Transcript Interest and Equivalence - The American University in Cairo

Interest and Equivalence
L. K. Gaafar
Interest and Equivalence
Example: You borrowed $5,000 from a bank and you
have to pay it back in 5 years. There are many ways the
debt can be repaid. (i = 0.08)
Plan
1:
Plan 2:
years.
Plan 3:
Plan 4:
At end of each year pay $1,000 principal plus interest due.
Pay interest due at end of each year and principal at end of five
Pay in five end-of-year payments ($1,252).
Pay principal and interest in one payment at end of five years.
All these plans are equivalent in the sense that the
sum of all outgoing cash flows at time 0 is $5,000.
Present Value
•
Plan 1: At end of each year pay $1,000 principal plus interest due
(a)
(b)
(c)
(d)
(e)
(f)
Yr.
1
2
3
4
5
Amnt. Owed
Begin. of Yr
$5,000
$4,000
$3,000
$2,000
$1,000
Int. Owed
0.08 * b
$400
$320
$240
$160
$80
$1,200
Total Owed
b+c
$5,400
$4,320
$3,240
$2,160
$1,080
Princip.
Payment
$1,000
$1,000
$1,000
$1,000
$1,000
$5,000
Total Payment
$1,400
$1,320
$1,240
$1,160
$1,080
$6,200
Present Value
•
Plan 2: Pay interest due at end of each year and principal at end of five years.
(a)
Yr.
1
2
3
4
5
(b)
(c)
Amnt. Owed
Begin. of Yr
Int. Owed
0.08 * b
$5,000
$5,000
$5,000
$5,000
$5,000
$400
$400
$400
$400
$400
$2,000
(d)
Total Owed
b+c
$5,400
$5,400
$ 5,400
$ 5,400
$ 5,400
(e)
(f)
Princip.
Payment
$0
$0
$0
$0
$5,000
$5,000
Total Payment
$400
$400
$400
$400
$5,400
$7,000
Present Value
•
Plan 3: Pay in five end-of-year payments
(a)
(b)
(c)
(d)
Yr.
Amnt. Owed
Begin. of Yr
Int. Owed
0.08 * b
Total Owed
b+c
1
2
3
4
5
$5,000
$4,148
$3,227
$2,233
$1,159
$400
$332
$258
$179
$93
$1,261
$5,400
$4,480
$3,485
$2,412
$1,252
(e)
(f)
Princip.
Payment
$852
$921
$994
$1,074
$1,159
$5,000
Total Payment
$1,252
$1,252
$1,252
$1,252
$1,252
$6,261
Present Value
•
Plan 4: Pay principal and interest in one payment at end of five years.
(a)
(b)
(c)
(d)
(e)
(f)
Yr.
1
2
3
4
5
Amnt. Owed
Begin. of Yr
$5,000
$5,400
$5,832
$6,299
$6,802
Int. Owed
0.08 * b
Total Owed
b+c
Princip.
Payment
$400
$432
$467
$504
$544
$2,347
$5,400
$5,832
$6,299
$6,802
$7,347
$0
$0
$0
$0
$0
$0
$0
$0
$5,000
$5,000
Compound Interest:
interest is charged on
the unpaid interest.
Total Payment
$7,347
$7,347
Present Value
Plan 1: At end of each year pay $1,000 principal plus interest due
(col. f) (col. b)
$ Owed
Year EOY Pay. Owed
1
$1,400 $5,000
$6,000
2
$1,320 $4,000
$5,000
3
$1,240 $3,000
$4,000
4
$1,160 $2,000
$3,000
5
$1,080 $1,000
$6,200 $15,000
$2,000
$ Owed
$1,000
$0
1
2
3
4
Time (Years)
5
Present Value
Plan 2: Pay interest due at end of each year and principal at end of five years.
Year
1
2
3
4
5
EOY Pay. Owed
$400 $5,000
$400 $5,000
$400 $5,000
$400 $5,000
$5,400 $5,000
$7,000 $25,000
$ Owed
$6,000
$5,000
$4,000
$3,000
$ Owed
$2,000
$1,000
$0
1
2
3
4
Time (Years)
5
Present Value
Plan 3: Pay in five end-of-year payments
Year
1
2
3
4
5
EOY Pay. Owed
$1,252 $5,000
$1,252 $4,148
$1,252 $3,227
$1,252 $2,233
$1,252 $1,159
$6,261 $15,767
$ Owed
$6,000
$5,000
$4,000
$3,000
$ Owed
$2,000
$1,000
$0
1
2
3
4
Time (Years)
5
Present Value
Plan 4: Pay principal and interest in one payment at end of five years.
Year
1
2
3
4
5
EOY Pay. Owed
$0 $5,000
$0 $5,400
$0 $5,832
$0 $6,299
$7,347 $6,802
$7,347 $29,333
$ Owed
$8,000
$7,000
$6,000
$5,000
$4,000
$3,000
$2,000
$1,000
$0
$ Owed
1
2
3
4
Time (Years)
5
Present Value
Summary of Payment Plans
(a)
(b)
(c)
Plan Total Paid Tot. Int. Area under
curve
1
$6,200 $1,200
$15,000
2
$7,000 $2,000
$25,000
3
$6,261 $1,261
$15,767
4
$7,347 $2,347
$29,333
(d)
Ratio:
(b)/(c)
0.08
0.08
0.08
0.08
Key Question. What common property do all four plans have?
Conclusion: interest rate = (total int. paid)/(area under curve)
or
total int. paid = (interest rate) * (area under curve)
Since the areas under the curve vary for the 4 plans, but the interest rate does
not, the total interest paid also varies.
Interest and Equivalence
Example: You borrowed $8,000 from a bank and you
have to pay it back in 4 years. There are many ways the
debt can be repaid. (i = 0.10)
Plan
1:
Plan 2:
years.
Plan 3:
Plan 4:
At end of each year pay $2,000 principal plus interest due.
Pay interest due at end of each year and principal at end of four
Pay in four end-of-year payments ($2,524).
Pay principal and interest in one payment at end of four years.
Example
With i = 10%, n = 4, find an equivalent uniform payment A for:
24000
18000
12000
6000
0
1
2
3
4
Possible Solution
This is a problem with decreasing costs instead of increasing costs. Note we can
write it as the DIFFERENCE of the following two diagrams, the second of which is
in the standard form we need, the first of which is a series of uniform payments.
1800
1200
-
0 600
A = 24000
We conclude now that:
A = 24000 – 6000 (A/G,10%,4) = 24000 –6000(1.381) = 15,714.
Possible Solution
24000
18000
12000
6000
0
0
1
2
3
4
24000
24000
24000
24000
1
2
3
4
6000
12000
18000
Nominal/Effective Rates
Suppose that a $1,000 lump-sum amount is invested for 10
years at a nominal interest rate of 6% compounded quarterly.
How much is it worth at the end of the tenth year?
Nominal/Effective Rates
Suppose that one has a bank loan for $10,000, which is to be
repaid at equal end-of-month installments for five years with
a nominal interest rate of 12% compounded monthly. What
is the amount of each payment?
Nominal/Effective Rates
What is the present worth of 8 end-of-quarter payment of
$1,000 each, if the nominal interest rate is 15% compounded
monthly?
Nominal/Effective Rates
Suppose that there exists a series of 10 end-of-year receipts of
$1,000 each. What is their equivalent worth as of the end of the
tenth year if the nominal interest rate is 12% compounded
quarterly? ($18,021)
Comparing Alternatives
The table below shows the cash flows associated with 6
mutually exclusive investment opportunities. If the MARR is
10%, identify the feasible alternatives and choose the best one.
All alternatives have a useful life of 10 years.
Alternative
A
B
C
D
E
F
Capital Investment (P)
-$900
-$1,500 -$2,500 -$4,000 -$5,000 -$7,000
Annual Revenue (A)
$150
$276
$400
$925
$1,125
$1,425
The key to solving this problem is to realize that each alternative provides us with an
opportunity to invest P for an annual return of A. Acceptable alternatives are those that
allow us to realize 10% or more on our investment (P). In other words, we are trying to
judge whether it is better to invest P at the rate of i (10%), or invest it for the return of A
every period (accept the alternative).
Present Worth (PW) Comparisons
Present worth evaluations assess the value of all
cash flows at time 0. In the given problem, we
are trying to evaluate whether an annual return
(A) realizes a specific return (profit, i) on an
initial investment (P).
The PW of the given cash flows is:
PW = -P + A (P/A, i, 10).
A
PW = ?
0
1
2
3
9
10
i = MARR = 10%
P
The best alternative is
the one with the highest
positive PW
A positive PW [A (P/A, i, 10) > P] indicates that the investment opportunity provides an annual
return that is higher than investing the amount P at the MARR (i). In other words, the given
return (A) realizes a higher profit than the targeted (i), and accordingly a favorable investment
opportunity for P. Another way to look at a positive PW is that the targeted return rate (i) may
be realized with a smaller investment (P-PW).
A PW of 0 indicates that the investment opportunity realizes exactly the targeted return rate (i),
and hence, a borderline investment opportunity.
A negative PW [A (P/A, i, 10) < P] indicates that the rate of return is less than the targeted (i),
and accordingly an unacceptable investment opportunity. In other words, it is better to invest
the amount P at the rate i than to invest it for an annual return of A.
For both cases when PW  0, the IRR method may be used to calculate the exact rate of
return.
Future Worth (FW) Comparisons
FW = ?
A
Future worth evaluations assess the value of all
cash flows at time n. In the given problem, we
are trying to evaluate whether an annual return
(A) realizes a specific return (profit, i) on an
initial investment (P).
The FW of the given cash flows is:
FW = -P(F/P, i, 10) + A (F/A, i, 10).
0
1
2
3
9
10
i = MARR = 10%
P
The best alternative is
the one with the highest
positive FW
A positive FW [A (F/A, i, 10) > P (F/P, i, 10)] indicates that the future value of the annuity (A)
is higher that that of the initial investment (P), and hence, a favorable investment opportunity.
In other words, getting A every period is better than investing P at the MARR (i). It also
means that the given return (A) realizes a higher profit than the targeted (i).
A FW of 0 indicates that the investment opportunity realizes exactly the targeted return rate (i),
and hence, a borderline investment opportunity.
A negative FW [A (F/A, i, 10) < P (F/P, i, 10)] indicates that the rate of return is less than the
targeted (i), and accordingly an unacceptable investment opportunity. In other words, it is
better to invest the amount P at the rate i than to invest it for an annual return of A.
For both cases when FW  0, the IRR method may be used to calculate the exact rate of
return.
Annual Worth (AW) Comparisons
A
AW = ?
Annual worth evaluations assess the annual
resultant of all cash flows. In the given problem,
1 2 3
9
0
we are checking that the offered annuity is at
least equal to the annuity we could get by
-P(A/P, i, 10)
investing P at the MARR (i). Accordingly, the
i = MARR = 10%
AW of the given cash flows is defined as:
P
The best alternative is
AW = -P(A/P, i, 10) + A.
the one with the highest
positive AW
10
A positive AW [A > P (A/P, i, 10)] indicates that the offered annuity is greater than what we
could get by investing P at the MARR (i), and hence, a favorable investment opportunity. In
other words, getting A every period is better than investing P at the MARR (i). It also means
that the given return (A) realizes a higher profit than the targeted (i).
An AW of 0 indicates that the investment opportunity realizes exactly the targeted return rate
(i), and hence, a borderline investment opportunity.
A negative AW [A < P (A/P, i, 10)] indicates that the rate of return is less than the targeted (i),
and accordingly an unacceptable investment opportunity. In other words, it is better to invest
the amount P at the rate i than to invest it for an annual return of A.
For both cases when AW  0, the IRR method may be used to calculate the exact rate of
Internal Rate of Return (IRR) Comparisons
A
IRR evaluations assess the rate of return at which
the net cash flow is equal to zero. It is the rate at
which the net worth (Present, Future, or Annual)
of all cash flows equals 0. A favorable
opportunity exists when IRR  MARR.
0
1
2
3
9
10
i (IRR) = ?
P
•Acceptable alternatives are those
for which IRR  MRR.
•Ranking Alternatives can only be
based on relative comparisons.
In our case, IRR is the actual interest at which a one time deposit P results in an annual return
of A.
Notice that:
•PW, FW, AW > 0 correspond to IRR > MARR,
•PW, FW, AW < 0 correspond to IRR < MARR,
•PW, FW, AW = 0 correspond to IRR = MARR.
In other words, a negative PW, FW, or AW does not necessarily mean a loss, but merely a
return rate that is smaller than the MARR.
Comparison Results
A
The table below shows the results of the various
calculations. Alternative C may be rejected
based on its negative net worth. Notice,
however, that this doesn’t mean it is unprofitable.
It only means that its profit (internal rate of
return) is less than the MARR.
0
1
2
3
9
P
Alternative
A
B
C
D
E
F
PW
21.69
195.90 -42.17 1683.72 1912.64 1756.01
FW
56.25
508.12 -109.39 4367.15 4960.89 4554.63
AW
3.53
31.88
-6.86
274.02
311.27
285.78
IRR
10.558% 12.961% 9.606% 19.097% 18.314% 15.566%
Among all remaining alternatives, E is the best followed by F, D, B, A, respectively.
PW, FW, AW results all point to the same ranking. IRR results, however, lead to a
different ranking. In the table above, while D has a higher IRR, it is applied to a smaller
investment (4,000), while the slightly lower rate of E gives a higher overall return when
applied to the higher investment (5,000), and hence, E is better than D. Accordingly,
IRR can only be used in relative comparisons (see next slide).
10
Relative Comparison Results (IRR)
A
In relative comparisons, alternatives are ranked
in an ascending order of their initial investments.
The IRR of the difference cash flows between the
first two alternatives is evaluated. If the resulting
IRR is greater than the MARR, the second
alternative is better and is used as a base for
further comparisons.
0
1
2
3
P
The relative IRR results in the table below agree with those of the net worth methods.
Alternative
B-A
D-B
E-D
F-E
F-D
IRR
16.401% 22.567% 15.098% 8.144% 10.558%
9
10
Payback (Payout) Period Method
R2
R1
Simple Payback (q):
Time value of money not considered.
Salvage value not considered.
Used as a rough estimate and a starting
point for discounted.
q
 ( Rk  Ek )  P  0, reducestoq 
k 1
0
E1
Rq-1
Rq’
q’-1 q’
2
1
S
E2
Eq-1
Eq
P
P
when only uniform annuitiesare involved
A
Discounted Payback (q’):
q'
 (R
k 1
k
 Ek )(P / F , i, k )  S ( P / F , i,q ' )  P  0
Example:
A piece of new equipment has been proposed by engineers to increase the productivity
of a certain manual welding operation. The investment cost is $25,000, and the
equipment will have a salvage value of $5,00 at the end of its life. Increased
productivity attributed to equipment will amount to $8,000 per year after taxes. Find
the payback period for this investment assuming MARR = 20%.
Payback Period- Example
A piece of new equipment has been proposed by engineers to increase the productivity
of a certain manual welding operation. The investment cost is $25,000, and the
equipment will have a salvage value of $5,00 at the end of its life. Increased
productivity attributed to equipment will amount to $8,000 per year after taxes. Find
the payback period for this investment assuming MARR = 20%.
S
A
Simple Payback (q):
P 25,000
q 
 3.125 4
A 8,000
Discounted Payback (q’):
4
Try 4
 (R
k 1
k
5
Try 5
 (R
k 1
k
0
1
2
q’-1 q’
P
 Ek )(P / F , i, k )  S ( P / F , i,4)  P   1878.86
 Ek )(P / F , i, k )  S ( P / F , i,5)  P  934.28
Bonds
Bonds are money guarantees that pay a fixed amount per period and a final amount at the end
of their life (maturity). This final amount is called the redemption value. Bonds are bought
for a value called the face value and usually pay a fixed amount every period that is a
percentage of the face value. A bond is said to be redeemed at par value when the redemption
value is the same as the face value.
Bonds pause an interesting problem when they are traded before their maturity as an
investment to earn a interest other than the one they provide.
Bonds
Find the current price of a 10-year bond paying 6% per year (payable semi-annually) that is
redeemed at par value, if bought by a purchaser to yield 10% per year. The face value of
the bond is $1,000. [761.16]
Bonds
A certain 8-year bond has a face value of $10,000. The bond pays a nominal interest rate of
8% every three months. A prospective buyer would like to earn a nominal interest of
10% per year. How much should this buyer pay for the bond? [$8,907.55]
Bonds
A bond with a face vale of $5,000 pays interest of 8% per year. This bond will be redeemed
at par value at the end of its 20-year life, an the first interest payment is due one year
from now.
a.
How much should be paid now for this bond in order to yield 10% per year? [$4,148.44]
b.
If the bond is purchased for $4,600, what annual yield would the buyer receive? [8.9%]
Benefit/Cost Ratio
For Evaluating Public Projects
B/C
PW (benefits)
PW ( B)

PW (total costs) I  PW ( S )  PW (O & M )
B/C
AW (benefits)
AW ( B)

AW (totalcosts) CR  AW (O & M )
Conventional B/C Ratio
B/C
PW ( B)  PW (O & M )
I  PW ( S )
Modified B/C Ratio
AW ( B)  AW (O & M )
B/C 
CR
Benefit/Cost Ratio
For Evaluating Public Projects
A city is considering extending the runways of its Municipal Airport so that
commercial jets can use the facility. The land can be purchased for $350,000.
Construction costs are projected to be $600,000, and the additional annual
maintenance costs for the runway are $22,500. The additional annual
maintenance costs of the new terminal are $75,000. The increase in flights will
require two new air traffic controllers at the cost of $100,000 per year. Annual
benefits of the runway extension are estimated as follows:
$325,000 from rental receipts.
$65,000 airport tax charged to passengers.
$50,000 convenience benefits.
$50,000 from tourism.
Use B/C ratio to justify the project, assuming a study period of 20 years and
an interest rate of 10%. [1.59, 2.62]