MANAGERIAL FINANCE 410
STUDENT LECTURE NOTE 3
I. Time Value of Money, A Review
A.
Forward : Students in this class are presumed to be intimately familiar with the concepts involved in calculating the Time Value of Money since these topics would have been covered in FIN 381 .
The purpose of reviewing these topics here is not to bore you to death. Rather, since different instructors may have alternate approaches to covering this material, this review is meant to “get us all on the same page of the playbook” and give us a common basis for covering future topics.
B.
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 ________ 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 additional 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.
C.
Intro to Time Value
1.
In any time value problem, there are five variables of interest. These are:


Interest rate (r),
1


Annual frequency (m),

Present Value (PV), an amount today, and

Future Value (FV), an amount at some future time.
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 WHICH OF THESE ARE
KNOWN—WHATEVER IS LEFT UNKNOWN
MUST BE WHAT YOU ARE TRYING TO FIND.
“We must look for consistency. Where there is a wont of it we must suspect deception.”
- Sherlock Holmes; ‘The Problem of Thor Bridge'
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).
D.
Future Value of a Single Payment
The general formula to determine the FV of a single payment is:
FV t
= PV
0
 






1
 r m

 t *m



= PV
0
* FVIF r%/m,t*m
. (3.1)
E.
The Present Value of a Single Payment
Solve Formula (3.1) for the PV of a single amount.
2

FV = PV * (1 + (r/m)) t*m

PV
0
=

(1

FV t
( r/m)) t * m

Thus, the general formula to determine the PV of a single payment is:

PV
0
= FV t



1
(1

( r/m)) t *m

 = FV t
* PVIF r%/m,t*m
. (3.2)
F.
Annuities
1.
An annuity is a series of ____ 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.
G.
Future Value of an Annuity
The general formula to determine the FV of an ordinary (=end-of-period) annuity is:
FV
A
= A



(1

( r/m)) t *m
(r/m)

1

 = A*(FVIFA r%/m,t*m
). (3.3)
3

H.
Finding Deposits to Accumulate a Future Sum
1.
In the preceding FV annuity example the amount of the periodic deposits were known (as well as r, t, and m). Equation (3.3) was used directly to calculate the Future Value of the Annuity.
2.
A different, but related problem is to identify a savings target, i.e., FV
A
will be known (as well as r, t, and m) and then calculate the periodic deposits, A (is unknown) required to achieve that target.
3.
Examples where this approach would be applied would be saving to make a down-payment on a home or car, saving up a college fund for your children or for a vacation. And of course, setting aside money to fund your retirement portfolio.
4.
Equation (3.3) serves as the starting point, but the unknown ‘A’ is the value that is solved for.
FV
A
= A * (FVIFA r%/m,t*m
)

A =


FV
FVIFA
A r%/m, t * m

 . (3.4)
I.
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

4.
The general formula to find the Present Value of an
Annuity is given below:
PV
A
= R






1



1
(1

( r/m)) t *m


(r/m)





= R

(PVIFA r%/m,t*m
) . (3.5)
I.
Finding Payments to Repay Borrowing Today
1.
In the preceding PV annuity problem the amount of the periodic payments were _____ (as well as r, t, and m). Equation (3.5) was used directly to calculate the Present Value of the Annuity.
2.
A different, but related problem is that an amount has been borrowed today, i.e., PV
A
, will be known (as well as r, t, and m) and then calculate the periodic repayments, R (is unknown) required to repay that amount.
3.
This problem would apply to any situation where you have borrowed some amount (the lump sum occurs today) that will be paid off in regular future payments. Examples are mortgage, vehicle, major appliance and home equity loan repayments.
4.
Equation (3.5) serves as the starting point, but the unknown ‘R’ is the value that is solved for.
PV
A
= R * (PVIFA r%/m,t*m
)

R =



PV
PVIFA
A r%/m, t * m



. (3.6)
5

J.
FV and PV of Annuities Due
1.
Up to this point we have always calculated the value of annuities based on the assumption that 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 higher than their end-of-period counterpart.
The future value of an annuity due is then:
FV
AD
= A * (FVIFA r%/m,t*m
) *



1



 r m






. (3.7)
And the present value of an annuity due is:
PV
AD
= R * (PVIFA r%/m,t*m
) *



1



 r m






. (3.8)
K.
Split Annuities
1.
In some annuity problems it may be the case that the original amount of the annuity increases or
6

decreases to a different level for a finite period of time.
2.
For example, a split annuity (investment) strategy can be used by people to generate an immediate, steady income stream while potentially stretching some retirement savings for the future. Basically, this strategy involves dividing the initial premium into an immediate fixed annuity contract and a deferred fixed annuity contract.
3.
To value either the PV or FV of a split annuity, the student needs to realize that a FVIFA or PVIFA interest factor is simply the sum of the relevant single payment FVIFs or PVIFs. If the single payment factors can be added together then they can be subtracted off.
Ex. 3.1
Midas Gold Mines is trying to evaluate how much to pay for a specific tract of land they are considering purchasing. Assuming they require a return of 20% on projects of comparable risk what is the most they should pay for the following projected cash flows?
Assume this is a 15-year investment and that cash flows are received at month-end.
Years 1-10: $50,000 per month
Years 11-15: $20,000 per month
A3.1: This is a present value problem since we wish to know how much to pay for the project today. The first ten years are an ordinary annuity and years
11-15 are the split part.
PV
(Years 1-10)
= $50k * PVIFA
20%/12, 10*12
7

= $50k * 51.74492364
= _____________.
PV
(Years 11-15)
= $20k*(PVIFA
20%/12, 15*12
- PVIFA
20%/12, 10*12
)
= $20k * (56.93799414 – 51.74492364)
= $20k * 5.19307050 = ___________.
Total PV = $2,587,246.18 + $103,861.41
= _____________.
L.
Growing Perpetuities
1.
A perpetuity is essentially the ultimate type of annuity in that it is a perpet ual ann uity (hence the name), i.e., it lasts forever.
2.
The typical perpetuity valuation assumes that the amount of the annuity to be received in the future is a fixed amount. The m (in Eq. (3.9) ) serves as a reminder that the periodicity of the future value amount and the discount rate must be converted to a consistent basis, i.e., the quarterly dividend on a fixed-dividend preferred stock would be discounted with the _________ rate. Alternatively, both the quarterly FV and quarterly discount rate could be annualised via multiplication by m.
PV
0
=


FV
1
/m r/m

 . (3.9)
3.
However, this fixed-amount FV is not a necessary feature as it is possible to calculate the value of a growing annuity also.
4.
If the annuity is growing at a constant annual (or whatever frequency m equals) growth rate then the
8

perpetuity formula needs to be modified by subtracting this growth rate from the discount rate.
Again, note that equation (3.10) reflects a consistent frequency.
PV
0
=





 




 r m



FV
1 m






 g m









. (3.10)
Ex. 3.2
Assume that Heather Vincent (F381 Fa’03, F382
Fa’04) has forecast free cash flows (FCFs) for ACM
Enterprises in an effort to determine the firm’s horizon value. Assume that ACME ’s weighted average cost of capital is 10% and that FCFs can be assumed to be growing at an annual rate of 4% into perpetuity. If the current (most recent) quarterly
(QFCF) FCF equals $247,524.75 verify that calculated firm value is the same on both: a) a quarterly basis and b) an annual basis.
A3.2a: Since the Current FCF is given it must first be converted to the Expected quarterly FCF. Then the annual discount and growth rates must be converted to a quarterly basis.
QFCF
1
= QFCF
0
*

 1
 g m

 = $247,524.75 *

 1

= _____________

________.
0.04
4


9

PV
0
=





 


0.10
4
$250k







0.04
4









= ______________.
A3.2b: Here, the Expected quarterly FCF needs to be converted to an annual basis which then is consistent with the annual discount and growth rates originally given in the problem.
PV
0
=



$250k * 4

0 .
10

0 .
04

 = ______________.
M.
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. The future value and present value of a single payment with continuous compounding/discounting are given below.
FV
Cont.
= PV
0
* e rt . (3.11)
PV
Cont.
= FV t
* e -rt . (3.12)
N.
The Implicit Interest Rate
10

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
(1.1) directly for ‘r”.
FV t
= PV
0
*


1
 r m t * m



1
 r m t * m
=




FV
PV
0 t 




1
 r m
=



FV t
PV
0



1
(t * m)

r =






 FV t
PV
0



1
(t * m)

1




* m. (3.13)
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
(and the periodic interest rate (r/m) must be an integer).
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.
11

O.
Effective Interest Rates
1.
How to convert the nominal, annual rate above to the “effective” rate?
2.
Note: The annual percentage yield (APY) that you see expressed in CD advertisements, for example, is the same as the effective rate. m
r e
= 1
 r

1 . (3.14) m
Homework Problems (For Practice)
HW Ex. 3.3
Dominic Parenté (F304 Sp’00, F305 Fa’00, F410
Fa’01) and his wife Chrissie relocated to sunny
Gainesville, Florida, after they both graduated from the
University of Toledo. In anticipation of their first-born child attending that university in Florida, they have started a college savings fund. Nick plans to have accumulated $75,000 by the end of 15 years. What weekly deposit will be required to accumulate this future sum if their investment account earns 8.25%?
HW A3.3: Use formula (3.4) and solve for the required deposit, A.
FV = A * (FVIFA r%/m,t*m
)

A =


FV
(FVIFA r%/m, t * m )

 .
Here: A = $75,000 ÷ (FVIFA
8.25%/52, 15*52
)
12

= $75,000 ÷


[( 1

( 0 .
0825 / 52 ))
15 * 52
]

1


( 0 .
0825 / 52 )
= $75,000 ÷ (1540.212676) = ______.
Heuristic check:
780 deposits

$48.69 = __________ < $75k.
Numerical Approach - Calculator Sequence:
.0825 [÷] 52 [=] [STO] 1 [+] 1 [=] [y x ] [(] 15 [x] 52 [)]
[=] [-] 1 [=] [÷] [RCL] 1 [=] [STO] 2 75000 [÷] [RCL] 2
[=]
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 15*52 8.25 ÷ 52
N I/Y
0
PV PMT
75000
FV
Solve for ___________
Keystrokes (assuming that [P/Y] is set for 1 period per year).
15 [x] 52 [=] [N] 8.25 [÷] 52 [=] [I/Y] 0 [PV] 75000
[FV] [CPT] [PMT]
General Excel Function: = PMT(rate,nper,pv,fv,type)
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.
13

SS Solution: = PMT($C36/$A36,$B36*$A36,$D36,$F36,0)
HW Ex. 3.4
In The Andy Griffith Show
, Ep. #39: “Mayberry
Goes Bankrupt”, Mr. Frank Myers’ has a $100 (face value) Confederate States of America bond issued in
1861. If this bond was redeemed in 1961 (100 years later) for $349,119.27 what was the stated interest rate on the bond (assume annual compounding)?
HW A3.4: The question asks directly for the growth rate between single payments. This is clearly an implicit interest rate question so use (3.13) . In this case m=1 since compounding is annual. However, the solutions shown would also take into account non-annual compounding.
r = 










$349,119.2
7
$100.00



1
(100 * 1)

1




* 1




= _____.
Numerical Approach - Calculator Sequence:
100 [x] 1 [=] [1/x] [STO] 1 349119.27 [÷] 100 [=] [y x ]
[RCL] 1 [=] [-] 1 [=] [x] 1 [=]
Financial Calculator – Calculator Approach:
Enter
Solve for
100 * 1
N I/Y
-100
PV
_______
0
PMT
349119.27
FV
14

Keystrokes (assuming that [P/Y] is set for 1 period per year).
100 [x] 1 [=] [N] 100 [+/-] [PV] 0 [PMT] 349119.27
[FV] [CPT] [I/Y] [x] 1 [=]
General Excel Function: =RATE(nper,pmt,pv,fv,type)
SS Soln: = (RATE($B28*$A28,$E28,-$D28,$F28,0))*$A28
HW Ex. 3.5
Eddie Bertoniere (F382 Fa’05) has just received a significant performance raise in his now highly-paid position as a securities analyst. To celebrate he is planning to purchase a 2008 Audi TT Roadster. The annual cost on his 4-year loan will be 3.9% and he has calculated that the monthly payments should be
$644.48. What amount will Eddie have paid to purchase this vehicle if he initially put a downpayment of $10k on the car?
HW A3.5: PV
A
= $644.48 * PVIFA
3.9%/12, 4*12
= $644.48




1





1
(1

( 0 .
039 /12))
4 * 12
(0.039/12)
= $644.48

(44.37672824)







= __________.
Thus, the cost of his roadster = $28,599.91 + $10k
15

= __________.
Numerical Approach - Calculator Sequence:
.039 [÷] 12 [=] [+] 1 [=] [y x ] [(] 4 [x] 12 [)] [=] [1/x] [+/-]
[+] 1 [=] [÷] [(] .039 [÷] 12 [)] [=] [x] 644.48
Financial Calculator – Calculator Approach:
Enter 4 * 12 3.9

12 644.48 0
N I/Y PV PMT FV
Solve for _________
Keystrokes (assuming that [P/Y] is set for 1 period per year).
4 [x] 12 [=] [N] 3.9 [÷] 12 [=] [I/Y] 644.48 [PMT] 0
[FV] [CPT] [PV]
General Excel Function: =PV(rate,nper,pmt,fv,type)
SS Solution: = PV($C20/$A20,$B20*$A20,$E20,$F20,0)
HW Ex. 3.6
Dr. Liping Zou (Fin 125.700 Sp’99, Fin 125.740
Fa’00) is planning a trip home to China to visit her parents. The airfare will be $1,389. Assuming she will pay for the ticket in six months how much does she need to deposit into her savings account today to be able to purchase her ticket if her account earns an annual rate of 4.5% and interest is compounded quarterly?
16

HW A3.6: Use Formula (3.2) to solve for unknown PV.
PV = $1,389 *




1



1
0 .
045
4
 
0 .
5 * 4




= $1,389 * (0.97787407)
= _________,
where PVIF
4.5%/4, 0.5*4
= 0.97787407.
Numerical Approach - Calculator Sequence:
0.045 [

] 4 [=] [+] 1[=] [y x ] [(] 0.5 [x] 4 [)] [=] [1/x] [x]
1389 [=]
Financial Calculator – Calculator Approach:
Enter 0.5*4 4.5

4
N I/Y PV
0
PMT
1389
FV
Solve for _________
Keystrokes (assuming that [P/Y] is set for 1 period per year).
0.5 [x] 4 [=] [N] 4.5 [

] 4 [=] [I/Y] 0 [PMT] 1389 [FV]
[CPT] [PV]
General Excel Function: = PV(rate,nper,pmt,fv,type)
SS Solution: = PV($C12/$A12,$B12*$A12,$E12,$F12,0)
17

HW Ex. 3.7
Jonathan Miller (F382 Fa’08) is 25 years old and he plans to have accumulated $1 million by the time he is
40, i.e., 15 years from now, through an investment portfolio he will develop himself. His goal is that the portfolio will earn a return of 20% p.a. (compounded bi-weekly). Assuming that Jonathan’s portfolio does achieve his return objective, what bi-weekly deposit will be required to attain his goal?
HW A3.7: Use equation (3.4) to solve for the bi-weekly deposit required.
A =



FVIFA
FV
A
20%/26, 15 * 26



= $1m




(1

(.
20 / 26 ))
15 * 26
(.20/26)

1



= $1m/(2451.316505) = ___________ = _______.
Numerical Approach - Calculator Sequence:
0.20 [

] 26 [=] [+] 1 [=] [y x ] [(] 15 [x] 26 [)] [=] [-] 1 [=]
[

] [(] .20 [

] 26 [)] [=] STO [1] 1,000,000 [

] RCL [1]
[=]
Financial Calculator – Calculator Approach:
Enter 15 * 26 20

26 0
N I/Y PV PMT
1000000
FV
Solve for _________
Keystrokes (assuming that [P/Y] is set for 1 period per year).
18

15 [x] 26 [=] [N] 20 [

] 26 [=] [I/Y] 0 [PV] 1000000
[FV] [CPT] [PMT]
General Excel Function: = PMT(rate,nper,pv,fv,type)
SS Solution: = PMT($C16/$A16,$B16*$A16,$D16,$F16,0)
HW Ex. 3.8
Dr. Wei-Huei “Wendy” Hsu (F&P 125.100 Sp’99) has written the following question for her Fin 125.700
MBA Corporate Finance class at Massey University’s
Wellington campus.
‘A firm’s chief financial analyst is evaluating leasing a new building. Assume that the purchase price
HW A3.8: 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 PV
AD
approach given in Formula ( 3.8
) is the place to start.
However, what Dr. Hsu wants the student to solve for is the lease payment (i.e., R), with the AD twist.
of the building would be $1.5 million and the fifteenyear lease terms will fully amortize this price. What is the estimated monthly, beginning-of-the-period payment she will calculate if the lessor earns a fixed-rate return of 8.25% p.a.? Round the answer to the nearest cent.’
What is the correct answer to this question?
19

PV
AD
= R * PVIFA r%/m,t*m
*



1



 r m






,

R =



PV
AD
PVIFA
8.25%/12,1 5 * 12
* (1

(r/m)) 


.
Here: R =


103
$ 1 , 500 , 000
.
0778683 * ( 1

(.
0825 / 12 ))

 = _________.
Numerical Approach - Calculator Sequence:
1 st Step: .0825 [÷] 12 [=] [+] 1 [=] [y x ] [(] 15 [x] 12 [)] [=]
[1/x] [+/-] [+] 1 [=] [÷] [(] .0825 [÷] 12 [)] [=] [STO] 1;
2 nd Step: .0825 [÷] 12 [=] [+] 1 [=] [x] [RCL] 1 [=] [STO] 2;
3 rd Step: 1500000 [÷] [RCL] 2
Financial Calculator – Calculator Approach: Need to first change the [2 nd ] [PMT] < = BGN > default from
END (end-of-period) to BGN (beginning-of-period).
Note: To change the payment period (END/BGN), press
[2 nd ] [BGN], then press [2 nd ] [SET]. Don’t forget to reset your calculator to [END] after this problem.
Enter 15 * 12 8.25

12 -1500000
N I/Y PV PMT
0
FV
Solve for __________
Keystrokes (assuming that [P/Y] is set for 1 period per year) and [END/BGN] is set for [BGN].
20

15 [x] 12 [=] [N] 8.25 [

] 12 [=] [I/Y] 1500000 [+/-]
[PV] 0 [FV] [CPT] [PMT]
General Excel Function: = PMT(rate,nper,pv,fv,type)
SS Solution: = PMT($C24/$A24,$B24*$A24,-$D24,$F24,1)
HW Ex. 3.9
If Tyra Navarre (F382 Sp’07, F495 Fa’07) deposits
$4,066.07 into an account that earns an annual interest rate of 7.5%, which is compounded daily. How much will she be able to withdraw at the end of 12 years?
HW A3.9: This is a future value of a single payment so use equation (3.1) .
FV
12
= $4,066.07
 






1

.
075
365
( 12 * 365 ) 


= $4,066.07 * (2.45937572)
= _____________ = _________, where: FVIF
7.5% / 365, 12*365
= (2.45937572).
Numerical Approach - Calculator Sequence:
0.075 [

] 365 [=] [+] 1 [=] [y x ] [(] 12 [x] 365 [)] [=] [x]
4066.07 [=]
Financial Calculator – Calculator Approach:
Enter 12*365 7.5

365 -4066.07 0
N I/Y PV PMT FV
Solve for 9999.99
21

Keystrokes (assuming that [P/Y] is set for 1 period per year).
12 [x] 365 [=] [N] 7.5 [

] 365 [=] [I/Y] 4066.07 [+/-]
[PV] 0 [PMT] [CPT] [FV]
General Excel Function: =FV(rate,nper,pmt,pv,type).
SS Solution: = FV($C8/$A8,$B8*$A8,$E8,-$D8,0)
In conclusion:
This implies that if somebody offers you $4,066.07 today or $9,999.99 at the end of 12 years, and your opportunity cost of funds = 7.5% (compounded daily), 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?
HW Ex. 3.10
<Your name here> and your spouse have recently sold your starter home and have netted $200k. You are interested in purchasing a 4BR, 3½BA custom home at
19222 Indian Ridge at The Lake at White Oak in Baton
Rouge. The stated purchase price equals $499,000 and you will use the sale proceeds as your downpayment.
If the mortgage terms are for a 25-year, 6.375%, fixedrate loan determine the monthly loan payment.
22

HW A3.10: 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 but you need to solve for R, so use formula (3.6) .
Second, what amount of the home purchase needs to be financed?
Amt Borrowed = Purchase Price – Downpayment.
Here: $499,000 - $200,000 = ________.
The monthly loan payment is then:
R = $299k ÷ (PVIFA
6.375%/12, 15*12
)
= $299k ÷







1




1
( 1

(.
06375 / 12 ))
25 * 12
.
06375
12










= $299k ÷ (149.8313099) = _________.
Heuristic Check:
$1,995.58 * 300 = ________ > $299,000.
Note: Why “A” in (3.4) is not the same as “R” in (3.6) .
If you had chosen to solve this (incorrectly) as an
FVIFA problem the answer you would have gotten is:
23

$407.14 (student to verify). CLEARLY, not the same as the correct answer above.
Numerical Approach - Calculator Sequence:
.06375 [÷] 12 [=] [STO] 1 [+] 1 [=] [y x ] [(] 25 [x] 12 [)]
[=] [1/x] [+/-] 1 [=] [÷] [RCL] 1 [=] [STO] 2 299000 [÷]
[RCL] 2 [=]
Financial Calculator – Calculator Approach:
Enter
Solve for
25 * 12 6.375 ÷ 12 -299000
N I/Y PV PMT
0
FV
_________
Keystrokes (assuming that [P/Y] is set for 1 period per year).
25 [x] 12 [=] [N] 6.375 [÷ ] 12 [=] [I/Y] 299000 [+/-]
[PV] 0 [FV] [CPT] [PMT]
General Excel Function: = PMT(rate,nper,pv,fv,type)
SS Solution: = PMT($C48/$A48,$B48*$A48,-$D48,$F48,0)
HW Ex. 3.11
Bethany Threeton (F382 Sp’05) is trying to get her start-up business off the ground and is looking for investment capital. If you lend her $25,000 today she has promised to make bi-monthly repayments of
$1,321.78 over a four-year period. Based on these two alternatives what nominal rate of return would Bethany be paying you? What effective rate would you be earning?
24

HW A3.11: In this problem both the present value compared to the annuity stream as well as the term (and m) are known. What is unknown is r, the relevant interest rate. When the implicit interest rate problem involves annuities, there is no direct
Equation Approach, and the relevant PVIFA table must be used.
PV
A
= R * PVIFA
X% / 6, 4*6

PVIFA =


$ 25 , 000 .
00
$ 1 , 321 .
78

 = _______.
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 =
24. It occurs under the 2% column, this is the bi-monthly rate, so the annual (nominal) rate equals 2% * 6 = 12%.
The effective rate is then found as follows:
r e
=


1

0 .
12
6


6

1 = _______.
Financial Calculator – Calculator Approach:
Enter 4 * 6
N I/Y
Solve for _______
-25000
PV
1321.78
PMT
0
FV
Keystrokes (assuming that [P/Y] is set for 1 period per year).
25

4 [x] 6 [=] [N] 25000 [+/-] [PV] 1321.78 [PMT] 0 [FV]
[CPT] [I/Y] [x] 6 [=]

Ans. = 12.00010293%.
In the last step, the bi-monthly, nominal rate of
2.00002% is converted to an annual basis by multiplying it by “m”.
General Excel Function: =RATE(nper,pmt,pv,fv,type)
SS Soln: =(RATE($B44*$A44,$E44,-$D44,$F44,0))*$A44
HW Ex. 3.12
Brittani Wester (F382 Fa’07) has agreed to serve as a
“guest” Finance Question Writer. The question she has written is as follows.
“Today is June 4, 2007 and a $10,000 Treasury Bill with a maturity of September 3, 2007 (i.e., maturity equals 91 days) is yielding a nominal rate of 4.68% p.a. (assume daily compounding and a 365-day year).
How much would you expect to pay for this bond today? Further, what effective interest rate would you have earned on this investment?”
What are the answers to these two questions?
HW A3.12
This is a present value of a single payment question, so use formula (3.2) .
PV = $10,000 * (PVIF
0.0468%/365, 365*(91/365)
)
26

= $10,000 *






1

.
0468
365
1
(( 91 / 365 ) * 365 )






= $10k * (0.9884006) = _________.
Numerical Approach - Calculator Sequence:
Do exponent first: 91 [÷] [365] [x] 365 [STO] 1
(This adjustment is (of course) trivial, but please to note that the formula will still work for periods that are less than one year.)
.0468 [÷] 365 [=] [+] 1 [=] [y x ] [RCL] 1 [=] [1/x] [x]
10000 [=]
Financial Calculator – Calculator Approach:
Enter (91/365)*365 4.68 ÷ 365 0 10000
Solve for
N I/Y PV
_______
PMT
Keystrokes (assuming that [P/Y] is set for 1 period per year).
91 [÷] 365 [x] 365 [=] [N] 4.68 [÷] 365 [=] [I/Y] 0
[PMT] 10000 [FV] [CPT] [PV]
General Excel Function: =PV(rate,nper,pmt,fv,type)
FV
27

SS Solution: = PV($C40/$A40,$B40*$A40,$E40,$F40,0)
The effective rate is then found using (3.14) as follows:
r e
=


1

0 .
0468
365


 365

1 = _________.
HW Ex. 3.13
Jana Simurkova (F381 Su’07, F382 Fa’07, F495
Fa’07) has wisely started saving for her retirement at the tender age of 23. She has identified a guaranteed annuity that will yield a return of 9.5% as long as she makes beginning of the week, bi-weekly deposits of
$125 for the next 36 years. Assuming that her goal is to have accumulated at least $1m dollars before she is
60, and that her investment does earn the promised return, will she achieve her goal and what will her account be worth after making these deposits?
HW A3.13: We want to know what the value of Jana’s account, after making equal, bi-weekly deposits is worth at the end of year 36, so this is clearly a FV of annuity problem, but with the annuity due twist.
FV
A
= $125.00 * (FVIFA
9.5%/26, 36*26
) *



1

= $125




(1

(.
095 / 26 ))
36 * 26
(.095/26)

1


 *


 r m



1







.095
26


= $125

(8040.69747)*(1.00365385) = ____________.
28

Numerical Approach - Calculator Sequence:
1 st Step: 0.095 [

] 26 [=] [+] 1 [=] [y x ] [(] 36 [x] 26 [)]
[=] [-] 1 [=] [

] [(] .095 [

] 26 [)] [=] [STO] 1;
2 nd Step: 0.095 [÷] 26 [=] [+] 1 [=] [x] [RCL] 1 [=] [STO]
2;
3 rd Step: 125 [x] [RCL] 2
Financial Calculator – Calculator Approach: Remember to re-set for BOP payments. [2 nd
Enter
] [BGN] [2
36 * 26 9.5

26 0 -125 nd ] [SET]
N I/Y PV PMT FV
Solve for ___________
Keystrokes (assuming that [P/Y] is set for 1 period per year).
36 [x] 26 [=] [N] 9.5 [

] 26 [=] [I/Y] 0 [PV] 125 [+/-]
[PMT] [CPT] [FV]
General Excel Function: = FV(rate,nper,pmt,pv,type)
SS Solution: = FV($C32/$A32,$B32*$A32,-$E32,$D32,1)
HW Ex. 3.14
According to an ABC news source the price for a 30second commercial during (the first-three quarters of)
Super Bowl XLI (2007) was $2.6 million. Assuming that the cost has grown at an annual rate of (exactly)
7.01953915% (continuously compounded) and that the number of intervals equals (t =) 22 what would the cost of a comparable 30-second commercial have been in
1985?
29

HW A3.14: The question provides the interest rate and the
FV between single payments. This is then a question where you need to find present value.
However there is continuous discounting so use equation (3.12) .
PV
Cont.
= FVt * e
-(r*t)
= $2.6m * e
-(.0701953915 * 22)
= $2.6m * 0.21346154
= ________.
Numerical Approach - Calculator Sequence:
.0701953915 [x] 22 [=] [+/-] [2nd] [e x ] [x] 2600000 [=]
Financial Calculator – Calculator Approach: n/a
General Excel Function (to find “e”): EXP(number)
SS Soln: =F52 * ( EXP(-B52*C52))
30