1. Equivalence and Compound interest
Anastasia Lidya M.

2. Economic Equivalence What do we mean by “economic equivalence?”
Why do we need to establish an economic equivalence?
How do we establish an economic equivalence?

3. Economic Equivalence
Economic equivalence exists between cash flows that have the same economic effect and could therefore be traded for one another.
Even though the amounts and timing of the cash flows may differ, the appropriate interest rate makes them equal.

4. Single-Payment Factors (F/P and P/F)
Objective:
Derive factors to determine the present or future worth of a cash flow
Cash Flow Diagram – basic format Fn
i% / period
n n P0
P0 = Fn1/(1+i)n →(P/F,i%,n) factor: Excel: =PV(i%,n,,F)
Fn = P0(1+i)n →(F/P,i%,n) factor: Excel: =FV(i%,n,,P)

5. Single-Payment Factors (F/P and P/F)
If you deposit P dollars today for N periods at i, you will have F dollars at the end of period N.
F dollars at the end of period N is equal to a single sum P dollars now, if your earning power is measured in terms of interest rate i.

6. Equivalence Between Two Cash Flows
Step 1: Determine the base period, say, year 5.
Step 2: Identify the interest rate to use.
Step 3: Calculate equivalence value.
$3,000
$2,042 5

7. Example 2.1: Equivalence
Various dollar amounts that will be economically equivalent to $3,000 in 5 years, given an interest rate of 8%

8. Required: To find P given A
Uniform-Series: Present Worth Factor (P/A) and Capital Recovery Factor(A/P)
Cash flow profile for P/A factor
n n n
$A per interest period
i% per interest period
Find P
Required: To find P given A
Cash flows are equal, uninterrupted and flow at the end of each interest period

9. (P/A) Factor Derivation
Setup the following:
Multiply by to obtain a second equation…
Subtract (1) from (2) to yield…
(1)
(2)
(3)

10. Example 2.2. Uniform Series: Find A, Given P, i, and N
Given: P = $250,000, N = 6 years, and i = 8% per year
Find: A
Formula to use:
Capital Recovery Factor

11. Example 2.3. – Deferred Loan Repayment
Given: P = $250,000, N = 6 years, and i = 8% per year, but the first payment occurs at the end of year 2
Find: A
Step 1: Find the equivalent amount of borrowing at the end of year 1:
Step 2: Use the capital recovery factor to find the size of annual installment:

12. Example 2.4. Uniform Series: Find P, Given A, i, and N
Present Worth Factor
Given: A = $10,576,923, N = 26 years, and i = 5% per year
Find: P
Formula to use:

13. Equivalent Future Worth
Equal Payment Series F
Equivalent Future Worth 1 2 N A A A P N 1 2 N

14. Equal-Payment Series Compound Amount Factor
Formula

15. An Alternate Way of Calculating the Equivalent Future Worth, F
A(1+i)N-2 A A A
A(1+i)N-1 N 1 2 1 2 N

16. Example 2.5. Uniform Series: Find F, Given i, A, and N
Given: A = $3,000, N = 10 years, and i = 7% per year
Find: F

17. Example 2.6. Handling Time Shifts: Find F, Given i, A, and N
Given: A = $3,000, N = 10 years, and i = 7% per year
Find: F
Each payment has been shifted to one year earlier, thus each payment would be compounded for one extra year

18. ANSI Standard Notation for Interest Factors
Standard notation has been adopted to represent the various interest factors
Consists of two cash flow symbols, the interest rate, and the number of time periods
General form: (X/Y,i%,n)
X represents what is unknown
Y represents what is known
i and n represent input parameters; can be known or unknown depending upon the problem

19. Notation - continued Example: (F/P,6%,20) is read as:
To find F, given P when the interest rate is 6% and the number of time periods equals 20.
In problem formulation, the standard notation is often used in place of the closed-form equivalent relations (factor)
Tables at the back of the text provide tabulations of common values for i% and n

20. Interpolation in Interest Tables
When using tabulated interest tables one might be forced to approximate a factor that is not tabulated

21. Arithmetic Gradient Factors (P/G) and (A/G)
A1+(n-1)G
Cash flow profile
A1+(n-2)G
Find P, given gradient cash flow G
A1+2G
A1+G
Base amount = A1
n n
CFn = A1 ± (n-1)G

22. Gradients have two components:
Gradient Example
$100
$200
$300
$400
$500
$600
$700
Gradients have two components:
1. The base amount and the gradient
2. The base amount (above) = $100/time period

23. © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved
Gradient Components
(n-1)G
(n-2)G
(n-3)G
Find P of gradient series 2G 1G 0G
Base amount = A / period
……..
n n n
Present worth point is 1 period to the left of the 0G cash flow
For present worth of the base amount, use the P/A factor (already known)
For present worth of the gradient series, use the P/G factor (to be derived)
Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6th Edition, 2005
© 2005 by McGraw-Hill, New York, N.Y All Rights Reserved

24. Example 2.7. – Linear Gradient: Find A, Given A1, G, i, and N
Given: A1 = $1,000, G = $300, N = 6 years, and i = 10% per year
Find: A

25. Example 2.8.Declining Linear Gradient Series
Given: A1 = $1,200, G = -$200, N = 5 years, and i = 10% per year
Find: F
Strategy: Since we have no interest formula to compute the future worth of a linear gradient series directly, we first find the equivalent present worth of the gradient series and then convert this P to its equivalent F.

26. Gradient Decomposition
As we know, arithmetic gradients are comprised of two components
Gradient component
2. Base amount
When working with a cash flow containing a gradient, the (P/G) factor is only for the gradient component
Apply the (P/A) factor to work on the base amount component
P = PW(gradient) + PW(base amount)

27. Equivalent A of gradient series
Use of the (A/G) Factor
A = G(A/G,i,n)
(n-1)G
(n-2)G
Find A, given gradient cash flow G
A A A A A 2G
Equivalent A of gradient series G
n n
CFn = (n-1)G

28. Geometric Gradient Series Factor
Cash flow series that starts with a base amount A1
Increases or decreases from period to period by a constant percentage amount
This uniform rate of change defines…
A GEOMETRIC GRADIENT
Notation:
g = the constant rate of change, in decimal form, by which future amounts increase or decrease from one time period to the next

29. Typical Geometric Gradient
A1(1+g)
A1(1+g)2
n n n
A1(1+g)n-1
Given A1, i%, and g%
Required: Find a factor (P/A,g%,i%,n) that will convert future cash flows to a single present worth value at time t = 0

30. Two Forms to Consider… To use the (P/A,g%,i%,n) factor Case: g = i
A1 is the starting cash flow
There is NO base amount associated with a geometric gradient
The remaining cash flows are generated from the A1 starting value
No tables available to tabulate this factor…too many combinations of i% and g% to support tables

31. Example 2.9. Retirement Plan – Saving $1 Million
Given: F = $1,000,000, g = 6%, i = 8%, and N = 20
Find: A1
Solution:

32. Nominal Versus Effective Interest Rates
Nominal Interest Rate:
Interest rate quoted based on an annual period
Effective Interest Rate:
Actual interest earned or paid in a year or some other time period

33. Nominal Versus Effective Interest Rates
Effective interest rate per year, ia, is the annual interest rate taking into account the effect of any compounding during the year.
In Example we saw that $100 left in the savings account for one year increased to $105.06, so the interest paid was $5.06. The effective interest rate per year, ia, is $5.06/$ = = 5.06%.

34. Financial Jargon 18% Compounded Monthly Interest Nominal period
interest rate
Annual
percentage
rate (APR)
Interest
period
18% Compounded Monthly

35. 18% Compounded Monthly What It Really Means? In words,
Interest rate per month (i) = 18%/12 = 1.5%
Number of interest periods per year (N) = 12
In words,
Bank will charge 1.5% interest each month on your unpaid balance, if you borrowed money.
You will earn 1.5% interest each month on your remaining balance, if you deposited money.
Question: Suppose that you invest $1 for 1 year at 18% compounded monthly. How much interest would you earn?

36. Effective Annual Interest Rate (Yield)
Formula:
r = nominal interest rate per year
ia = effective annual interest rate
M = number of interest periods per year
Example:
18% compounded monthly
What It really Means
1.5% per month for 12 months or
19.56% compounded once per year

37. Practice Problem Solution:
Suppose your savings account pays 9% interest compounded quarterly.
Interest rate per quarter
Annual effective interest rate (ia)
If you deposit $10,000 for one year, how much would you have?

38. Compounding Semi-annually Compounding Quarterly
Nominal and Effective Interest Rates with Different Compounding Periods
Effective Rates
Nominal Rate
Compounding Annually
Compounding Semi-annually
Compounding Quarterly
Compounding Monthly
Compounding Daily 4%
4.00%
4.04%
4.06%
4.07%
4.08% 5
5.00
5.06
5.09
5.12
5.13 6
6.00
6.09
6.14
6.17
6.18 7
7.00
7.12
7.19
7.23
7.25 8
8.00
8.16
8.24
8.30
8.33 9
9.00
9.20
9.31
9.38
9.42 10
10.00
10.25
10.38
10.47
10.52 11
11.00
11.30
11.46
11.57
11.62 12
12.00
12.36
12.55
12.68
12.74

39. Why Do We Need an Effective Interest Rate per Payment Period?
Whenever payment and compounding periods differ from
each other, one or the other must be transformed so that
both conform to the same unit of time.
Payment period
Interest period
Payment period
Interest period

40. Effective Interest Rate per Payment Period (i)
Functional Relationships among r, i, and ia, where interest is calculated based on 9% compounded monthly and payments occur quarterly
Formula:
C = number of interest periods per payment period
K = number of payment periods per year
CK = total number of interest periods per year, or M
r/K = nominal interest rate per payment period

41. Effective Interest Rate per Payment Period with Continuous Compounding
Example: 12% compounded continuously
(a) effective interest rate per quarter
(b) effective annual interest rate
Formula: With continuous compounding

42. Case 0: 8% compounded quarterly Payment Period = Quarter
Interest Period = Quarterly
1st Q
2nd Q
3rd Q
5th Q
1 interest period
Given r = 8%,
K = 4 payments per year
C = 1 interest period per quarter
M = 4 interest periods per year

43. Case 1: 8% compounded monthly Payment Period = Quarter
Interest Period = Monthly
1st Q
2nd Q
3rd Q
5th Q
3 interest periods
Given r = 8%,
K = 4 payments per year
C = 3 interest periods per quarter
M = 12 interest periods per year

44. Case 2: 8% compounded weekly Payment Period = Quarter
Interest Period = Weekly
1st Q
2nd Q
3rd Q
5th Q
13 interest periods
Given r = 8%,
K = 4 payments per year
C = 13 interest periods per quarter
M = 52 interest periods per year

45. Case 3: 8% compounded continuously Payment Period = Quarter
Interest Period = Continuously
1st Q
2nd Q
3rd Q
5th Q
 interest periods
Given r = 8%,
K = 4 payments per year

46. Summary: Effective Interest Rates per Quarter at Varying Compounding Frequencies
Case 0
Case 1
Case 2
Case 3
8% compounded quarterly
8% compounded monthly
8% compounded weekly
8% compounded continuously
Payments occur quarterly
2.000% per quarter
2.013% per quarter
2.0186% per quarter
2.0201% per quarter

47. Exercise
A loan shark lends money on the following terms: "If I give you $50 on Monday, you owe me $60 on the following Monday." (a) What nominal interest rate per year (r) is the loan shark charging? (b) What effective interest rate per year (ia)is he charging? (c) If the loan shark started with $50 and was able to keep it, as well as all the money he received, out in loans at all times, how much money would he have at the end of one year?

48. answer Effective interest rate per year (ia) = F = P(F/P, i, n)
Therefore, i= 20% per week
Nominal interest rate per year = 52 weeks x 0,20 = 10,40 = 1040%
Effective interest rate per year (ia) =

49. Exercise
On January 1, a woman deposits $5000 in a' credit union that pays 8% nominal annual interest, compounded quarterly. She wishes to withdrawall the money in five equal yearly sums ,beginning December 31 of the first year. How much should she withdraw each year?

50. Continuous compounding

51. Continuous compounding
A man deposited $500 per year into a credit union that paid 5% interest, compounded annually. At the end of 5 years, he had $2763 in the credit union. How much would he have if the institution paid 5% nominal interest, compounded continuously?