and calculating:

(3) Yes, the stream is an annuity.

(4) The condition for using the nominal rate is that the compounding is done on a yearly basis i.e then the nominal rate can be used in the calculations. This is because when the compounding period is other than a year, then using the nominal rate will result in undervaluation of the Future Value.

m.

(1) Future Value of the investment in bank:

PV=$850

r=6.76649p.a. (or more appropriately 0.018538% per day)

t=15 months (or more appropriately 457 days: 12 months (365 days) + 3 months (31+30+31=92 days))

Future Value of buying the note: $1,000

(2) Present Value of the investment in bank: $850

Present Value of the buying the note:

FV=$1,000

r=6.76649p.a. (or more appropriately 0.018538% per day)

t=15 months (or more appropriately 457 days: 12 months (365 days) + 3 months (31+30+31=92 days))

Rearranging and calculating;

(3) EAR of investment in the bank: 7%p.a.

EAR of investment in the note can be found using the formula:

FV=$1,000

PV=$850

t=457days

r=? (This will be the daily compounding rate, not the EAR)

 or

Now we find the EAR:

Yes, we should buy the note. All of the above calculations, (1), (2) and (3), point to the same conclusion investing in the note is more worthy than investing in the bank. The FV of the note is higher, the PV of the note is higher and the EAR of the note is higher too making the investment in the note more profitable.

BIBLIOGRAPHY

Block, Hirt (2005). Foundations of Financial Management (11th ed.) New York: McGraw-Hill.  Chapters 9 and 14.

Ross, E. (2006). Fundamentals of Corporate Finance (6th ed.) New York: Westerfield and Jordan.  Chapter 5.

Time Value of Money. Retrieved April 20, 2008, from Leasing and Time Value of Money Web site: http://www.executivecaliber.ws/sys-tmpl/timevalueofmoney/

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