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Lecture

Chap003.doc


Department
Finance
Course Code
FIN 612
Professor
Cynthia Holmes

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Chapter 03 - Mortgage Loan Foundations: The Time Value of Money
Solutions to Questions - Chapter 3
Mortgage Loan Foundations: The Time Value of Money
Question 3-1
What is the essential concept in understanding compound interest?
The concept of earning interest on interest is the essential idea that must be understood in the compounding
process and is the cornerstone of all financial tables and concepts in the mathematics of finance.
Question 3-2
How are the interest factors (IFs) Exhibit 3-3 developed? How may financial calculators be used to calculate IFs
in Exhibit 3-3?
Computed from the general formula for compounding for monthly compounding for various combinations of “i”
and years. FV = PV x (1+i)n. Calculators can be used by entering $ 1 for PV, the desired values for n and i and
solving for FV.
Question 3-3
What general rule can be developed concerning maximum values and compounding intervals within a year? What
is an equivalent annual yield?
Whenever the nominal annual interest rates offered on two investments are equal, the investment with the more
frequent compounding interval within the year will always result in a higher effective annual yield. An equivalent
annual yield is a single, annualized discount rate that captures the effects of compounding (and if applicable,
interest rate changes).
Question 3-4
What does the time value of money (TVM) mean?
Time value simply means that if an investor is offered the choice between receiving $1 today or receiving $1 in the
future, the proper choice will always be to receive the $1 today, because that $1 can be invested in some
opportunity that will earn interest. Present value introduces the problem of knowing the future cash receipts for an
investment and trying to determine how much should be paid for the investment at present. When determining
how much should be paid today for an investment that is expected to produce income in the future, we must apply
an adjustment called discounting to income received in the future to reflect the time value of money.
Question 3-5
How does discounting, as used in determining present value, relate to compounding, as used in determining future
value? How would present value ever be used?
The discounting process is a process that is the opposite of compounding. To find the present value of any
investment is simply to compound in a “reverse” sense. This is done by taking the reciprocal of the interest factor
for the compound value of $1 at the interest rate, multiplying it by the future value of the investment to find its
present value.
Present value is used to find how much should be paid for a particular investment with a certain future value at a
given interest rate.
Question 3-6
What are the interest factors (IFs) in Exhibit 3-9? How are they developed? How may financial calculators be
used to calculate IFs in Exhibit 3-9?
Compound interest factors for the accumulation of $1 per period, e.g., $1 x [1 + (1+i) + (1+i)2 …] etc.
Calculators
may be used by entering $ 1 values for PMT, entering the desired values for n and i then solving for FV.
Question 3-7
What is an annuity? How is it defined? What is the difference between an ordinary annuity and an annuity due?
An annuity is a series of equal deposits or payments.
An ordinary annuity assumes payments or receipts occur at the end of a period.
An annuity due assumes deposits or payments are made at the beginning of the period.
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Chapter 03 - Mortgage Loan Foundations: The Time Value of Money
Question 3-8
How must one discount a series of uneven receipts to find PV?
Each periodic cash receipt or payment must be discounted individually then summed to obtain present value. That
is: PV= CF1 (1/1 + i)1 + CF2 (1/1+ i)2.+ CFn (1/1 + i)n where CF is cash inflow and i equals the discount rate.
Question 3-9
What is the sinking-fund factor? How and why is it used?
A sinking-fund factor is the reciprocal of interest factors for compounding annuities. These factors are used to
determine the amount of each payment in a series needed to accumulate a specified sum at a given time. To this
end, the specified sum is multiplied by the sinking-fund factor.
Question 3-10
What is an internal rate of return? How is it used? How does it relate to the concept of compound interest?
The internal rate of return integrates the concepts of compounding and present value. It represents a way of
measuring a return on investment over the entire investment period, expressed as a compound rate of interest. It
tells the investor what compound interest rate the return on an investment being considered is equivalent to.
Solutions to Problems - Chapter 3
Mortgage Loan Foundations: The Time Value of Money
Problem 3-1
a) Future Value = FV(n,i,PV,PMT)
= FV (7yrs, 6%, $12,000, 0)
= $18,044 (annual compounding)
b) Future Value = FV(n,i,PV,PMT)
= FV (28 quarters, 9% ÷ 4, $12,000, 0)
= $22,375 (quarterly compounding)
c) Equivalent annual yield: (consider one year only)
Future Value of (a) = FV(n,i,PV,PMT)
= FV (1yr, 6%, $12,000, 0)
= $12,720
($12,720 - $12,000) / $12,000 = 6.00% effective annual yield
Future Value of (b) = FV(n,i,PV,PMT)
= FV (1yr, 9%, $12,000, 0)
= $13,117
($13,117 - $12,000) / $12,000 = 9.31% effective annual yield
Alternative (b) is better because of its higher effective annual yield.
Problem 3-2
Investment A: 6% compounded monthly
Future Value of A = FV(n,i,PV,PMT)
= FV (12 mos., 6% ÷ 12, $25,000, 0)
= $26,542 (monthly compounding)
Investment B: 7% compounded annually
Future Value of B = FV(n,i,PV,PMT)
= FV (1yr, 7%, $25,000, 0)
= $26,750 (annual compounding)
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Chapter 03 - Mortgage Loan Foundations: The Time Value of Money
Investment B should be chosen over A. Investment B pays 7% compounded annually and is the better choice because it
provides the greater future value.
Problem 3-3
Find the future value of 24 deposits of $5,000 made at the end of each 6 months. Deposits will earn an annual rate of 8.0%,
compounded semi-annually.
Future Value = FV(n,i,PV,PMT)
= FV (24 periods, 8% ÷ 2, 0, $5,000)
= $195,413
Note: Total cash deposits are $5,000 x 24 = $120,000. Total interest equals $75,413 or ($195,413 - $120,000). The
$120,000 represents the return of capital (initial principal) while the $75,413 represents the interest earned on the capital
contributions.
Find the future value of 24 beginning-of-period payments of $5,000 at an annual rate of 8.0%, compounded semi-annually
based on an annuity due.
Future Value = FV(n,i,PV,PMT)
= FV (25 periods, 8% ÷ 2, 0, $5,000)
= $208,230
Note: n is changed to 25 because the deposits are made at the beginning of each period. Therefore, the first deposit will be
compounded 25 times whereas if the 1st deposit was made at the end of the period it would be compounded only 24 times.
This pattern holds true for each deposit made. The second deposit would be compounded 24 times and the last deposit
would be compounded once. This example illustrates the difference between and annuity due (beginning of period deposits)
and an ordinary annuity (end of period deposits).
Problem 3-4
Find the future value of quarterly payments of $1,250 for four years, each earning an interest rate of 10 percent annually,
compounded quarterly.
Future Value = FV(n,i,PV,PMT)
= FV (16 periods, 10% ÷ 4, 0, $1,250)
= $24,225
Problem 3-5 (REV)
End of Year Amount Deposited FV(n,i,PV,PMT) Future Value
1 $2,500 FV(4 yrs, 15%,$2,500, 0) $4,372.52
2 $0 FV(3 yrs, 15%,0, 0) $0.00
3 $750 FV(2 yrs, 15%, $750, 0) $991.88
4 $1,300 FV(1 yr, 15%, $1,300, 0) $1,495.00
5 $0 $0.00
Total Future Value = $6,859.40
The investor will have $6,859.40 on deposit at the end of the 5th year.
*Each deposit is made at the end of the year.
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