Additional Solved Problems
Lump Sum
Future Value
The Problem
– You've received a $40,000 legal settlement.
Your great-uncle recommends investing it for
retirement in 27-years by “rolling over” oneyear certificates of deposit (CDs)
– Your local bank has 3% 1-year CDs
– How much will your investment be worth?
– Comment.
Categorization
– Your capital gains will be reinvested. There is
no cash-flow from the settlement for 27 years,
so this is a lump sum problem.
– There is some uncertainty in the cash flows
because interest rate are static for just the first
year, but we assume that it will be 3% until you
retire
– If you are unable to shelter your earnings, the
IRS will want their cut
Data Extraction
•
•
•
•
PV = $40,000
i = 3% (or 3% * (1- marginal tax rate)?)
n = 27-years
FV = ?
Solution by Equation
F  P(1  i ) n
 40,000(1  0.03) 27
 $ 88,851.56
Calculator Solution
N
I
PV
27
3% 40,000
PMT FV
0
?
$88,851.56
Comments
• Your great uncle's a financial idiot
• Given a 27-year investment, you should
either
– Invest the money more aggressively to
accumulate the money you need to survive, or
– Live! Blow the money on that red
convertible!
3 Additional Solved Problems
Lump Sum
Interest Rate
Problem 1
• If you have five years to increase your
money from $3,287 to $4,583, at what
interest rate should you invest?
Algebraic Solution
1
n
F
F  P (1  i )  i     1
P
n
1
5
4583 

i
  1  0.0687  6.87%
 3287 
Problem 2
• An investment you made 12-years ago
is today worth its purchase price. It has
never paid a dividend.
• Closer inspection reveals that the share
price has been highly periodic, moving
from $150 when purchased, to $300 in
the next year, to $75 in the next, back to
$150, before repeating
Cyclical Price Movement
300
250
Price ($)
200
150
100
50
0
0
2
4
6
Year
8
10
12
12-Year and Average Returns
EndCF  StartCF  Dividends
StartCF
150  150  0

0
150
Compare with Average HPR
HoldingPer iodReturn 
1 300  150 75  300 150  75
150  75 





12  150
300
75
75 
1
100%  75%  100%    100%  41.67%
12
Comments
– Here we have the average holding period return
being 41.67% per year, while the security has
returned you nothing over the whole period!
– Averages seduce us with their intuitiveness
– The correct average to have used was the
geometric average of return factors, not the
arithmetic average of return rates
Averages Must be Meaningful 1
– You walk 1 mile at 2 mph and another at 3
mph. What was your average speed? (2+3)/2 =
2.5 mph.
– NO!
– The first leg lasts 1/2 hour, and the second leg
lasts 1/3 hours, total 5/6 hours.
– So average speed is 2/(5/6) = 2.4 mph.
Averages Must be Meaningful 2
– A little analysis shows that the correct mean for
the walker is the harmonic mean
– The correct mean for the return problem may
be shown to be the geometric mean of the
(1+return)’s
– The appropriate mean requires thought
Problem 3
– In 1066 the First Duke of Oxbridge was
awarded a square mile of London for his
services in assisting the conquest the England.
The 30th Duke wished to live a faster paced
life, and sold his holding in 1966 for
£5,000,000,000. Examination of original
project’s cost showed only the entry “1066 a.d.:
to repair armor, £5”
– What was rate of capital appreciation ?
Categorization
– We may assume that the Dukes lived quite well
from leasing land to their tenants, but we are
not interested in the revenue cash flows here,
just the capital cash flows
– There is a present cash flow, a future cash flow,
and no annuity payments, so the problem is the
return on a lump-sum invested for a number of
periods
Data Extraction
•
•
•
•
PV = 10
FV = 5,000,000,000
n = (1966 - 1066) = 1900
i=?
Solution by Equation
F  P1  i 
n
1
n
F

 n    1
P
1
1900
5000000000 
n  
 1
5


 1.096667999%  1.10%
Solution by Calculator
n
i
1900 ?
1.09666%
PV Pmt FV
-5
0
5,000,000,000
Comments
• Note that a capital gain of only 1.1% per
year results in a huge value over time
• Time plus return is very potent
• The real issue here is what is missing,
namely the revenue streams
Additional Solved Problems
Lump Sum
Number of periods
The Problem
• How many years would it take for an
investment of $9,284 to grow to $22,450
if the interest rate is 7% p.a. ?
• p.a. = per annum = per year
Categorization
– This is a lump sum problem asking for a
solution in terms of time. Most of these
problems are useful models of reality if
expressed in real terms, not nominal terms
– In any nominal situation, the terminal $22,450
will not be a constant, but will depend on the
unknown time
– We will assume that the numbers and rates are
in real terms
Data Extraction
•
•
•
•
PV = $9,284
FV = $22,450
i = 7% p.a.
n=?
Solution by Equation
F  P1  i 
n
F

ln 
P
n  
ln1  i 
 22,450 
ln

9,284 
n 
 13.05 years
ln1  0.07 
Additional Solved Problems
Lump Sum
Present Value
The Problem
– If investment rates are 1% per month, and you
have an investment that will produce $6,000
one hundred months from now, how much is
your investment worth today?
Categorization
– This is the most basic of financial situations,
and involves finding the present value of a
future payment given no periodic payments
– The issue of risk is a little fuzzy. It is assumed
that the rate given is for the project’s risk
category
Data Extraction
•
•
•
•
FV = $6000
PV = ?
n = 100 months
i = 1%
Solution by Equation
F
n
F  P (1  i )  P 

F
*
(
1

i
)
(1  i ) n
6,000
P
 $ 2,218.27
100
(1  0.01)
n
Calculator Solution
N
I
PV
PMT
100
1%
?
0
-2,218.27
FV
6000
Additional Solved Problems
Lump Sum
Special Case: Doubling
Rule of 72
The Problem
• Consider the following simple example:
– Sol Cooper Investments have offered you a
deal. Invest with them and they will double
your investment in 10 years. What interest rate
are they offering you?
– We could solve this using
•
but this is over-kill
i F P 1
n
Data Extraction
• Doubling
• n = 10
• i=?
Some Algebra
F  P1  i 
n
F
F


ln 
ln 
P
P
n   
ln1  i 
 i  1  i 3 1  i 5

2 
 
  
  
 2  i  3  2  i  5  2  i 

F

ln 
2  i   F 
P

n

ln 
i 
2i
P
2 * 

2i
2  0.08  2 
To double, i  8% : n * i 
ln   0.72
2
1
Solution by Equation
i * n  0.72
72
i  %  7.20%
10
Accurate answer
1
n
1
2x 

i     1  210  1  7.16%
 x 
The Secret Reveled
– Now you have seen the derivation of the rule
of 72, you are now able to produce your own
personal rules. Example:
• “The Rule of a Magnitude”
To increase your wealth by 10 times, the product of
interest and time is 240, that is about (2.08/2)*ln(10)
Example, how long will it take to increase your money ten
times, given interest rates of 10%?
N = 240/10 = 24 years, real answer is 24.16 years
How good is the Rule of 72?
– We have derived a rule using approximation
methods, but have no idea how accurate it is
– There are two tests we could apply
• we could take some range, and determine the
absolute maximum error of the rule in that range
• we could simply graph the error
– Graphs are fun:
Doubling your Money
100.00%
90.00%
80.00%
70.00%
rule
algebra
Interest
60.00%
50.00%
40.00%
30.00%
20.00%
10.00%
0.00%
0
5
10
15
Years
20
25
30
Absolute Error
5.00%
0.00%
0
5
10
15
% Error
-5.00%
-10.00%
error
-15.00%
-20.00%
-25.00%
-30.00%
Years
20
25
30
Graph of Rule of 72 Error
– The high error in a part of the graph that does
not interest us is hiding the error in the part that
does. We have two choices
• plot absolute error on a log scale
• truncate the graph and re-scale
– Truncation is fun
Absolute Error
0.10%
0.05%
0.00%
5
10
15
20
% Error
-0.05%
-0.10%
-0.15%
-0.20%
-0.25%
-0.30%
-0.35%
-0.40%
Years
25
30
Another Example
– You are a stockbroker wishing to persuade a
young client to reconsider her $50,000 invested
in 3%-CDs.
– Your client believes that stock mutual funds
will return about 12% for the foreseeable
future, but is averse to the volatility risks. Her
money will remain fully invested for the next
48 years.
Step 1
– The first step requires the calculation of how
long is required to obtain a single doubling
• CDs: 72/3 = 24 years to double
• Mutual fund: 72/12 = 6 years to double
Step 2
– The second step requires the calculation of how
many doublings will occur during the lives of
the investments
• CDs: 48/24 = 2 doublings
• Mutual fund: 48/6 = 8 doublings
Step 3
– The third step calculates the value of the
investment in 48 years
– CDs: 2 doublings of $50,000
• = $200,000
– Mutual fund: 8 doubling of $50,000
• =256 * $50,000
• =$12,800,000 in 48 years
Conclusion
– We shall discover that her risk is smaller than
she imagines, but she will be about 64 times
more wealthy if she accepts that risk
– Using the accurate method, her respective
wealths are $206,613 and $11,519,539,
– The lesson is to start to invest early, and
accept some risk
Growth at 3 and 12 %
– The following graph shows her wealth
increases over 10 years at a 3% and 12%
• The graph was cut at 10 years because the 12% rate
of growth is so large that it dwarfs the 3% growth,
making the graph meaningless
Growth of $50,000 for 10 Years
@ 3% and 12%
Ten Years Growth @ 3% & 12%
180000
160000
Value at end of Holding Period
140000
CD
Stock
120000
100000
80000
60000
40000
20000
0
0
1
2
3
4
5
6
Holding Period in Years
7
8
9
10
Log Transformation of Y-Axis
– A common way to plot two such cash flows on
the same graph is to use a semi-log graph. This
prevents scale problems from hiding one of the
graphs
– Note that the two graphs appear to be straight
lines, and this is in fact the case
Growth of $50,000 at 3% and
12% for 48 Years (Log Scale)
48 Years Log scale
CD
Stock
Value at end of Holding Period (Log Scale)
100000000
10000000
1000000
100000
10000
0
5
10
15
20
25
30
Holding Period in Years
35
40
45
50
What is the use of the Rule?
– A significant source of avoidable error in
financial calculations results from blindly
“running the numbers” without reviewing them
for empirical reasonableness
– It is a good practice to estimate values before
computing them
– The rule of 72 is one tool that sometimes gives
you “numerical feel” of a problem
• Your reaction to learning the rule of 72 is
– “Why bother, I’ve got the latest and best HP
financial calculator.”
• In a business meeting, the unilateral
drawing of a financial calculator has a
chilling effect on your opponents
flexibility in a negotiation
– It is amazing how many real problems you
can solve in your head using the rule of 72
Additional Solved Problems
Irregular Cash Flows
Backwards Method
The Problem
– You have been offered a video business, and
estimate that video rental technology will be
obsolete in 8 years when cable bandwidths and
video compression will permit “movie-ondemand.” You require a 20% return on this
class of risk. The cash flows, starting 1-year
from now,are: 90, 110, 140, 140, 130, 90, 70,
30 (thousands of $s)
A Faster Method of Discounting
– This is basically a present value of a lump sum
repeated 8-times
– The most straightforward method would be to
crunch the answer or use an Excel worksheet
– A good method to use on a calculator is the
following algorithm:
A Faster Method of Discounting
(Continued)
– Input the last cash flow, and divide by the
interest factor to “bring it to” a year earlier
– Iterate:
• Add this discounted cash flow to the cash flow that
is already there, and discount the total for another
period by dividing by the interest factor. Stop when
you reach the current time
• Doing this is a lot simpler than it sounds
Data and Computation:
Backwards
year
8
7
6
5
4
3
2
1
0
Flow
Accumulation
$30.00
$30.00
$70.00
$97.27
$90.00
$178.43
$130.00
$292.21
$140.00
$405.64
$140.00
$508.77
$110.00
$572.52
$90.00
$610.47
$0.00
$554.97
Equations
rate
RateFactor
0.1
=1+rate
year
8
7
6
5
4
3
2
1
0
Flow
30
70
90
130
140
140
110
90
0
Accumulation
=0 + B8
=C8/RateFactor + B9
=C9/RateFactor + B10
=C10/RateFactor + B11
=C11/RateFactor + B12
=C12/RateFactor + B13
=C13/RateFactor + B14
=C14/RateFactor + B15
=C15/RateFactor + B16
Data and Computation:
Traditional
year_
0
1
2
3
4
5
6
7
8
Sum
Flow_
Discounted
$0.00
$0.00
$90.00
$81.82
$110.00
$90.91
$140.00
$105.18
$140.00
$95.62
$130.00
$80.72
$90.00
$50.80
$70.00
$35.92
$30.00
$14.00
$554.97
Equations
year_
0
1
2
3
4
5
6
7
8
Sum
Flow_
0
90
110
140
140
130
90
70
30
Discounted
=Flow_*(1+rate)^-year
=Flow_*(1+rate)^-year
=Flow_*(1+rate)^-year
=Flow_*(1+rate)^-year
=Flow_*(1+rate)^-year
=Flow_*(1+rate)^-year
=Flow_*(1+rate)^-year
=Flow_*(1+rate)^-year
=Flow_*(1+rate)^-year
=SUM(C19:C27)
Calculator Solution
The computation a BAII+ calculator is
30/1.1+ 70/1.1+ 90/1.1+ 130/1.1+
140/1.1+ 140/1.1+ 110/1.1+ 90/1.1=
The solution is $554.97
Calculators differ in the way they string
computations, you may need to add “=“
after the dollar amounts
See the savings on computational time!
Comments
– It is particularly useful to know the backwards
method when the yield curve is not flat. (Use
the forward rates). The level of computation
savings are even greater in this case
Additional Solved Problems
Deceptive Interest Rates
The Problem
• Advertisement:
– American Classic Cars! Finance Special!
Sprite Conversion! Now Only $15,000!
Just $1,000 Down, and 3-years to pay!
Only 3% per year! (Compounded monthly
with your good credit.)
The Problem (Continued)
• Classic Car News has an almost
identical car advertised for $9,000, but it
needs $3,000 of work to match the
condition of the car offered by ACC.
• What implied rate of interest, (per year,
compounded yearly) would you be
paying if you purchased the car from
ACC?
Explanation
• When purchasing from Smart, you are
buying a bundle of financing and car
• To un-bundle the package, you use the
cost of acquiring the competing car
– Cash value of car = $9,000 + $3,000 =
$12,000
• Next, determine the cash flows associated
with the financed car
Calculator
N
36
I
PV
PMT
3%/12 ($15,000
?
=
-$1,000) -407.14
0.25%
=
$14,000
FV
0
Analysis Continued
• The equivalent value of each cash flow is
•
•
•
•
$(12,000-1,000)
-407.14
…
-407.14 (36 equal payments in all)
Calculator (Continued)
N
I
PV
PMT
(monthly)
36 3%/12 = ($15,000?
0.25% $1,000) = -407.14
$14,000
FV
36
0
?
($12,000- -407.14
1.6419% $1,000) =
$11,000
0
True Interest Rates
i  1.6419 *12  19.70% p.a. compounded monthly
i  1.01641912  1  21.58% p.a. compounded p.a.
The true interest rate on this loan is much higher than that
in the advertisement
An enterprising lady sold jewelry in a factory where
she worked. The people she sold to were poor credit
risks. She gave them interest-free loans, one third down.
She marked up her prices to cost + 200%. No Risk!
Series of Annuities
• The next problem evaluates a project that
has a sequence of annuities
– The method of solution is to evaluate each
annuity to the date one year before its first
cash flow, and then to discount these lump
sum equivalent amounts to today’s date
– The cash flow feature of a financial calculator
may also be used
The Problem
• Expected cash flows from a project
requiring a 20% return
• Years
Cash Flow Each Year
• 0
$(20,000,000)
• 1 to 5
$3,000,000
• 6 to 30
$2,000,000
• 31 to 49
$1,000,000
• 50
$(2,000,000)
Present Values of Components
From
To
0
1
6
31
50
Rate
0
5
30
49
50
Amount
PV
(20,000,000) (20,000,000)
3,000,000
8,971,836
2,000,000
3,976,649
1,000,000
20,404
(2,000,000)
(220)
0.2 Sum
(7,031,331)
Excel Equations
From
0
1
6
31
50
To
0
5
30
49
50
Amount
-20000000
3000000
2000000
1000000
-2000000
Rate
0.2 Sum
PV
=Amount/Rate/(1+Rate)^(From-1)*(1-(1/(1+Rate)^(To-From+1)))
=Amount/Rate/(1+Rate)^(From-1)*(1-(1/(1+Rate)^(To-From+1)))
=Amount/Rate/(1+Rate)^(From-1)*(1-(1/(1+Rate)^(To-From+1)))
=Amount/Rate/(1+Rate)^(From-1)*(1-(1/(1+Rate)^(To-From+1)))
=Amount/Rate/(1+Rate)^(From-1)*(1-(1/(1+Rate)^(To-From+1)))
=SUM(E5:E9)
Note
• A single lump sum is just a degenerate
annuity. The above equations made use of
this fact
• The project is not at all attractive at the
given rate
• At what discount rate does the project
become attractive?
Present Value of Project v Rate
50,000,000
Present Value
40,000,000
30,000,000
20,000,000
10,000,000
0
0.00%
5.00%
10.00%
(10,000,000)
Discount Rate
15.00%
20.00%
The Problem
• What is the present value of the following
project? The cash flow starts in year 1
• $20,000, $20,000, $20,000, $20,000,
$20,000, $20,000, $20,000, $15,000,
$20,000, $20,000, $20,000, $20,000,
$20,000, $20,000, $20,000.
• The discount rate is 12% p. a.
Analysis
• This project is basically an annuity with a
hiccup.
– Add $5,000 to the hiccup,
– Evaluate the annuity, and then
– Subtract the PV of the $5,000
Algebraic Solution
pmt   1 
1  
Padj _ ann 

i   1  i 
n




15
20,000   1  
1  

   136,217.29

0.12   1.12  
Ppert
Padj _ ann  Ppert
 5,000

 2,019.42
8
1.12
This is the project’s value
 $ 134,197.87
The Problem
• Mary will retire in 12-years, has $100,000
saved, and will put $12,000 into an account
(at the end of every year) until she retires.
• She will take a $20,000 cruse in year-5.
• She expects to live 8 years after she retires,
and will leave $30,000 to “bury her.” What
will be her retirement income?
• The bank pays 3%
Key to Solution
– After Mary’s wake, there is no money left. The
future value of all her cash flows is the zero.
The present value of all cash flows must also be
zero
– We will discount all flows to the current year
– You may prefer to use Mary’s retirement or
death day as the reference
Solution Outline
•
•
•
•
•
0 = 100000 +
12000*PVIFA(3%, 12-years) 20000*PVIF(3%, 5-years) X*PVIFA(3%, 8-years)*PVIF(3%, 12-years)
- 30000*PVIF(3%, 20-years)
Solution by Equation


12000
12
0  100000 
1  1.03
0.03
X
8
-5
20000*( 1.03 ) 
1  1.03 * ( 1.03 )-12 
0.03
30000*( 1.03 )- 20


int
Solution using Excel
0 ret
-37694
year CF
Pert
bal
0
100000
1
12000
115000
2
12000
130450
3
12000
146364
4
12000
162754
5
12000
-20000
159637
6
12000
176426
7
12000
193719
8
12000
211531
9
12000
229876
10
12000
248773
11
12000
268236
12
12000
288283
13
-37694
259237
14
-37694
229320
15
-37694
198506
16
-37694
166767
17
-37694
134076
18
-37694
100404
19
-37694
65722
20
-37694
-30000
0
This was set to an arbitrary
value, and then solved for
This is set to zero
using the Tools
“Solve” function