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2. Content of the Chapter Interest Rate: important
terminologies Nominal and Effective Rate of Interest Effective
Annual Interest Rate Converting Nominal rate into Effective Rate
Calculating Effective Interest rates Equivalence Relations: PP and
CP Continuous Compounding Varying Intrest Rates
3. Previous Learning Our learning so for is based one interest
rate thats compounded annually Interest rates on loans, mortgages,
bonds & stocks are commonly based upon interest rates
compounded more frequently than annually When amount is compounded
more than once annually, distinction need to be made between
nominal and effective rate of interests
4. Interest Rate: important terminologies New time-based
definitions to understand and remember Interest period (t) period
of time over which interest is expressed. For example, 1% per
month. Compounding period (CP) The time unit over which interest is
charged or earned. For example,10% per year, here CP is a year.
Compounding frequency (m) Number of times compounding occurs within
the interest period t. For example, at i = 10% per year, compounded
monthly, interest would be compounded 12 times during the one year
interest period.
5. Examples of interest rate Statements Annual interest rate of
8% compounded monthly Here interest period (t) = 1 year compounding
period (CP) = 1 month compounding frequency (m) = 12 Annual
interest rate of 6% compounded weekly Here interest period (t) = 1
year compounding period (CP) = 1 week compounding frequency (m) =
52
6. Different Interest Statements Interest rates can be quoted
in many ways: Interest equals 6% per 6-months Interest is 12%
(annually) Interest is 1% per month Interest is 12.5% per year,
compounded monthly You must read the various ways to state interest
and to do calculations.
7. Nominal Interest Rate A nominal interest rate is denoted by
(r) It does not include any consideration of the compounding of
interest(frequency) It is given as: r = interest rate per period x
number of compounding periods Nominal rates are all of the form r %
per time period
8. Examples: Nominal Interest Rate 1.5% per month for 24 months
Same as: (1.5%)(24)= 36% per 24 months 1.5% per month for 12 months
Same as (1.5%)(12 months) = 18% per year 1.5% per month for 6
months Same as: (1.5%)(6 months) = 9% per 6 months or semi annual
period 1% per week for 1 year Same as: (1%)(52 weeks) = 52% per
year
9. Summary: Nominal Interest Rate A nominal rate do not
reference the frequency of compounding They all have the format r%
per time period Nominal rates can be misleadingHow? Annual interest
of $80 on a $1,000 investment is a nominal rate of 8% whether the
interest is paid in $20 quarterly instalments, in $40 semi-annual
instalments, or in an $80 annual payment? We need an alternative
way to quote interest rates. The true Effective Interest Rate is
then applied.
10. Effective Interest Rate (EIR) Effective interest rates (i)
take accounts of the effect of the compounding EIR are commonly
express on an annual basis (however any time maybe used) EIR rates
are mostly of the form: r % per time period, CP-ly (Compounding
Period) Nominal rates are all of the form r % per time period
11. Examples : Effective Interest Rates Quote: 12 percent
compounded monthly is translated as: 12% is the nominal rate
compounded monthly conveys the frequency of the compounding
throughout the year For this quote there are 12 compounding periods
within a year.
12. Examples: Effective Interest Rate Some times, Compounding
period is not mentioned in Interest statement For example, an
interest rate of 1.5% per month .. It means that interest is
compounded each month; i.e., Compounding Period is 1 month.
REMEMBER: If the Compounding Period is not mentioned it is
understood to be the same as the time period mentioned with the
interest rate.
13. Example Compounding Period Statement 1. 10% per year 2. 10
% per year compounded monthly 3. 4. 3% per quarter compounded daily
1.5% per month compounded monthly What it is ? 1. CP = not stated
but its a year 1. Effective rate per year 2. 2. Effective rate per
year CP = Stated, CP= month 3. CP= stated, CP= day 3. Effective
rate per quarter 4. CP=stated, CP=month 4. Effective rate per month
IMPORTANT: Nominal interest rates are essentially simple interest
rates. Therefore, they can never be used in interest formulas.
Effective rates must always be used hereafter in all interest
formulas
14. Some other names for NIR and EIRs Interest on Credit Cards,
loans and house mortgages sometime use term Annual Percentage
Rate(APR) for interest payment.its same as Nominal Interest Rate
For example: An APR of 15% is the same as a nominal 15% per year or
a nominal 1.25% on a monthly basis. Returns for investments,
certificates of deposits and saving accounts commonly use Annual
Percentage Yield (APY) which is same as Effective Interest Rate
Remember: the effective rate is always greater than or equal to the
nominal rate, and similarly APY > APR .. Why ?
15. Converting Nominal rate into Effective Rate per CP So for,
we always used t and CP values of 1 year so compounding frequencies
was always m=1, which make them all effective rate of interest (For
EIR , Interest period and Compounding period should be same) But
thats not always the case, we may have situation in which t has
different value than CP in that case we need to find effective rate
of interest per compounding period Effective rate per CP can be
determined from nominal rate by using following relation Effective
rate per CP = r % per time period t = r m compounding periods per t
m Where: CP is compounding period, t is the basic time unit of the
interest, m is the frequency of compounding and r is nominal
interest rate
16. Example: Calculating Effective Interest rates Three
different bank loan rates for electric generation equipment are
listed below. Determine the effective rate on the basis of the
compounding period for each rate (a) 9% per year, compounded
quarterly (b) 9% per year, compounded monthly (c) 4.5% per 6
months, compounded weekly
17. Example: Calculating Effective Interest rates per CP a. 9%
per year, compounded quarterly. b. 9% per year, compounded monthly.
c. 4.5% per 6 months, compounded weekly. The principle amount
change in each period for EIR here, in case of nominal it will
remain same in each case
18. More About Interest Rate Terminology Sometimes it is not
obvious whether a stated rate is a nominal or an effective rate.
Basically there are three ways to express interest rates
19. Effective Annual Interest Rates When we talk about Annual
we consider year as the interest period t , and the compounding
period CP can be any time unit less than 1 year Nominal rates are
converted into Effective Annual Interest Rates (EAIR) via the
equation: ia = (1 + i)m 1 where ia = effective annual interest rate
i = effective rate for one compounding period (r/m) m = number
times interest is compounded per year
20. Example For a nominal interest rate of 12% per year,
determine the nominal and effective rates per year for (a)
quarterly, and ia = (1 + i)m 1 (b) monthly compounding Solution:
where ia = effective annual interest rate i = effective rate for
one compounding period (r/m) m = number times interest is
compounded per year (a) Nominal r per year = 12% per year Nominal r
per quarter = 12/4 = 3.0% per quarter Effective i per year = (1 +
0.03)4 1 = 12.55% per year (b) Nominal r per month = 12/12 = 1.0%
per month Effective I per year = (1 + 0.01)12 1 = 12.68% per
year
21. r = 18% per year, compounded CP-ly
22. Effective Interest Rates for any Time Period The following
relation of Effective Annual Interest Rates ia = (1+i)m 1 can be
generalize for determining the effective interest rate for any time
period (shorter or longer than 1 year). i = (1 + r / m)m 1 where i
= effective interest rate for any time period r = nominal rate for
same time period as i m = no. times interest is compd in period
specified for i
23. Example: Effective Interest Rates For an interest rate of
1.2% per month, determine the nominal and effective rates (a) per
quarter, and i = (1 + r / m)m 1 (b) per year Solution: (a) Nominal
rate (r) per quarter = (1.2)(3) = 3.6% per quarter Effective rate
(i) per quarter = (1 + 0.036/3)3 1 = 3.64% per quarter (b) Nominal
rate (r) per year = (1.2)(12) = 14.4% per year Effective rate (i)
per year = (1 + 0.144 / 12)12 1 = 15.39% per year
24. Economic Equivalence: From Chapter 1 Different sums of
money at different times may be equal in economic value at a given
rate Rate of return = 10% per year $110 Year 0 1 $100 now 1 $100
now is economically equivalent to $110 one year from now, if the
$100 is invested at a rate of 10% per year Economic Equivalence:
Combination of interest rate (rate of return) and time value of
money to determine different amounts of money at different points
in time that are economically equivalent .. Compounding/Discounting
(F/P, P/F, F/A, P/G etc.)
25. Equivalence Relations: Payment Period(PP) & Compounding
Period(CP) The payment period (PP) is the length of time between
cash flows (inflows or outflows) r = nominal 8% per year,
compounded semi-annually CP 6 months 0 1 PP 1 month 2 3 CP 6 months
4 5 6 7 8 9 10 11 12 Months
26. Equivalence Relations: Payment Period(PP) and Compounding
Period It is common that the lengths of the payment period and the
compounding period (CP) do not coincide To do correct calculation
Interest rate must coincide with compounding period It is important
to determine if PP = CP, PP >CP, or PP CP PP < CP
27. Case I: When PP>CP for Single Amount for P/F or F/P Step
1: Identify the number of compounding periods (M) per year Step 2:
Compute the effective interest rate per payment period (i) i = r/M
Step 3: Determine the total number of payment periods (n) Step 4:
Use the SPPWF or SPCAF with i and N above
28. Example Determine the future value of $100 after 2 years at
credit card stated interest rate of 15% per year, compounded
monthly. Solution: P = $100, r = 15%, m = 12 EIR /month = 15/12 =
1.25% n = 2 years or 24 months Alternative Method i = (1 + r/m)m 1
= (1+0.15/12)12 1 = 16.076 F = P(F/P, i, n) F = P(F/P, i, n) F =
P(F/P, 0.0125, 24) F = P(F/P, 0.16076, 2) F = 100(F/P, 0.0125, 24)
F = 100(1.3474) F = 100(1.3474) F = $134.74 F = 100(1.3456) F =
$134.56 The results are slightly different because of the rounding
off 16.076% to 16.0%
29. Factor Values for Untabulated i or n There are 3 ways to
find factor values for untabulated i or n values 1. Use formula 2.
Use spreadsheet function 3. Linearly interpolate in interest tables
Formula or spreadsheet function is fast & accurate
Interpolation is only approximate
30. Factor Values for Untabulated i or n Factor value axis f2 f
Linear assumption unknown f1 X1 Required X X2 i or n axis Absolute
Error = 2.2215 2.2197 = 0.0018
31. Case II: When PP >CP for Series for P/A or F/A For
series cash flows, first step is to determine relationship between
PP and CP Determine if PP CP, or if PP < CP When PP CP, the only
procedure (2 steps) that can be used is as follows: First, find
effective i per PP Example: if PP is in quarters, must find
effective i/quarter Second, determine n, the number of A values
involved Example: quarterly payments for 6 years yields n = 46 = 24
You can use than the standard P = A(P/A, i , n) or F = A(F/A, i, n)
etc.
32. Example: PP >CP for Series for P/A or F/A For the past 7
years, Excelon Energy has paid $500 every 6 months for a software
maintenance contract. What is the equivalent total amount after the
last payment, if these funds are taken from a pool that has been
returning 8% per year, compounded quarterly? Solution: Compounding
Period (CP) = Quarter & PP = 6 months r = 8 % per year or 4%
per 6 months & m=2/ quarter PP > CP Effective rate (i) per 6
months = (1+r/m)m -1 i= (1+0.04/2)2 1 => 4.04% Since, total time
is 7 years and PP is 6 months we have total 7x2=14 payments F =
A(F/A, i, n) F = 500(F/A, 0.0404, 14) F = 500(18.3422) =>
$9171.09
33. Case III: Economic Equivalence when PP< CP If a person
deposits money each month into a savings account where interest is
compounded quarterly, do all the monthly deposits earn interest
before the next quarterly compounding time? If a person's credit
card payment is due with interest on the 15th of the month, and if
the full payment is made on the 1st, does the financial institution
reduce the interest owed, based on early payment? Anyone ? CP: 3
months = 1 quarter 0 1 PP 1 month 2 3 4 5 6 7 8 9 10 11 12
Months
34. Case III: Economic Equivalence when PP< CP Two policies:
1. Inter-period cash flows earn no interest (most common) 2.
inter-period cash flows earn compound interest positive cash flows
are moved to beginning of the interest period (no interest earned)
in which they occur and negative cash flows are moved to the end of
the interest period (no interest paid) cash flows are not moved and
equivalent P, F, and A values are determined using the effective
interest rate per payment period
35. Example 4.11: Example: Clean Air Now (CAN) Company Last
year AllStar Venture Capital agreed to invest funds in Clean Air
Now (CAN), a start-up company in Las Vegas that is an outgrowth of
research conducted in mechanical engineering at the University of
NevadaLas Vegas. The product is a new filtration system used in the
process of carbon capture and sequestration (CCS) for coal-fired
power plants. The venture fund manager generated the cash flow
diagram in Figure in $1000 units from AllStars perspective.
Included are payments (outflows) to CAN made over the first year
and receipts (inflows) from CAN to AllStar. The receipts were
unexpected this first year; however, the product has great promise,
and advance orders have come from eastern U.S. plants anxious to
become zero-emission coal-fueled plants. The interest rate is 12%
per year, compounded quarterly, and AllStar uses the no-inter
period-interest policy. How much is AllStar in the red at the end
of the year?
36. Example 4.11: Example: Clean Air Now (CAN) Company The
venture fund manager generated the cash flow diagram in $1000 units
from AllStars perspective as given below. Included are payments
(outflows) to CAN made over the first year and receipts (inflows)
from CAN to AllStar. The receipts were unexpected this first year;
however, the product has great promise, and advance orders have
come from eastern U.S. plants anxious to become zero-emission
coal-fueled plants. The interest rate is 12% per year, compounded
quarterly, and AllStar uses the no-inter period-interest policy.
How much is AllStar in the red at the end of the year?
37. Example: Clean Air Now (CAN) Company Given cash flows
Positive Cash flows (inflows) at the start of CP period Negative
Cash flows (outflows) at the end of CP period
38. Example: Clean Air Now (CAN) Company Solution: Effective
rate per quarter = 12/4 = 3% Now F = 1000[-150(F/P, 3%, 4)-200(F/P,
3%, 3) +(180-175 )(F/P, 3%, 2)+ 165(F/P, 3%, 1)-50] F = $ (-262111)
Investment after one year
39. Continuous Compounding If compounding is allowed to occur
more and more frequently, the compounding period becomes shorter
and shorter, and m (the number of compounding periods per payments)
increases Continuous compounding is present when the duration of
CP, the compounding period, becomes infinitely small and m , the
number of times interest is compounded per period, becomes
infinite. Businesses with large numbers of cash flows each day
consider the interest to be continuously compounded for all
transactions.
40. Continuous Compounding We have effective interest rate
generalize formula as follows: i = (1 + r/m)m 1 Taking m limits
tends to infinity and simplifying the equation we get the following
expression for continuous effective interest rate i = er 1
41. Example: Continuous Compounding Example: If a person
deposits $500 into an account every 3 months at an interest rate of
6% per year, compounded continuously, how much will be in the
account at the end of 5 years? Solution: Payment Period: PP = 3
months Nominal rate per three months: r = 6% /4 = 1.50% Continuous
Effective rate per 3 months: i = e0.015 1 = 1.51% F =
500(F/A,1.51%,20) = $11,573 Practice: Example 4.12 & 4.13
42. Varying Interest Rates Interest rate does not remain
constant full life time of a project In order to do incorporate
varying interest rates in our calculations, normally, engineering
studies do consider average values that do care of these
variations. But sometimes variation can be large and having
significant effects on Present or future values calculated via
using average values Mathematically, varying interest rates can be
accommodated in engineering studies
43. Varying Interest Rates When interest rates vary over time,
use the interest rates associated with their respective time
periods to find P The general formula for varying interest rate is
given as: P = F1(P/F, i1, 1) + F2(P/F, i1, 1)(P/F, i2, 1) + .. + Fn
(P/F, i1, 1)(P/F, i2, 1) (P/F, in, 1) For single F or P only the
last term of the equation can be used. For uniform series replace F
with A
45. Problem 4.57: Varying Interest Rates Calculate (a) the
Present value (b) the uniform Annual worth A of the following Cash
flow series P=? i=10% 1 2 3 4 5 6 7 i=14% 8 Year 0 $100 $100 $100
$100 $100 $160 $160 $160 P = F1(P/F, i1, 1) + F2(P/F, i1, 1)(P/F,
i2, 1) + .. + Fn (P/F, i1, 1)(P/F, i2, 1) (P/F, in, 1) P = 100(P/A,
10%, 5) + 160 (P/A, 14%, 3) (P/F, 10%, 5) = 100(3.7908) +
160(2.3216)(0.6209) = $609.72
46. Problem 4.57: Varying Interest Rates (b) the uniform Annual
worth A of the following Cash flow series P = 609.72 i=10% 1 2 3 4
5 6 7 i=14% 8 Year 0 A=? P = 100(P/A, 10%, 5) + 150 (P/A, 14%, 3)
(P/F, 10%, 5) = 100(3.7908) + 160(2.3216)(0.6209) = $609.72 609.72
= A(3.7908) + A(2.3216)(0.6209) A = 609.72 / 5.2323 A = $ 116.53
per year