PROBLEMS IN BUSINESS FINANCE 382
STUDENT LECTURE NOTE 1A Fa13CR
I. Time Value of Money, A Review
A.Fundamental Concept: A dollar you receive today is
worth ____ than that same dollar received tomorrow.
1. This is because in the interval you could have
earned interest on the dollar received today.
2. So, to be indifferent between an amount today and
an amount to be received later, the later amount
must include some __________ compensation in
the form of interest.
3. This fact means that you ______ add and subtract,
or otherwise accurately compare dollar amounts
which occur at different points in time.
B.Intro to Time Value
1. In any time value problem, there are five variables
of interest. These are:
 Term, in years (t);
 Periodicity, i.e., periods per year (m);
 Annual Interest rate (r%);
 Present Value (PV0), an amount today (at Year
0); and
 Future Value (FVt), an amount at some future
time, (in Year t).
2. The first step in solving any time value problem
seems obvious, but often isn’t, and is without
doubt, THE MOST IMPORTANT PART. This is
IDENTIFYING WHAT IS __________.
1
3. A second step is deciding which approach to use,
i.e.,:
a. Numerical (using the appropriate formula and a
calculator—will work with any calculator),
b. Financial Calculator (assumes you have one),
or
c. Excel Spreadsheet (need to have a computer).
C.Future Value of a Single Payment
The general formula to determine the FV of a single
payment is:
  r   t*m 
FVt = PV0  1      = PV0 * FVIFr/m,t*m. (1A.1)
  m   
Ex. 1A.1
Suppose Ashleigh Smith (F381, Spr’10, F382, Fa’10)
deposits $1,521.12 into an account that earns an annual
interest rate of 8.5%, which is compounded hourly
(Yes!). How much will she be able to withdraw at the
end of 14 years?
A. 1A.1 FV14
  .085   (14*8760 ) 
= $1,521.12  1  


  8760  

= $1,521.12 * (3.28706209)
= $____________ = $_______,
where: FVIF 8.5%/8760,14*8760 = (3.28706209).
Numerical Approach - Calculator Sequence:
2
0.085 [] 8760 [=] [+] 1 [=] [yx] [(] 8760 [x] 14 [)] [=]
[x] 1521.12 [=]
Financial Calculator – Calculator Approach:
Note, before making any calculations, you need to check
whether or not the [P/Y] is set for monthly payments
(this is the default for the TI BA II Plus calculator) or
not.1
Enter
14*8760 8.58760 -1521.12 0
N
I/Y
PV
PMT
Solve for
FV
5000.02
Keystrokes (assuming that [P/Y] is set for 1 period per
year).
14 [x] 8760 [=] [N] 8.5 [] 8760 [=] [I/Y] 1521.12 [+/-]
[PV] 0 [PMT] [CPT] [FV]
General Excel Function: =FV(rate,nper,pmt,pv,type).
SS Solution: =FV($M9/$K9,$L9*$K9,$O9,-$N9,0)
In conclusion:
This implies that if somebody offers you $ 1521.12
today or $5,000 at the end of 14 years, and your
opportunity cost of funds = 8.5% (compounded
1
With a BAII Plus calculator, this is done using the following steps. To enter
compounding periods per year, press [2nd] [P/Y] (m?) [ENTER]. Then press [2nd]
[QUIT] to return to standard calculator mode. To check whether the compounding
periods are set correctly, press [2nd] [P/Y], it should show whatever you have set.
3
hourly), that you would be indifferent between the two
alternatives.
Instructive Query: Suppose your opportunity cost of
funds was 9%, do you prefer the amount today or to
receive the amount in the future?
D.The Present Value of a Single Payment
Suppose we know that a certain amount will be
received in the future, and we want to know what it
is worth to us in today's dollars given a certain
opportunity cost. This is equivalent to asking the
question of what amount deposited ______ would
grow to this known future amount? The process of
finding PVs of future amounts is known as
DISCOUNTING.
Solve Formula (1A.1) for the PV of a single amount.
FV = PV * (1 + (r/m))t*m
 PV0 =
FVt
(1  (r/m))t*m 
Thus, the general formula to determine the PV of a
single payment is:


1
 PV0 = FVt  
= FVt * PVIFr/m,t*m. (1A.2)
t*m 
(1

(
r/m))


Ex. 1A.2
Eddie Bertoniere (F382, Fa’05) knows he will need
$2,000 at the end of three years. How much will he
4
need to deposit into his bi-monthly compounded, 5%
p.a. account today?
A. 1A.2 Use Formula (1A.2) to solve for unknown PV.

1
PV = $2,000 * 

0.05
 1 
6
 


3*6 


= $2,000 * (0.861243115)
= $_________,
where PVIF5%/6,3*6 = 0.861243115.
Numerical Approach - Calculator Sequence:
0.05 [] 6 [=] [+] 1[=] [yx] [(] 3 [x] 6 [)] [=] [1/x] [x]
2000 [=]
Financial Calculator – Calculator Approach:
Enter
Solve for
3*6 56
N
I/Y
PV
-1722.4862
0
PMT
2000
FV
Keystrokes (assuming that [P/Y] is set for 1 period per
year).
3 [x] 6 [=] [N] 5 [] 6 [=] [I/Y] 0 [PMT] 2000 [FV]
[CPT] [PV]
General Excel Function: =PV(rate,nper,pmt,fv,type)
5
SS Solution: =PV($M13/$K13,$L13*$K13,$O13,$P13,0)
E. Annuities
1. An annuity is a ______ of equal payments which
are either made or received with a regular
frequency.
2. General Rule: FV or PV of a stream of payments is
equal to the sum of the individual FVs or PVs in
the stream.
3. Important Assumption: Payments are made or
received at the ____ of the period.
4. To identify whether an annuity is a future or
present value the most important question IS
WHEN DOES THE LUMP SUM OCCUR?
5. In the FUTURE VALUE OF AN ANNUITY case
deposits are made on a regular basis to
ACCUMULATE A FUTURE SUM.
F. Future Value of an Annuity
The general formula to determine the FV of an
ordinary (=end-of-period) annuity is:
 (1  (r/m)) t*m  1
FVA = A  
.
(r/m)


(1A.3)
Ex. 1A.3
Boyd “Beau” Mothe III (F382, Sp’12) is 25 years old
and expects to retire at age 65, in 40 years. Assume
Beau makes weekly deposits of $50.24 into a balanced
Vanguard mutual-fund portfolio earning an average,
overall return of 6.5% p.a., (compounded weekly) at
6
the end of each week for the next 40 years. How much
will his portfolio be worth when Beau retires at 65?
A. 1A.3
FVa = $50.24 * (FVIFA6.5%/52,40*52)


 (1  (.065 / 52)) 40*52  1 
= $50.24  

(.065/52)


= $50.24  (9953.51633) = $__________.
Numerical Approach - Calculator Sequence:
0.065 [] 52 [=] [+] 1 [=] [yx] [(] 40 [x] 52 [)] [=] [-] 1
[=] [] [(] .065 [] 52 [)] [=] [x] 50.24 [=]
Financial Calculator – Calculator Approach:
Enter
40 * 52 6.5  52
N
I/Y
0
PV
Solve for
-50.24
PMT
FV
500,064.66
Keystrokes (assuming that [P/Y] is set for 1 period per
year).
40 [x] 52 [=] [N] 6.5 [] 52 [=] [I/Y] 0 [PV] 50.24 [+/-]
[PMT] [CPT] [FV]
General Excel Function: =FV(rate,nper,pmt,pv,type)
SS Solution: =FV($M17/$K17,$L17*$K17,-$O17,$N17,0)
7
G. Present Value of an Annuity
1. In a PRESENT VALUE OF AN ANNUITY
problem the LUMP SUM OCCURS TODAY.
Annuity payments are then made in the future.
2. Best example to help the student visualize PV of an
Annuity is to think of borrowing money to buy a
home (for example).
3. When do you get the loan proceeds?  ______!
When do the annuity repayments occur?  On a
regular basis over a future period of time.
4. The general formula to find the Present Value of an
Annuity is given below:
 1

1

  (1  ( r/m)) t*m  
PVA = R 
 = R  (PVIFAr%,t). (1A.4)
(r/m)




Ex. 1A.4
Joseph Vandigo (F382, Spr’06) is financing a home
mortgage from Hancock Bank. Suppose that the biweekly loan payments equal $517.63 and the loan
term is 25 years. If the loan’s fixed interest rate is
5.95% how much has Joe borrowed to purchase this
home?
PVA = $517.63 * PVIFA 5.95%/26,25*26
8
  1

1

 
25* 26  
(1

(
0
.
0595
/26))


= $517.63  


(0.0595/26)




= $517.63  (338.0781804)
= $__________.
Numerical Approach - Calculator Sequence:
.0595 [÷] 26 [=] [+] 1 [=] [yx] [(] 25 [x] 26 [)] [=] [1/x] [+/-]
[+] 1 [=] [÷] [(] .0595 [÷] 26 [)] [=] [x] 517.63
Financial Calculator – Calculator Approach:
25 * 26 5.95  26
N
I/Y
Enter
Solve for
PV
-174,999.41
517.63
PMT
0
FV
Keystrokes (assuming that [P/Y] is set for 1 period per
year).
25 [x] 26 [=] [N] 5.95 [÷] 26 [=] [I/Y] 517.63 [PMT] 0
[FV] [CPT] [PV]
General Excel Function: PV(rate,nper,pmt,fv,type)
SS Solution: =PV($M21/$K21,$L21*$K21,$O21,$P21,0)
H. FV and PV of Annuities Due
1. Up to this point we have always calculated the
value of annuities based on the assumption that
9
payments are made or received at the ____ of
the period.
2. However, in the real world, in many situations,
beginning-of-the-period (BOP) payments are
more realistic, i.e., lease and mortgage
payments.
3. Fortunately, the necessary adjustment to account
for BOP payments is quite easy. Think about
how a future or present value is affected by
making deposits or receiving payments one
period sooner. Each deposit made earns interest
for an extra period (FV) and each payment
received is discounted one less period (PV).
Thus, both FV and PV of an annuity due will be
________ than their end-of-period counterpart.
The future value of an annuity due is then:
  r 
FVAD = A * (FVIFAr%,t) * 1     .
  m 
(1A.5)
And the present value of an annuity due is:
  r 
PVAD = R * (PVIFAr%,t) * 1     .
  m 
(1A.6)
Ex. 1A.5
Yessica Rizelia (F382, Sp’04, F695, Fa’04) is her
firm’s chief financial analyst and she is evaluating
leasing a new building. Assume that the purchase price
of the building would be $500,000 and the ten-year
lease terms will fully amortize this price. What is the
10
estimated monthly, beginning-of-the-period payment
Yessica will calculate if the lessor earns a fixed-rate
return of 7.5% p.a.?
A.1A.5. In this example the lease payments are BOM, so
this is a Present Value of an Annuity problem
with an Annuity Due chaser. Thus, the PVAD
approach given in Formula (1A.6) is the place
to start. However, what Yessica is solving for is
the lease payment (i.e., R), with the AD twist.
  r 
PVAD = R * PVIFA r%/m,t*m * 1     ,
  m 


PVAD
 R= 
.
PVIFA
*
(1

(r/m))
7.5%/12,10 *12




$500,000
Here: R = 
 = $________.
84
.
24474271
*
(
1

(.
075
/
12
))


Numerical Approach - Calculator Sequence:
1st Step: .075 [÷] 12 [=] [+] 1 [=] [yx] [(] 10 [x] 12 [)] [=]
[1/x] [+/-] [+] 1 [=] [÷] [(] .075 [÷] 12 [)] [=] [STO] 1;
2nd Step: .075 [÷] 12 [=] [+] 1 [=] [x] [RCL] 1 [=] [STO] 2;
3rd Step: 500000 [÷] [RCL] 2
Financial Calculator – Calculator Approach: Need to
first change the [2nd] [PMT] < = BGN> default from
END (end-of-period) to BGN (beginning-of-period).
11
Note: To change the payment period (END/BGN), press
[2nd] [BGN], then press [2nd] [SET]. Don’t forget to reset your calculator to [END] after this problem.
10 * 12 7.5  12
N
I/Y
Enter
-500000
PV
Solve for
PMT
5,898.2246
0
FV
Keystrokes (assuming that [P/Y] is set for 1 period per
year) and [END/BGN] is set for [BGN].
10 [x] 12 [=] [N] 7.5 [] 12 [=] [I/Y] 500000 [+/-] [PV] 0
[FV] [CPT] [PMT]
General Excel Function: PMT(rate,nper,pv,fv,type)
SS Solution: PMT($M25/$K25,$L25*$K25,-$N25,$P25,1)
I. The Implicit Interest Rate
1. Finding the interest rate that is “implied” between
the present value and future value of a single
payment may be done simply by solving equation
(1A.1) directly for ‘r”.
  r 
FVt = PV0 * 1    
  m 
t *m
  r 
 1    
  m 
 FV 
  r 
 1     =  t 
  m 
 PV0 
12
1
(t * m)
t *m
 FV 
= t
 PV0 
1


(t * m)


 FV

 r =  t 
 1 * m.
PV
  0 

(1A.7)
2. With annuity problems, finding the implicit
interest rate is much easier using the Financial
Calculator Function or Excel Function compared
to using the Equation Approach. The latter
approach requires having the appropriate interest
factor table available.
3. If using the Equation Approach you need to look
up the relevant FVIFA or PVIFA in the body of
the table, in the row which equals t * m. The factor
will appear under the column which will be for r 
m. To convert the “m” period rate into the annual
“nominal” (vs. effective) it needs to be multiplied
by m as the last step.
Ex. 1A.6
Chad Notariano (F382, Sp’11) has found that the price
for a 30-second commercial during the (first-three
quarters of the) 2006 Super Bowl was $2.5 million.
He’s also found that the same 30-seconds cost
$550,000 in 1985. What quarterly, compounded
growth rate in cost (per second) does this represent
assuming that this represents 21 return intervals?
A.1A.6. The question asks directly for the cost growth
rate between single payments. This is clearly an
implicit interest rate question so use (1A.7).
13
1

 
(21* 4)


 1 * 4 = _______%.
r =    $2.5m 
  $550k
 


 
 
Numerical Approach - Calculator Sequence:
21 [x] 4 [=] [1/x] [STO] 1, 2500000 [÷] 550000 [=] [yx]
[RCL] 1 [=] [-] 1 [=] [x] 4 [=]
Financial Calculator – Calculator Approach:
Enter
Solve for
21 * 4
N
I/Y
1.818877
-550000
PV
0
PMT
2500000
FV
Keystrokes (assuming that [P/Y] is set for 1 period per
year).
21 [x] 4 [=] [N] 550000 [+/-] [PV] 0 [PMT] 2500000
[FV] [CPT] [I/Y] [x] 4 [=]
In the last step, the quarterly rate of 1.818877% is
converted to an annual ‘nominal’ rate by multiplying it
by ‘m’.
General Excel Function: RATE(nper,pmt,pv,fv,type)
SS Soln: =RATE($L29*$K29,$O29,-$N29,$P29,0)*$K29
Addendum: How to convert the nominal, annual rate
above to the “effective” rate?
14
m
  r 
re = 1      1.
  m 
(1A.8)
4
Here:
  0.0727551  
re =  1  
   1 = ________%.
4

 
J. Continuous Compounding/Discounting
1. This is the logical extension of what happens when
compounding or discounting takes place as
frequently as possible, i.e., it is continuous.
2. The most practical application is that continuous
compounding is assumed in the pricing of many
derivative instruments, most notably in the widelyused Black-Scholes option pricing model (We will
be seeing this FIN 384). The future value and
present value of a single payment with continuous
compounding/discounting are given below.
FVCont. = PV0 * ert;
(1A.9)
PVCont. = FVt * e-rt;
(1A.10)
where:
e = e1 = 2.718281828. BA II+ CS: 1 [2nd] [ex]
Additional Practice Problems
Ex. 1A.7
Hailee Brasington (F382, Fa’11) has identified a
special investment account that guarantees a 9% return
p.a. if she makes $200 deposits at the beginning of
15
each month for the next 30 years. What will her
account be worth at the end of this period?
A.1A.7
  r 
FVa = $200.00 * (FVIFA 9%/12,30*12) * 1    
  m 


 (1  (.09 / 12))30*12  1    .09  
= $200  
 * 1    
(.09/12)

   12  
= $200  (1830.743483) * (1.0075) = $__________.
Numerical Approach - Calculator Sequence:
1st Step: 0.09 [] 12 [=] [+] 1 [=] [yx] [(] 30 [x] 12 [)] [=]
[-] 1 [=] [] [(] .09 [] 12 [)] [=] [STO] 1;
2nd Step: 0.09 [÷] 12 [=] [+] 1 [=] [x] [RCL] 1 [=] [STO] 2;
3rd Step: 200 [x] [RCL] 2
Financial Calculator – Calculator Approach: Remember
to re-set for BOP payments. [2nd] [BGN] [2nd] [SET]
Enter
30 * 12
N
9  12
I/Y
0
PV
Solve for
-200
PMT
FV
368,894.81
Keystrokes (assuming that [P/Y] is set for 1 period per
year).
30 [x] 12 [=] [N] 9 [] 12 [=] [I/Y] 0 [PV] 200 [+/-]
[PMT] [CPT] [FV]
16
General Excel Function: =FV(rate,nper,pmt,pv,type)
SS Solution: =FV($M33/$K33,$L33*$K33,-$O33,$N33,1)
Ex. 1A.8
Suppose Brennan Brupbacher (F382, Sp’10) wishes to
have accumulated $40,000 to afford a downpayment
on a house by the end of 4 years. What weekly deposit
will be required to accumulate this future sum if his
investment account earns 5.2%?
A1A.8. Use formula (1A.3) and solve for the required
deposit, A.


FV
FV = A * (FVIFAr%,t)  A = 
.

 (FVIFAr%, t) 
Here: A = $40,000 ÷ (FVIFA 0.1%,208)
[(1  (0.052 / 52)) 52*4 ]  1
= $40,000 ÷ 

(0.052 / 52)


= $40,000 ÷ (231.0852153) = $______.
Heuristic check:
208 deposits  $173.10 = $36,004.8 < $40k.
Numerical Approach - Calculator Sequence:
.052 [÷] 52 [=] [STO] 1 [+] 1 [=] [yx] [(] 4 [x] 52 [)] [=]
[-] 1 [=] [÷] [RCL] 1 [=] [STO] 2 40000 [÷] [RCL] 2 [=]
Financial Calculator – Calculator Approach:
Enter
4 * 52 5.2 ÷ 52
17
0
40000
N
I/Y
PV
Solve for
PMT
-173.096318
FV
Keystrokes (assuming that [P/Y] is set for 1 period per
year).
4 [x] 52 [=] [N] 5.2 [÷] 52 [=] [I/Y] 0 [PV] 40000 [FV]
[CPT] [PMT]
General Excel Function: = PMT(rate,nper,pv,fv,type)
SS Solution: =PMT($M37/$K37,$L37*$K37,$N37,$P37,0)
Ex. 1A.9
Assume that on Dennis Haydell’s (F382, Fa’09) 7th
birthday his grandparents gave him a special
investment certificate that was worth $10,000 on the
day he turned 23 (16 years later) as long as he earned
an “A” grade in his Finance 382 class (which he
clearly did). Assuming their investment was to earn an
annual return of 6.75%, compounded daily, what did
his grandparents originally pay for this certificate?
A.1A.9. This is a present value of a single payment
question, so use formula (1A.2).
PV = $10,000 * (PVIF 0.0675/365,365*16)




1

= $10,000 * 
16 *365 

.
0675




 1  


   365  

18
= $10k * (0.3396295) = $________.
Numerical Approach - Calculator Sequence:
.0675 [÷] 365 [=] [+] 1 [=] [yx] [(] 16 [x] 365 [)] [=] [1/x]
[x] 10000 [=]
Financial Calculator – Calculator Approach:
Enter
16 * 365 6.75 ÷ 365
N
I/Y
Solve for
PV
-3396.29
0
10000
PMT
FV
Keystrokes (assuming that [P/Y] is set for 1 period per
year).
365 [x] 16 [=] [N] 6.75 [÷] 365 [=] [I/Y] 0 [PMT] 10000
[FV] [CPT] [PV]
General Excel Function: PV(rate,nper,pmt,fv,type)
SS Solution: =PV($M41/$K41,$L41*$K41,$O41,$P41,0)
Ex. 1A.10
Congratulations, Jessica Poret (F382, Sp’09) you have
matched four numbers in the Louisiana State Lottery.
Jessica has two choices as to how she may receive her
prize, described below. What is the implied rate of
return between the two alternatives assuming the
annual return is compounded bi-monthly?
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Alternative 1: Jessica will receive an after-tax prize of
$65,000, today.
Alternative 2: Jessica would receive an after-tax prize
of $2,518.63 at the end of every other
month for the next five years.
A.1A.10. In this problem both the present value (of
Alternative 1) compared to the annuity stream
(of Alternative 2), as well as the term (and m)
are known. What is unknown is r, the relevant
interest rate. As described above in reference
to Ex. 1A.6, when the implicit interest rate
problem involves annuities, there is no direct
Equation Approach, and the relevant PVIFA
table must be used.
PVA = R * PVIFA X%/6,5*6
 $65,000.00 
 PVIFA = 
= 25.80768116.

 $2,518.63 
To find the answer using the equation approach, you
need to be able to look up the PVIFA factor (calculated
above) in the appropriate table in the row where t*m =
30. It occurs under the 1% column, this is the bi-monthly
rate, so the annual (nominal) rate equals 1% * 6 = 6%.
Financial Calculator – Calculator Approach:
Enter
Solve for
5*6
N
I/Y
1.00001
-65000
PV
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2518.63
PMT
0
FV
Keystrokes (assuming that [P/Y] is set for 1 period per
year).
5 [x] 6 [=] [N] 65000 [+/-] [PV] 2518.63 [PMT] 0 [FV]
[CPT] [I/Y] [x] 6 [=]  Ans. = ___________%.
In the last step, the bi-monthly, nominal rate of
1.00001% is converted to an annual basis by multiplying
it by “m”.
General Excel Function: RATE(nper,pmt,pv,fv,type)
SS Soln: =RATE($L45*$K45,$O45,-$N45,$P45,0)*$K45
Ex. 1A.11
Jonathan Miller (F382, Fa’08) and his spouse have
managed to save up $50,000 that they are going to use
to purchase a home at 39725 River Oaks Drive,
Ponchatoula. Assume the purchase price equals
$235,000. If the mortgage terms are for a 15-year,
5.75%, fixed-rate loan calculate their monthly loan
payments.
A.1A.11. When does the lump sum occur? Clearly, if
you borrow money you receive the proceeds of
the loan now. Therefore, this is a PV of an
annuity question so use formula (1A.4) and
solve for R.
First, need to solve (1A.4) for R.
 PVA 
PVA = R * (PVIFAr%,t)  R = 
.

PVIFA
r%,
t


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Second, what amount of the home purchase needs to be
financed?
Amt Borrowed = Purchase Price – Downpayment.
Here: $235,000 - $50,000 = $_________.
The monthly loan payment is then:
R = $185k ÷ (PVIFA 5.75%/12, 15*12)
 

1   1

15*12
  (1  (.0575 / 12))



= $185k ÷ 
 .0575 






 12 




= $185k ÷ (120.4224293) = $________.
Heuristic Check:
$1,536.26 * 180 = ___________ > $185,000.
Note: Why “A” in (1A.3) is not the same as “R” in
(1A.4). If you had chosen to solve this (incorrectly) as
an FVIFA problem the answer you would have gotten
is: $649.80 (student to verify). CLEARLY, not the
same as the correct answer above.
Numerical Approach - Calculator Sequence:
.0575 [÷] 12 [=] [STO] 1 [+] 1 [=] [yx] [(] 15 [x] 12 [)]
[=] [1/x] [+/-] 1 [=] [÷] [RCL] 1 [=] [STO] 2 185000 [÷]
[RCL] 2 [=]
Financial Calculator – Calculator Approach:
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Enter
15 * 12
N
5.75 ÷ 12
I/Y
-185000
PV
Solve for
PMT
1536.25866
0
FV
Keystrokes (assuming that [P/Y] is set for 1 period per
year).
15 [x] 12 [=] [N] 5.75 [÷ ] 12 [=] [I/Y] 185000 [+/-]
[PV] 0 [FV] [CPT] [PMT]
General Excel Function: = PMT(rate,nper,pv,fv,type)
SS Solution: =PMT($M49/$K49,$L49*$K49,-$N49,$P49,0)
Ex. 1A.12
According to an ABC news source the price for a 30second commercial during (the first-three quarters of)
Super Bowl XLV (2011) was $3.0 million. Assuming
that the cost has grown at an annual rate of (exactly)
6.52480495% (continuously compounded) and that the
number of intervals equals (t =) 26 what would the cost
of a comparable 30-second commercial have been in
1985?
A.1A.12. The question provides known FV and the
interest rate between single payments. This is
then a question where you need to find present
value. However there is continuous
discounting so use equation (1A.10).
PVCont. = FV26 * e -(r*t) = $3.0m * e -(.0652480495 * 26)
= $3.0m * 0.18333333
= $________.
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Numerical Approach – Calculator Sequence:
.0652480495 [x] 22 [=] [+/-] [2nd] [ex] [x] 3000000 [=]
Financial Calculator – Calculator Approach: n/a
General Excel Function (to find “e”): EXP(number)
SS Soln: =F52*(EXP(-B52*C52))
RULES FOR POSITIVE AND NEGATIVE
SIGNS IN EXCEL FUNCTIONS
Note: These rules apply generally to annuity problems
because in single payment problems FV (PV) is always
positive (negative) whether in the argument or answer.
The Jessica Poret and Kelly Rollins (F382, Sp’09) Rule:
In every TVM equation there is one negative
property. PV is Always negative, unless it is equal
to zero, in this case PMT has to be negative. FV is
Always positive!
The Dr. Meyer Rules (inspired by Jessica Poret and
Kelly Rollins, (above)):
If the Lump Sum is FV and known (unknown) then
PMT is negative in the answer (argument).
If the Lump Sum is PV and known (unknown) then
PMT is positive in the argument (answer).
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