Math of finance with exercises

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Lecture-7Lecture-7
Mathematics of FinanceMathematics of Finance
or concepts of valueor concepts of value
WITH EXCERCISEWITH EXCERCISE
Muhammad Shafiq
University of Balochistan, Quetta
forshaf@gmail.com
www.Slideshare.net/forshaf

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The Time Value of MoneyThe Time Value of Money
The Interest Rate
Simple Interest
Compound Interest
Amortizing a Loan

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The time value of money (TVM) is the idea
that money available at the present time is
worth more than the same amount in the
future due to its potential earning capacity.
This core principle of finance holds that,
provided money can earn interest, any
amount of money is worth more the sooner
it is received.
Why does money haveWhy does money have
time value?time value?

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Obviously, $10,000 today$10,000 today.
You already recognize that there is
TIME VALUE TO MONEYTIME VALUE TO MONEY!!
The Interest RateThe Interest Rate
Which would you prefer -- $10,000$10,000
todaytoday or $10,000 in 5 years$10,000 in 5 years?

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Types of InterestTypes of Interest
Compound InterestCompound Interest
Interest paid (earned) on any previous
interest earned, as well as on the
principal borrowed (lent).
Simple InterestSimple Interest
Interest paid (earned) on only the original
amount, or principal borrowed (lent).

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1. Read problem thoroughly
2. Determine if it is a PV or FV problem
3. Create a time line
4. Put cash flows and arrows on time line
5. Determine if solution involves a single
CF, annuity stream(s), or mixed flow
6. Solve the problem
7. Check with financial calculator (optional)
Steps to Solve Time ValueSteps to Solve Time Value
of Money Problemsof Money Problems

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Simple Interest FormulaSimple Interest Formula
FormulaFormula SI = P0(i)(n)
SI: Simple Interest
P0: Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods

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SI = P0(i)(n)
= $1,000(.07)(2)
= $140$140
Simple Interest ExampleSimple Interest Example
Assume that you deposit $1,000 in an
account earning 7% simple interest for
2 years. What is the accumulated
interest at the end of the 2nd year?

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FVFV = P0 + SI
= $1,000 + $140
= $1,140$1,140
Future ValueFuture Value is the value at some future
time of a present amount of money, or a
series of payments, evaluated at a given
interest rate.
Simple Interest (FV)Simple Interest (FV)
What is the Future ValueFuture Value (FVFV) of the
deposit?

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The Present Value is simply the
$1,000 you originally deposited.
That is the value today!
Present ValuePresent Value is the current value of a
future amount of money, or a series of
payments, evaluated at a given interest
rate.
Simple Interest (PV)Simple Interest (PV)
What is the Present ValuePresent Value (PVPV) of the
previous problem?

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Future value is the value of an asset at a
specific date. It measures the nominal
future sum of money that a given sum of
money is "worth" at a specified time in the
future assuming a certain interest rate, or
more generally, rate of return; it is the
present value multiplied by the
accumulation function.

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FVFV11 = PP00 (1+i)1
= $1,000$1,000 (1.07)
= $1,070$1,070
You earned $70 interest on your $1,000
deposit over the first year.
This is the same amount of interest you
would earn under simple interest.
Future ValueFuture Value Compound InterestCompound Interest::
Single Deposit (Formula)Single Deposit (Formula)

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FVFV11 = P0(1+i)1
FVFV22 = P0(1+i)2
General Future ValueFuture Value Formula:
FVFVnn = P0 (1+i)n
or FVFVnn = P0 (FVIFFVIFi,n) -- See Table ISee Table I
General Future ValueGeneral Future Value
FormulaFormula
General Future ValueGeneral Future Value
FormulaFormula
etc.

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Calculation based on Future value Interest Factor Table 1:
FVFV55 = $10,000 (FVIFFVIF10%, 5) =
$10,000 (1.611) =
$16,110$16,110 [Due to Rounding]
Problem SolutionProblem SolutionProblem SolutionProblem Solution
Julie Miller wants to know how large her deposit of $10,000$10,000 today will
become at a compound annual interest rate of 10% for 5 years5 years.
Calculation based on general formula: FVFVnn = P0 (1+i)n
FVFV55 = $10,000 (1+ 0.10)5
= $16,105.10$16,105.10

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At the end of three years, how much is an initial deposit of At the end of three years, how much is an initial deposit of
$100 worth, assuming a compound annual interest rate of $100 worth, assuming a compound annual interest rate of
(i) 100 percent? (ii) 10 percent (iii) zero percent?(i) 100 percent? (ii) 10 percent (iii) zero percent?
FVn = P0(1 + i)n
(i) FV3 = $100(2.0)3
= $100(8) = $800
(ii) FV3 = $100(1.10)3 = $100(1.331) = $133.10
(iii) FV3 = $100(1.0)3 = $100(1) = $100

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Future value example:Future value example:
At the end of five years, how much is an initial $500 deposit followed At the end of five years, how much is an initial $500 deposit followed
by five year-end, annual $100 payments worth, assuming a compound by five year-end, annual $100 payments worth, assuming a compound
annual interest rate of (i) 10 percent? annual interest rate of (i) 10 percent? (ii) 5 percent? (iii) 0 percent?(ii) 5 percent? (iii) 0 percent?
FVn = P0(1 + i)n
; FVAn = R[([1 + i]n - 1)/i]
(i) FV5 = $500(1.10)5
= $500(1.611) = $ 805.50
FVA5 = $100[([1.10]5
- 1)/(.10)] =
$100(6.105) = 610.50
$1,416.00
(ii) FV5 = $500(1.05)5
= $500(1.276) = $ 638.00
FVA5 = $100[([1.05]5
- 1)/(.05)] =
$100(5.526) = 552.60
$1,190.60
(iii) FV5 = $500(1.0)5
= $500(1) = $ 500.00
FVA5 = $100(5)* = 500.00
$1,000.00
*[Note: We had to invoke hospital's rule in
the special case where i = 0; in short,
FVIFAn = n when i = 0.]

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In 1790 John Jacob Astor bought approximately an acre of landIn 1790 John Jacob Astor bought approximately an acre of land
on the east side of Manhattan Island for $58. Astor, who wason the east side of Manhattan Island for $58. Astor, who was
considered a shrewd investor, made many such purchases. Howconsidered a shrewd investor, made many such purchases. How
much would his descendants have in 2009, if instead of buyingmuch would his descendants have in 2009, if instead of buying
the land, Astor had invested the $58 at 5 percent compoundthe land, Astor had invested the $58 at 5 percent compound
annual interest?annual interest?
Being a little creative,
we can express our problem as follows:
FV219 = P0 × (1 + i)219
= P0 × (1 + i)50 × (1 + i)50 × (1 + i)50 × (1 + i)50 × (1 + i)19
= $58 × 11.467 × 11.467 × 11.467 × 11.467 × 2.527
= $58 × 43,692.26 = $2,534,151.08
Given the current price of land in New York City, Astor’s one-acre purchase seems
to have passed the test of time as a wise investment. It is also interesting to note
that with a little reasoning we can get quite a bit of mileage out of even a basic
table.

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How do you determine the future value (present value)How do you determine the future value (present value)
of an investment over a time span that contains aof an investment over a time span that contains a
fractional period (e.g., 11/4 years)?fractional period (e.g., 11/4 years)?
All you do is alter the future value (present value)
formula to include the fraction in decimal form.
Let’s say that you invest $1,000 in a savings account
that compounds annually at 6 percent and want to
withdraw your savings in 15 months (i.e.,1.25 years).
Since FVn = P0(1 + i )n
, you could withdraw the following
amount 15 months from now:
FV1.25 = $1,000(1 + 0.06)1.25
= $1,075.55

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We will use the ““Rule-of-72Rule-of-72””..
Double Your Money!!!Double Your Money!!!
Quick! How long does it take to
double $5,000 at a compound rate
of 12% per year (approx.)?

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Approx. Years to Double = 7272 / i%
7272 / 12% = 6 Years6 Years
[Actual Time is 6.12 Years]
The “Rule-of-72”The “Rule-of-72”
Quick! How long does it take to
double $5,000 at a compound rate
of 12% per year (approx.)?

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Want to Double Your Money?Want to Double Your Money?
The “Rule of 72” Tells You HowThe “Rule of 72” Tells You How
Bill Veeck once bought the Chicago White Sox baseball team franchise for $10 million and then sold it 5
year slater for $20 million. In short, he doubled his money in 5 years. What compound rate of return did
Veeck earn on his investment? A quick way to handle compound interest problems involving doubling your
money makes use of the “Rule of 72.” This rule states that if the number of years, n, for which an investment
will be held is divided into the value 72, we will get the approximate interest rate, i, required for the
investment to double in value. In Veeck’s case, the rule gives
72/n = I or 72/5 = 14.4%
Alternatively, if Veeck had taken his initial investment and placed it in a savings account earning 6 percent
compound interest, he would have had to wait approximately
12 years for his money to have doubled:
72/i = n or 72/6 = 12 years
Indeed, for most interest rates we encounter, the “Rule of 72” gives a good approximation of the interest rate
– or the number of years – required to double your money. But the answer is not exact. For example, money
doubling in 5 years would have to earn at a 14.87 percent compound annual rate [(1 + 0.1487)5 = 2]; the
“Rule of72” says 14.4 percent. Also, money invested at 6 percent interest would actually require only 11.9
years to double [(1 + 0.06)11.9 = 2]; the “Rule of 72” suggests 12.
However, for ballpark-close money-doubling approximations
that can be done in your head, the “Rule of 72” comes in pretty handy.

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Assume that you need $1,000$1,000 in 2 years.2 years.
Let’s examine the process to determine
how much you need to deposit today at a
discount rate of 7% compounded
annually.
0 1 22
$1,000$1,000
7%
PV1PVPV00
Present ValuePresent Value
Single Deposit (Graphic)Single Deposit (Graphic)

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PVPV00 = FVFV11 / (1+i)1
PVPV00 = FVFV22 / (1+i)2
General Present ValuePresent Value Formula:
PVPV00 = FVFVnn / (1+i)n
or PVPV00 = FVFVnn (PVIFPVIFi,n) -- See Table IISee Table II
General Present ValueGeneral Present Value
FormulaFormula
etc.

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Calculation based on general formula:
PVPV00 = FVFVnn / (1+i)n
PVPV00
= $10,000$10,000 / (1+ 0.10)5
=
= $6,209.21$6,209.21
Calculation based on present value Interest Factor Table I:
PVPV00 = $10,000$10,000 (PVIFPVIF10%, 5)
= $10,000$10,000 (.621)
= $6,210.00$6,210.00 [Due to Rounding]
Mr. X wants to know how large of a deposit to make so thatMr. X wants to know how large of a deposit to make so that
the money will grow tothe money will grow to $10,000$10,000 in 5 years at a discount rate ofin 5 years at a discount rate of 10%10%..

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P0 = FVn[1/(1 + i)n]
i) $100[1/(2)3] = $100(.125) = $12.50
(ii) $100[1/(1.10)3] = $100(.751) = $75.10
(iii) $100[1/(1.0)3] = $100(1) = $100
Exercise for present valueExercise for present value
$100 at the end of three years is worth how much today, assuming a$100 at the end of three years is worth how much today, assuming a
discount rate of (i) 100 percent? (ii) 10 percent? (iii) 0 percent?discount rate of (i) 100 percent? (ii) 10 percent? (iii) 0 percent?

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Unknown Number of CompoundingUnknown Number of Compounding
(or Discounting) Periods(or Discounting) Periods
For example, how long would it take for an investment of $1,000 to grow to $1,900 if we
invested it at a compound annual interest rate of 10 percent? Because we know both the
investment’s future and present value, the number of compounding (or discounting) periods
(n) involved in this investment situation can be determined by rearranging either a basic
future value or present value equation. Using future value, we get
FVn = P0(FVIF10%,n)
$1,900 = $1,000(FVIF10%,n)
FVIF10%,n = $1,900/$1,000 = 1.9
Reading down the 10% column in Table, we look for the future value interest factor (FVIF) in
that column that is closest to our calculated value. We find that 1.949 comes closest to 1.9,
and that this number corresponds to the 7-period row. Because 1.949 is a little larger than
1.9, we conclude that there are slightly less than 7 annual compounding periods implicit in
the example situation. For greater accuracy, simply rewrite FVIF10%,n as (1 + 0.10)n, and
solve for n as follows:
(1 + 0.10)n = 1.9
n(ln 1.1) = ln 1.9
n = (ln 1.9)/(ln 1.1) = 6.73 years

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Unknown Interest (or Discount) Rate.Unknown Interest (or Discount) Rate.
Let’s assume that, if you invest $1,000 today, you will receiveLet’s assume that, if you invest $1,000 today, you will receive
$3,000 in exactly 8 years. The compound interest (or$3,000 in exactly 8 years. The compound interest (or
discount) rate implicit in this situation can be found bydiscount) rate implicit in this situation can be found by
rearranging either a basic future value or present valuerearranging either a basic future value or present value
equation.equation.
FV8 = P0(FVIFi,8)
$3,000 = $1,000(FVIFi,8)
FVIFi,8 = $3,000/$1,000 = 3
Reading across the 8-period row in Table 3.3, we look for the future value interest factor (FVIF) that
comes closest to our calculated value of 3. In our table, that interest factor is 3.059 and is found in the
15% column. Because 3.059 is slightly larger than 3, we conclude that the interest rate implicit in the
example situation is actually slightly less than 15 percent. For a more accurate answer, we simply
recognize that FVIFi,8 can also be written as (1 + i )8, and solve directly for i as follows:
(1 + i )8 = 3
(1 + i ) = 31/8 = 30.125 = 1.1472
i = 0.1472 OR
=14.72%

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Vernal Equinox wishes to borrow $10,000 for three years. AVernal Equinox wishes to borrow $10,000 for three years. A
group of individuals agrees to lend him this amount if hegroup of individuals agrees to lend him this amount if he
contracts to pay them $16,000 at the end of the three years.contracts to pay them $16,000 at the end of the three years.
What is the implicit compound annual interest rate implied byWhat is the implicit compound annual interest rate implied by
this contract (to the nearest whole percent)?this contract (to the nearest whole percent)?
$10,000 = $16,000(PVIFx%,3)
(PVIFx%,3) = $10,000/$16,000 = 0.625
Going to the PVIF table at the back of the book and looking across
the row for n = 3, we find that the discount factor for 17 percent is
0.624 and that is closest to the number above.

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Joe Hernandez has inherited $25,000 and wishes to purchase anJoe Hernandez has inherited $25,000 and wishes to purchase an
annuity that will provide him with a steady income over the next 12annuity that will provide him with a steady income over the next 12
years. He has heard that the local savings and loan association isyears. He has heard that the local savings and loan association is
currently paying 6 percent compound interest on an annual basis. If hecurrently paying 6 percent compound interest on an annual basis. If he
were to deposit his funds, what year-end equal-dollar amount (to thewere to deposit his funds, what year-end equal-dollar amount (to the
nearest dollar) would he be able to withdraw annually such that henearest dollar) would he be able to withdraw annually such that he
would have a zero balance after his last withdrawal 12 years fromwould have a zero balance after his last withdrawal 12 years from
now?now?
$25,000 = R(PVIFA6%,12) = R(8.384)
R = $25,000/8 .384 = $2,982

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You have been offered a note with four years to maturity,You have been offered a note with four years to maturity,
which will pay $3,000 at the end of each of the four years.which will pay $3,000 at the end of each of the four years.
The price of the note to you is $10,200. What is the implicitThe price of the note to you is $10,200. What is the implicit
compound annual interest rate you will receive (to thecompound annual interest rate you will receive (to the
nearest whole percent)?nearest whole percent)?
$10,000 = $3,000(PVIFAx%,4)(PVIFAx%,4) = $10,200/$3,000 = 3.4
Going to the PVIFA table at the back of the book and looking across
the row for n = 4, we find that the discount factor for 6 percent
is 3.465, while for 7 percent it is 3.387. Therefore, the note has
an implied interest rate of almost 7 percent.

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Sales of the Aman Company were $500,000 this year, and theySales of the Aman Company were $500,000 this year, and they
are expected to grow at a compound rate of 20 percent for theare expected to grow at a compound rate of 20 percent for the
next six years. What will be the sales figure at the end of each ofnext six years. What will be the sales figure at the end of each of
the next six years?the next six years?
Year Sales
1 $ 600,000 = $ 500,000(1.2)
2 720,000 = 600,000(1.2)
3 864,000 = 720,000(1.2)
4 1,036,800 = 864,000(1.2)
5 1,244,160 = 1,036,800(1.2)
6 1,492,992 = 1,244,160(1.2)

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The Habib Bark limited is considering the purchase of a debarking machine that is
expected to provide cash flows as follows:
END OF YEAR
Years ------------------ 1 2 3 4 5
Cash flow ----------- $1,200 $2,000 $2,400 $1,900 $1,600
END OF YEAR
Years------------------- 6 7 8 9 10
Cash flow ------------- $1,400 $1,400 $1,400 $1,400 $1,400
If the appropriate annual discount rate is 14 percent, what is the present value of this cash-flow
stream?
Solution is on next slide

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solution of last presentation slide
Present Value
Year Amount Factor at 14% Present Value
1 $1,200 .877 $1,052.40
2 2,000 .769 1,538.00
3 2,400 .675 1,620.00
4 1,900 .592 1,124.80
5 1,600 .519 830.40
Subtotal (a) ........................... $6,165.60
1-10 (annuity) 1,400 5.216 $7,302.40
1-5 (annuity) 1,400 3.433 -4,806.20
Subtotal (b) ........................... $2,496.20
Total Present Value (a + b) ............ $8,661.80

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Perpetuity.Perpetuity.
A perpetuity is an ordinary annuity whose payments or receipts continue forever. The
ability to determine the present value of this special type of annuity will be required when
we value perpetual bonds and preferred stock, a look back to PVAn in should help us to
make short work of this type of task. Replacing n in Eq with the value infinity (∞) gives us
PVA∞ = R[(1 − [1/(1 + i )∞])/i ] (3.12)
Because the bracketed term – [1/(1 + i)∞] – approaches zero, we can rewrite Eq. as
PVA∞ = R[(1 − 0)/i ] = R(1/i )
or simply
PVA∞ = R/i
Thus the present value of a perpetuity is simply the periodic receipt (payment) divided by
the interest rate per period. For example, if $100 is received each year forever and the
interest rate is 8 percent, the present value of this perpetuity is $1,250 (that is, $100/0.08).

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What is the aggregate present value of $500 received atWhat is the aggregate present value of $500 received at
the end of each of the next three years, assuming athe end of each of the next three years, assuming a
discount rate of (i) 4 percent? (ii) 25 percent?discount rate of (i) 4 percent? (ii) 25 percent?
PVAn = R[(1 -[1/(1 + i)n])/i]
(i) $500[(1 -[1/(1 + .04)3])/.04] = $500(2.775) = $1,387.50
(ii) $500[(1 -[1/(1 + .25)3])/.25] = $500(1.952) = $ 976.00

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Types of AnnuitiesTypes of Annuities
Ordinary AnnuityOrdinary Annuity: Payments or receipts
occur at the end of each period.
Annuity DueAnnuity Due: Payments or receipts
occur at the beginning of each period.
An AnnuityAn Annuity represents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.

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You need to have $50,000 at the end of 10 years. To accumulateYou need to have $50,000 at the end of 10 years. To accumulate
this sum, you have decided to save a certain amount at thethis sum, you have decided to save a certain amount at the endend
of each of the next 10 years and deposit it in the bank. The bankof each of the next 10 years and deposit it in the bank. The bank
pays 8 percent interest compounded annually for long-termpays 8 percent interest compounded annually for long-term
deposits .How much will you have to save each year (to thedeposits .How much will you have to save each year (to the
nearest dollar)?nearest dollar)?
Sol: $50,000 = R(FVIFA8%,10) = R(14.486)
R = $50,000/14.486 = $3,452

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Annuity DueAnnuity Due
In contrast to an ordinary annuity, where payments or receipts occur
at the end of each period, an annuity due calls for a series of equal
payments occurring at the beginning of each period. Luckily, only a
slight modification to the procedures already outlined for the
treatment of ordinary annuities will allow us to solve annuity due
problems.
the future value of the three-year annuity due is simply equal to the
future value of a comparable three-year ordinary annuity compounded
for one more period. Thus the future value of an annuity due at i
percent for n periods (FVADn) is determined as:
FVADn = R(FVIFAi,n)(1 + i )

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Hint on Annuity ValuationHint on Annuity Valuation
The future value of an ordinary
annuity can be viewed as
occurring at the endend of the last
cash flow period, whereas the
future value of an annuity due
can be viewed as occurring at
the beginningbeginning of the last cash
flow period.

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You need to have $50,000 at the beginning of 10 years. ToYou need to have $50,000 at the beginning of 10 years. To
accumulate this sum, you have decided to save a certain amountaccumulate this sum, you have decided to save a certain amount
at theat the endend of each of the next 10 years and deposit it in the bank.of each of the next 10 years and deposit it in the bank.
The bank pays 8 percent interest compounded annually for long-The bank pays 8 percent interest compounded annually for long-
term deposits. As you deposit a certain amount at theterm deposits. As you deposit a certain amount at the beginningbeginning
of each of the next 10 years. Now, how much will you have toof each of the next 10 years. Now, how much will you have to
save each year (to the nearestsave each year (to the nearest
dollar)?dollar)?
$50,000 = R(FVIFA8%,10)(1 + .08) = R(15.645)
R = $50,000/15.645 = $3,196

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Take Note
Whether a cash flow appears to occur at the
beginning or end of a period often depends on your
perspective, however. (In a similar vein, is midnight
the end of one day or the beginning of the next?)
Therefore, the real key to distinguishing between
the future value of an ordinary annuity and an
annuity due is the point at which the future value is
calculated. For an ordinary annuity, future value is
calculated as of the last cash flow. For an annuity
due, future value is calculated as of one period after
the last cash flow.

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present value of anpresent value of an
annuity dueannuity due
The determination of the present value of an annuity due at i
percent for n periods (PVADn) is best understood by example.
Figure 3.7 illustrates the calculations necessary to determine both
the present value of a $1,000 ordinary annuity at 8 percent for
three years (PVA3) and the present value of a $1,000 annuity due
at 8 percent for three years (PVAD3). the present value of a three-
year annuity due is equal to the present value of a two-year
ordinary annuity plus one nondiscounted periodic receipt or
payment. This can be generalized as follows:
PVADn = R(PVIFAi,n−1) + R
= R(PVIFAi,n−1 + 1)

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FVADFVADnn = R(1+i)n
+ R(1+i)n-1
+
... + R(1+i)2
+ R(1+i)1
= FVAFVAnn (1+i)
Overview View of anOverview View of an
Annuity Due -- FVADAnnuity Due -- FVAD
R R R R R
0 1 2 3 n-1n-1 n
FVADFVADnn
i% . . .
Cash flows occur at the beginning of the period

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FVADFVAD33 = $1,000(1.07)3
+
$1,000(1.07)2
+ $1,000(1.07)1
= $1,225 + $1,145 + $1,070
= $3,440$3,440
Example of anExample of an
Annuity Due -- FVADAnnuity Due -- FVAD
$1,000 $1,000 $1,000 $1,070
0 1 2 33 4
$3,440 = FVAD$3,440 = FVAD33
7%
$1,225
$1,145

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PVAPVA33 = $1,000/(1.07)1
+
$1,000/(1.07)2
+
$1,000/(1.07)3
= $934.58 + $873.44 + $816.30
= $2,624.32$2,624.32
Ordinary Annuity -- PVAOrdinary Annuity -- PVA
$1,000 $1,000 $1,000
0 1 2 33 4
$2,624.32 = PVA$2,624.32 = PVA33
7%
$ 934.58
$ 873.44
$ 816.30
Cash flows occur at the end of the period

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PVADPVADnn = R/(1+i)0
+ R/(1+i)1
+ ... + R/(1+i)n-1
= PVAPVAnn (1+i)
Overview of anOverview of an
Annuity Due -- PVADAnnuity Due -- PVAD
R R R R
0 1 2 n-1n-1 n
PVADPVADnn
R: Periodic
Cash Flow
i% . . .

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PVADPVADnn = $1,000/(1.07)0
+ $1,000/(1.07)1
+
$1,000/(1.07)2
= $2,808.02$2,808.02
Annuity Due -- PVADAnnuity Due -- PVAD
$1,000.00 $1,000 $1,000
0 1 2 33 4
$2,808.02$2,808.02 = PVADPVADnn
7%
$ 934.58
$ 873.44

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PVADPVADnn = R (PVIFAi%,n)(1+i)
PVADPVAD33 = $1,000 (PVIFA7%,3)(1.07) = $1,000
(2.624)(1.07) = $2,808$2,808
Valuation Using Table IVValuation Using Table IV
Period 6% 7% 8%
1 0.943 0.935 0.926
2 1.833 1.808 1.783
3 2.673 2.624 2.577
4 3.465 3.387 3.312
5 4.212 4.100 3.993

49
Julie Miller will receive the set of cash
flows below. What is the Present ValuePresent Value
at a discount rate of 10%10%?
Mixed Flows ExampleMixed Flows Example
0 1 2 3 4 55
$600 $600 $400 $400 $100$600 $600 $400 $400 $100
PVPV00
10%10%

50
1. Solve a “piece-at-a-timepiece-at-a-time” by
discounting each piecepiece back to t=0.
2. Solve a “group-at-a-timegroup-at-a-time” by first
breaking problem into groups
of annuity streams and any single
cash flow group. Then discount
each groupgroup back to t=0.
How to Solve?How to Solve?

51
““Piece-At-A-Time”Piece-At-A-Time”
0 1 2 3 4 55
$600 $600 $400 $400 $100$600 $600 $400 $400 $100
10%
$545.45$545.45
$495.87$495.87
$300.53$300.53
$273.21$273.21
$ 62.09$ 62.09
$1677.15$1677.15 == PVPV00 of the Mixed Flowof the Mixed Flow

52
““Group-At-A-Time” (#1)Group-At-A-Time” (#1)
0 1 2 3 4 55
$600 $600 $400 $400 $100$600 $600 $400 $400 $100
10%
$1,041.60$1,041.60
$ 573.57$ 573.57
$ 62.10$ 62.10
$1,677.27$1,677.27 == PVPV00 of Mixed Flowof Mixed Flow [Using Tables][Using Tables]
$600(PVIFA10%,2) = $600(1.736) = $1,041.60
$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57
$100 (PVIF10%,5) = $100 (0.621) = $62.10

53
““Group-At-A-Time” (#2)Group-At-A-Time” (#2)
0 1 2 3 4
$400 $400 $400 $400$400 $400 $400 $400
PVPV00 equals
$1677.30.$1677.30.
0 1 2
$200 $200$200 $200
0 1 2 3 4 5
$100$100
$1,268.00$1,268.00
$347.20$347.20
$62.10$62.10
PlusPlus
PlusPlus

54
Defining the calculator variables:
For CF0: This is ALWAYS the cash flow occurring
at time t=0 (usually 0 for these problems)
For Cnn:* This is the cash flow SIZE of the nth
group of cash flows. Note that a “group” may only
contain a single cash flow (e.g., $351.76).
For Fnn:* This is the cash flow FREQUENCY of the
nth group of cash flows. Note that this is always a
positive whole number (e.g., 1, 2, 20, etc.).
Solving the Mixed FlowsSolving the Mixed Flows
Problem using CF RegistryProblem using CF Registry
* nn represents the nth cash flow or frequency. Thus, the
first cash flow is C01, while the tenth cash flow is C10.

55
General Formula:
FVn = PVPV00(1 + [i/m])mn
n: Number of Years
m: Compounding Periods per Year
i: Annual Interest Rate
FVn,m: FV at the end of Year n
PVPV00: PV of the Cash Flow today
Semiannual and Other CompoundingSemiannual and Other Compounding
PeriodsPeriods

56
Julie Miller has $1,000$1,000 to invest for 2
years at an annual interest rate of
12%.
Annual FV2 = 1,0001,000(1+ [.12/1])(1)(2)
= 1,254.401,254.40
Semi FV2 = 1,0001,000(1+ [.12/2])(2)(2)
= 1,262.481,262.48
Impact of FrequencyImpact of Frequency

57
Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2)
= 1,266.771,266.77
Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2)
= 1,269.731,269.73
Daily FV2 = 1,0001,000(1+[.12/365])(365)(2)
= 1,271.201,271.20
Impact of FrequencyImpact of Frequency

58
Suppose you were to receive $1,000 at the end of 10Suppose you were to receive $1,000 at the end of 10
years. If your opportunity rate is 10 percent, what is theyears. If your opportunity rate is 10 percent, what is the
present value of this amount if interest is compounded (a)present value of this amount if interest is compounded (a)
annually? (b) quarterly? (c) continuously?annually? (b) quarterly? (c) continuously?
Amount Present Value Interest Factor Present Value
$1,000 1/(1 + .10)10
= .386 $386
1,000 1/(1 + .025)40
= .372 372
1,000 1/e(.10)(10)
= .368 368

59
Effective Annual Interest Rate
The actual rate of interest earned
(paid) after adjusting the nominal
rate for factors such as the number
of compounding periods per year.
(1 + [ i / m ] )m
- 1
Effective AnnualEffective Annual
Interest RateInterest Rate

60
Wander co has a $1,000 CD at the
bank. The interest rate is 6%
compounded quarterly for 1 year.
What is the Effective Annual
Interest Rate (EAREAR)?
EAREAR = ( 1 + 6% / 4 )4
- 1
= 1.0614 - 1 = .0614 or 6.14%!6.14%!
BW’s EffectiveBW’s Effective
Annual Interest RateAnnual Interest Rate

61
Present (or Discounted)Present (or Discounted)
Value.Value.
When interest is compounded more than once a year, the formula for calculating present value must be
revised along the same lines as for the calculation of future value. Instead of dividing the future cash
flow by (1 + i)n as we do when annual compounding is involved, we determine the present value by 3
The Time Value of Money PV0 = FVn /(1 + [i /m])mn
where, as before, FVn is the future cash flow to
be received at the end of year n, m is the number of times a year interest is compounded, and i is the
discount rate. We can use Eq. (for example, to calculate the present value of $100 to be received at the
end of year 3 for a nominal discount rate of 8 percent compounded quarterly:
PV0 = $100/(1 + [0.08/4])(4)(3)
= $100/(1 + 0.02)12 = $78.85
If the discount rate is compounded only annually, we have
PV0 = $100/(1 + 0.08)3 = $79.38
Thus, the fewer times a year that the nominal discount rate is compounded, the greater the present
value. This relationship is just the opposite of that for future values.

62
Effective Annual InterestEffective Annual Interest
RateRate
The actual rate of interest earned (paid) after adjusting the nominal rate for
factors such as the number of compounding periods per year.
effective annual interest rate = (1 + [i /m])m − 1
For example, if a savings plan offered a nominal interest rate of 8
percent compounded quarterly on a one-year investment, the effective
annual interest rate would be
(1 + [0.08/4])4 − 1 = (1 + 0.02)4 − 1 = 0.08243

63
1. Calculate the payment per period.
2. Determine the interest in Period t.
(Loan balance at t-1) x (i% / m)
3. Compute principal paymentprincipal payment in Period t.
(Payment - interest from Step 2)
4. Determine ending balance in Period t.
(Balance - principal paymentprincipal payment from Step 3)
5. Start again at Step 2 and repeat.
Steps to Amortizing a LoanSteps to Amortizing a Loan

64
Usefulness of AmortizationUsefulness of Amortization
2.2. Calculate Debt OutstandingCalculate Debt Outstanding -- The
quantity of outstanding debt
may be used in financing the
day-to-day activities of the firm.
1.1. Determine Interest ExpenseDetermine Interest Expense --
Interest expenses may reduce
taxable income of the firm.

65
Julie Miller is borrowing $10,000$10,000 at a
compound annual interest rate of 12%.
Amortize the loan if annual payments are
made for 5 years.
Step 1: Payment
PVPV00 = R (PVIFA i%,n)
$10,000$10,000 = R (PVIFA 12%,5)
$10,000$10,000 = R (3.605)
RR = $10,000$10,000 / 3.605 = $2,774$2,774
Amortizing a Loan ExampleAmortizing a Loan Example

66
The Happy Hang Glide Company is purchasing a building andThe Happy Hang Glide Company is purchasing a building and
has obtained a $190,000 mortgage loan for 20 years. The loanhas obtained a $190,000 mortgage loan for 20 years. The loan
bears a compound annual interest rate of 17 percent and callsbears a compound annual interest rate of 17 percent and calls
for equal annual installment payments at the end of each offor equal annual installment payments at the end of each of
the 20 years. What is the amount of the annual payment?the 20 years. What is the amount of the annual payment?
$190,000 = R(PVIFA17%,20) = R(5.628)
R = $190,000/5.628 = $33,760

67
Amortizing a Loan ExampleAmortizing a Loan Example
End of
Year
Payment Interest Principal Ending
Balance
0 --- --- --- $10,000
1 $2,774 $1,200 $1,574 8,426
2 2,774 1,011 1,763 6,663
3 2,774 800 1,974 4,689
4 2,774 563 2,211 2,478
5 2,775 297 2,478 0
$13,871 $3,871 $10,000
[Last Payment Slightly Higher Due to Rounding]

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