1. Chapter 4 TIME VALUE OF MONEY
ENGINEERING ECONOMIC
(BPK30902)
Chapter 4 TIME VALUE OF MONEY
HJ. ZUIKARNAIN HJ. DAUD

2. TIME VALUE OF MONEY Introduction Simple Interest Compound Interest
The Concept Equivalence
Cash Flow-Diagram Table
Applications of Future and Present Values

3. Learning Outcomes Differentiate between types of interest rate
Understand & measures the interest rate in the economy
Explain the concepts of economic equivalence & interest rates
Understand the role of time value in engineering economics

4. Learning Outcomes
Explain the mechanics of compounding, that is, how money grows over time when it is invested.
Discuss the relationship between compounding & bringing money back to the present.
Define an ordinary annuity & calculate its compounding or future value.
Differentiate between an ordinary annuity & an annuity due, and determine the future and present value of an annuity due.
Determine the future or present value of a sum when there are non-annual compounding periods.
Determine the present value of an uneven stream of payments.

5. Introduction

6. Introduction The Role Of Time Value In Engineering Economics
The timing of cash flows has important economic consequences in recognizing the time value of money
For example, Ringgit today is worth more than a Ringgit that will be received at some future date
If today we have RM100 that can buy a eqiupment, is not the same in the future if the cost of the equipment has incresed

7. Interest = amount owed now – original amount
Introduction
Interest: The Cost Of Money
Interest is the difference between an ending amount of money and the beginning amount
Interest = amount owed now – original amount
For example, ending amount owed is RM110,000 & Beginning amount owed is RM100,000, therefore; Interest = RM10,000 (RM110,000 – RM100,000)
When money is borrowed the interest paid is the charge to the borrower for the use of the lender’s property
When money is lent or invested, the interest earned is the lender’s gain from providing a good to another use
interest paid
interest earned

8. Introduction Market Interest Rate No Banking Institution
Interest rate quoted by financial institutions
For example, Base Lending Rate ( BLR ) in Malaysia.  Update 16/10/2012 No
Banking Institution
With Effect From
BLR (% p.a.) 1.
CIMB Bank Berhad
04/04/2011
6.35 2.
Malayan Banking Berhad
10/05/2011
6.60 3.
OCBC Bank (Malaysia) Berhad
11/05/2011 4.
Public Bank Berhad 5.
RHB Bank Berhad

9. Simple Interest

10. Simple Interest
Simple interest is calculated using the principal only, ignoring any interest accrued in preceding interest periods
Interest = (principal) (number of periods) (interest rate)
P n i%
When applied, total interest “I” may be found by
I = ( P ) ( n ) ( i )
where
P = principal amount lent or borrowed
n = number of interest periods ( e.g. years )
i = interest rate per interest period

11. Simple Interest
Accumulated interest from prior periods is not used in calculations for the following periods.
Simple interest is normally used for a single period of less than a year, such as 30 or 60 days.
Example: You borrow RM10,000 for 3 years at 5% simple annual interest
interest = (P )( i )(n) = (10,000)(0 .05)(3) = 1,500
Example 2: You borrow RM10,000 for 60 days at 5% simple interest per year (assume a 365 day year).
interest = (P )( i )(n) = (10,000)(0 .05)(60/365) = 

12. Compound Interest

13. Compound Interest Compound interest means interest on top of interest
The interest accrued for each interest period is calculated on the principal plus the total amount of interest accumulated in all previous periods
Interest = (principal + all accrued interest) (interest rate)
In general formula form:
Total due after a number of years:
principal (1+ interest rate) number of years

14. Period Amount Owed Beginning of period (RM)
Compound Interest
principal (1+ interest rate) number of years n
Period Amount Owed Beginning of period (RM)
Interest Amount for
Period 10%) (RM)
Amount Owed
at end of period (RM) 1
1,000
100
1,100 2
110
1,210 3
121
1,331

15. Compound Interest
Compound interest is calculated each period on the original principal  and all  interest accumulated during past periods
Although the interest may be stated as a yearly rate, the compounding periods can be yearly, semi-annually, quarterly, or even continuously
You can think of compound interest as a series of back-to-back simple interest contracts
The interest earned in each period  is added to the principal of the previous period to become the principal for the next period. 

16. Compound Interest
For example, you borrow RM10,000 for three years at 5% annual interest compounded annually:
interest year 1 = (P)(i)(n)
= (10,000)(0.05)(1)
= 500 interest year 2 =  (p2 = p1 + i1)(i)(n)
= (10, ) (0.05)(1)
= 525 interest year 3 = (p3 = p2 + i2) (i)(n)
= (10, ) (0.05)(1) =
Total interest earned over the three years 
= = RM1,  
Compare this to RM1,500 earned over the same number of years using simple interest.

17. Interest Comparison
The power of compounding can have an astonishing effect on the accumulation of wealth.  
This table shows the results of making a one-time investment of RM10,000  for 30 years using 12% simple interest, and 12% interest compounded yearly and quarterly.
Type of Interest
Principal Plus Interest Earned
Simple
46,000.00
Compounded Yearly
299,599.22
Compounded Quarterly
347,109.87

18. Nominal Interest Rate (APR)
Financial institutions generally quote interest rates, termed nominal interest rates, in annual terms without the effects of compounding.
Also known as Annual Percentage Rate (APR)
The terms annual percentage of rate (APR), nominal APR describe the interest rate for a whole year (annualized), rather than just a monthly fee/rate, as applied on a loan, mortgage loan, credit card, etc. It is a finance charge expressed as an annual rate
The nominal APR is calculated as the rate, for a payment period, multiplied by the number of payment periods in a year

19. Nominal Interest Rate (APR)
The nominal interest rate is the periodic interest rate times the number of periods per year;
For example, a nominal annual interest rate of 12% based on monthly compounding means a 1% interest rate per month (compounded).
If a credit card issuer quotes an APR rate of 18 percent, monthly, this means
interest rate of 1.5 percent per month,
APR = 12 (1.5%) = 18%
i = r/M
M = number of compounding periods in a year
r = Annual interest rate

20. Effective Interest Rate (APY)
Definition: Actual interest paid on a loan, or earned on a deposit account, depending on the frequency of compounding or effect of inflation.
It is different from the nominal rate of interest which ignores compounding and other factors.
The effective interest rate is the interest rate on a loan or financial product restated from the nominal interest rate as an interest rate with annual compound interest payable in arrears
It is used to compare the annual interest between loans with different compounding terms (daily, monthly, annually, or other)
Annual percentage yield or effective annual yield is the analogous concept used for savings or investment products, such as a certificate of deposit.

21. Effective Interest Rate (APY)
Since any loan is an investment product for the lender, the terms may be used to apply to the same transaction, depending on the point of view.
is a true costs of loans & are much more useful than nominal interest rates, as they incorporate the effect of compounding
i.e. = (1 + r/m)mn-1
ie = effective rate a year
r = nominal rate a year
m = frequency of compounding in a year
Example, the effective interest rate a year
i.e. = ( /2)2-1
= ( )2-1
= 6.09%

22. Examples Monthly compounding Daily compounding
Examples when number of m is different
Given the APR is 12 percent per year, find the annual effective interest rate if m is as below;
Yearly compounding
ie = 12%
Semiannually Compounding
ie = ( /2)2-1
= 12.36%
Quarterly compounding
ie = ( /4)4-1
= 12.55%
Monthly compounding
ie = ( /12) 12-1
= 12.68%
Daily compounding
ie = ( /365)365-1
= 12.74%

23. Examples Given an interest rate of 12% compounded monthly. Find
the effective monthly interest rate
ie = (1 + r/m)mn-1
ie = ( /12)12*1/12-1
= 1%
the effective quarterly interest rate
ie = ( /12)12*1/4-1
= 3.03%
the effective semiannual interest rate
ie = ( /12)12*1/2-1
= 6.15%

24. Examples Given an interest rate of 12% compounded quarterly. Find
the effective monthly interest rate
ie = ( /4)4*1/12-1
= 0.99%
the effective semiannual interest rate
ie = ( /4)4*1/2-1
= 6.09%

25. The Concept Equivalence

26. Economic Equivalence
The equivalence we talk about in engineering economic analysis is the equivalence of monetary value. 
If two cash flows have the same monetary value, we say they are economically equivalent. 
In other words we are economically indifferent to either one of the two alternatives. 
This is not to say that other considerations are unimportant in making a choice between two alternatives; but for engineering economic analysis we will limit ourselves to an economic equivalence perspective only.

27. 4 General Principles of Economic Equivalence
a common basis for comparison is found by converting all cash flows to a single point in time, which is generally the present period (present worth) or the future period (future worth)
conversion is made to the present - present worth (P)
converted in future - future worth (F) 
The conversion of the cash flow can also be made to a point in time between the start and end of the project
Principle 2
specifies that equivalence depends on the interest rate used in the analysis
If the interest rate changes, equivalence needs to be recalculated

28. 4 General Principles of Economic Equivalence
deals with the case of multiple-period cash flow, in which the payment and/or receipts are dispersed through multiple periods during the life time of the project
Principle 3 is a corollary to Principle 1; a multiple period cash flow needs a common time basis for further analysis
Principle 4
states that equivalence is independent of point-of- view. 
For example, it shouldn't matter whether you are taking the banker's or the borrower's point of view.  
For Principle 4 to hold, the two basic assumptions described above need to be satisfied

29. Cash Flow-Diagram Table

30. Cash Flow
The future cash flow from any projects will become the basis of economics studies. Varies pattern of cash flow:
Single amount
Either currently held or expected at some future date.
Examples include RM3 000 today and RM to be received at the end of 4 years.
A uniform series/ Annuity
Engineering projects of often entail uniform series cash flows.
Uniform series means that the cash flows exist at every period and the amount is the same.
For examples, car-lease payment, home- mortgage loan are
Mixed stream cash flow
Mixed stream is a stream of unequal periodic cash flows that reflect no particular pattern.

31. Cash Flow P Present sum i Interest rate
Cash flows are described as the inflows and outflows of money, occurs during specified periods of time, such as 1 month or 1 year
Inflows : Cash receipts revenue
Outflows: Cash disbursements expenses & costs
Some notations:
P Present sum
i Interest rate
n number of interest compounding period
F Final sum/future sum

32. Cash Flow Samples of Cash inflow estimates: Revenues
Operating cost reductions
Asset salvage value
Receipts from stock and bond sales.
Saving or return of corporate capital funds
Samples of Cash outflow estimates:
Purchase of assets
Operating costs
Loan interest
Income taxes
Net cash flow = receipts - disbursements
= cash inflows - cash outflows
Example = Annual Revenue – Annual operating cost - Annual tax
Net cash flow = RM20,000 - RM8,000 - RM2,000
= RM10,000

33. Cash Flow Diagram Notation
n = 5 I = 10% initial investment = RM50,000.
CASH FLOW DIAGRAM NOTATION 1 2 3 4
5 = N
P =RM50,000
A = RM10,000
i = 10% per year

34. Constructing a Cash Flow Diagram
The time line is a horizontal line divided into equal periods such as days, months, or years. 
Each cash flow, such as a payment or receipt, is plotted along this line at the beginning or end of the period in which it occurs. 
Funds that you pay out such as savings deposits or lease payments are negative cash flows that are represented by arrows which extend downward  from the time line with their bases at the appropriate positions along the line. 
Funds that you receive such as proceeds from a mortgage or withdrawals from a saving account are positive cash flows represented by arrows extending upward from the line.

35. Example
You are 40 years old and have accumulated RM50,000 in your savings account.  You can add RM100 at the end of each month to your account which pays an annual interest rate of 6% compounded monthly.  Will you be able to retire in 20 years?
The time line is divided into 240 monthly periods (20 years times 12 payments per year) since the payments are made monthly and the interest is also compounded monthly.   The RM50,000 that you have now (present value) is a negative cash outflow  since you will treat it as though you were just now depositing it into the account.    It is represented with a downward pointing arrow with its base at the beginning of the first period.  The 240 monthly RM100 deposits  are also negative outflows  represented with downward pointing arrows placed at the end of each period.  Finally you will withdraw some unknown amount (the future value) after 20 years.  Represent this positive  inflow with an upward pointing arrow with its base at the very end of the last period.

36. Questions & Problems
Construct a cash flow diagram for the following: RM5,000 outflow at time zero, RM1,500 per year inflow in years 1 through 5 at an interest rate of 10% per year, and an unknown future amount in year 5.
Construct a net cash flow diagram for the following situations
the sum of RM40,000 is invested at time zero, resulting in revenues of RM19,000 per year and RM2,000 in annual expenses over the next four years.
A lease is signed that requires monthly payments of RM2,000 over the next three years.
A loan is taken out for RM70,000 and paid back through seven annual payments of RM12,000.
A machine is purchased for RM1 million, operated for six years at the cost of RM50,000 per year, and sold for RM 60,000 at the end of year 6.

37. Questions & Problems
East Pahang Company got a RM150 million contract to provide information technology services for a department of the Pahang Education. Draw the cash flow diagram, assuming a five year contract with equal sized end-of-year payments.
Is interest an expense or revenue? Explain.
What is the difference between simple and compound interest?
Assume that a nominal interest rate is 10% per year compounded monthly. Count the effective rate (ie) for a quarter of the year.
Miss Jun borrows RM30,000 with 5% simple interest rate. Find the total loan after 5 years.

38. Questions & Problems
Affnan borrows RM35,000 from a RBC bank. Assuming that loan is charged compounded interest rate at 4% per year. Find his total loan at the end of year 3.
In order to restructure company’s debt, Telecom Malaysia decided to pay off one of its short term liability. If the company borrowed the money 1 year ago at an interest rate of 7% per year and the total cost of repaying the loan was RM25 million, what was the amount of the original loan?
If your credit card calculates interest based on 12.5% APR,
What are your monthly interest rate and annual effective interest rate?
If your current outstanding balance is RM4,000 and you skip payments for three months, what would be the total balance three months from now?

39. Questions & Problems the amount of the payment, and
Three years ago, RRCO Sdn. Bhd. invested RM200,000 in a certificate of deposit that paid simple interest of 9% per year. Now the company plans to invest the total amount accrued in another security that pays 10% per year compound interest. How much will the new certificate be worth 2 years from now?
In order to build a new factory facility, the regional distributor for Multi-Valves Company borrowed RM0.8 million at 9% per year interest. If the company repaid the loan in a lump sum amount after 3 years, what was
the amount of the payment, and
the amount of interest.
Assume that nominal interest rate for loan is 12% compounded monthly. Calculate an effective interest rate per year.

40. Questions & Problems
An interest rate of 10.5% compounded quarterly is advertised. Find
the effective monthly interest rate
the effective quarterly interest rate
the effective semiannual interest rate
the effective annual interest rate
An interest rate of 10% compounded monthly is advertised. Find
Calculate an effective interest rate per year. If given the interest rate 14% compounded quarterly?
Which of the following represents a better investment, 12.5% compounded semiannually or 2% per quarter?

41. Questions & Problems
Which of the following represents a cheaper cost, 6.5% per quarter or 9% compounded monthly?
Suppose that you make quarterly deposits in a savings account that earns 8% interest compounded monthly. Compute the effective interest rate per quarter.
Calculate the semiannually payment on a loan of RM30,000 for five years at a nominal interest rate of 9% compounded monthly.
What is the future worth of RM6,000 deposited at the end of each six-month period for 6 years at 6% compounded semiannually?

42. Applications of Future & Present Values

43. INTEREST RATE FACTORS Future value - single sums (F/P, i, n)
Present value (P/F, i, n)
Future value of an annuity (F/A, i, n)
Present value of an annuity (P/A, i, n)
Equal-payment series Sinking Fund Factor (A/F, i, n)
Equal-payment series Capital-Recovery Factor (A/P, i, n)
Uneven Cash flow

44. Formula Of Present Value & Future Value For Single Cash

45. Future Worth Future Worth - single sums (F/P, i, n)
Question - What is F, given P?
If you deposit RM100 in an account earning 6%, how much would you have in the account after 1 year?
F = P (F/P i, n )
F = 100 (F/P .06, 1 )
F = P (1 + i)n
F = 100 (1.06)1 = RM106
100 1
106
Thus, FWn = PW ( l + i)n
Where;
n = the number of years during which the compounding occurs
i = the annual interest (or discount) rate
PW = the present value or original amount invested at the beginning of the first period
FWn = the future value of the investment at the end of n years

46. Future Worth
Future Value of Single Cash Flow Future value can be computed by the following formula:
FVn = PV (1 + r)n
Where
FV = Future value
PV = Present value
r = Rate of Interest
n = Number of periods
E.g.
Let us calculate the future value of an investment of RM2,000 compounded annually at the rate of 12%, after 4 years period
FV = RM2,000 x ( )4 = RM3,147.04

47. Future Worth
Frequent Compounding: Interest is compounded often more than once a year. In such cases, the formula for FV becomes: FV n = PV (1 + (r / m)) n x m
Where:
m = Number of total compounding periods in a year
If compounded semi-annually, m=2
If compounded quarterly, m = 4 and so on.

48. Future Worth Frequent Compounding: Example:
Let us assume that the interest of 12% is compounded semi- annually; rest of details being the same, the future value after 4 years would be: m = 2
FV = RM2,000 x (1 + (0.12 / 2)) 4 x 2
FV = RM3,187.70
Thus, when interest is compounded yearly once, the FV is only RM3, whereas if it is compounded twice a year, the FV is RM3,
Similarly, if interest is compounded quarterly or monthly, FV would accordingly be greater.

49. Future Worth
Future value using Simple Interest If no interest is earned on the interest on the investment, it is called as simple interest. The future value of an investment in such cases would be calculated by the following formula: FVn = PV (1 + [n x r])
Where
n = Number of years
r = Interest Rate
The future value using a simple interest would obviously be lower than the future value using compound interest as there is no interest earned on the interest portion of the investment.
E.g An investment of RM10,000, if invested at 13% simple interest rate will in 6 years be: FV = RM10,000 x (1 + [6 X 0.13])
FV = RM17,800

50. Example
If you deposit RM100 in an account earning 6%, how much would you have in the account after 5 years?
F = P (1 + i)n
F = 100 (1.06)5 = RM133.82
If you deposit RM100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years?
F = P (1 + i/m) mn
F = 100 (1.015)20 = RM134.68
If you deposit RM100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years?
F= P (1 + i/m) mn
F = 100 (1.005)60 = RM134.89

51. Present Worth Present Worth (P/F, i, n) Question - What is P, given F?
The value today of a future cash flow or series of cash flows.
Or,
The value in today's Ringgit of a sum of money to be received in the future, involves nothing other than inverse compounding.
Mathematically, the present value of a sum of money to be received in the future can be determined with the following equation:
PW = FWn / (1+i)n or FWn (1+i)-n
where:
n = the number of years until payment will be received
i = the opportunity rate or discount rate
PW = the present value of the future sum of money
FWn = the future value of the investment at the end of n years

52. Present Worth Finding Present Worth (PWs) is called discounting.
Discounting is the process of finding the present value of a cash flow or a series of cash flows; discounting is the reverse of compounding.
Discounting means removing the interest that is imbedded in the future cash amounts are often referred to as a discounted cash flow technique
PV calculations involve the compounding of interest means that any interest earned is reinvested and itself will earn interest at the same rate as the principal (“earn interest on interest”

53. Components Present Value (PW) Calculation
Present value amount (PW)
Future value amount (FW)
Length of time before the future value amount occurs (n)
Interest rate used for discounting the future value amount (i)
PW = FW/ (1+i)n
(or)
PW = FW (1 + i)-n
(or)    
PW = FW x [ 1 ÷ (1 + i)n ]  

54. Example
What is the PV of RM800 to be received 10 years from today if our discount rate is 10%
PW = 800/(1.10)10 = RM308.43
If you planning to buy a motorcycle that costs you RM years from today, how much that you have to save in your saving account today if bank pay you at a rate of 5% annually?
PW = F/(1+i)n
PW = F(1+i)-n
= 5000 (1+.05)-3
= RM4,319.19

55. Example
Let's assume we are to receive RM100 at the end of two years. How do we calculate the present value of the amount, assuming the interest rate is 8% per year compounded annually?
The following timeline depicts the information we know, along with the unknown component (PW):
PW=? 1
FW=100
n = 2, I = 8%
I year

56. Example Calculation Using the PW Formula
The present value formula for a single amount is:  
PW = FW (1 + i)-n       (or)       PW = FW x [ 1 ÷ (1 + i)n ]  
Using the second version of the formula, the solution is:
PV = FV x [ 1 ÷ (1 + i)n ] PW = 100 x [ 1 ÷ ( )2 ] PW = 100 x [ 1 ÷ (1.08)2 ] PW = 100 x [ 1 ÷ ] PW = 100 x [ ] ← PV factor PW = RM85.73
The answer, RM85.73, tells us that receiving RM100 in two years is the same as receiving RM85.73 today, if the time value of money is 8% per year compounded annually. ("Today" is the same concept as "time period 0.")

57. Formula Of Interest For Similar Series Of Present Value & Future Value

58. Annuity
Annuity is a series of payments of an equal amount at fixed intervals for a specified number of periods.
For example, RM100 at the end of each of the next three years is a three-year annuity. The payments are given the symbol A or PMT.
i) Future value of an annuity (F/A, i, n)
Question - What is F, given A?
If you invest RM1,000 each year at 8%, how much would you have in the account after 3 years?
Equation:
F = A (F/A, i, n)
FW = A @ Fn = A { [ (1+i)n – 1] / i }
FW = 1,000 { [ (1.08)3 – 1] / 0.08 }
FW = 1,000 {3.2464}
FW = RM3,246.40

59. Example
If you invest RM1,000 each year at 8% compounded semiannually, how much would you have in the account after 3 years?
Find the annual effective rate first,
ie = ( /2)2*1 -1
= 8.16%
FW = 1,000 { [ (1.0816)3 – 1] / }
FW = 1,000 { }
FW = RM3,251.46
If you invest RM1,000 every six-month at 8% compounded semiannually, how much would you have in the account after 3 years?
FW = 1,000 ( /2)2*3
FW = 1,000 { [ (1.04)6 – 1] / 0.04 }
FW = 1,000 {6.633}
FW = RM

60. Future Value – Annuity
An annuity is a stream of equal annual cash flows occurring at regular intervals of time.
When the cash flows occur at the end of each period, the annuity is called as an ordinary annuity or a deferred annuity.
When the cash flows occur at the beginning of each period, the annuity is called as an annuity due.
The Future Value of an Annuity is given by the following formula:
FV An = A (1 + r)n -1 + A (1 + r)n A
= A [(1 + r)n -1] / r
Where
FV An = Future value of an annuity at time n
A = Annuity periodic amount
r = Interest rate
n = Number of periods

61. Example
A person deposits RM10,000 annually in a bank for 5 years which earns an interest of 9%, compounded annually. Let us calculate the future value of his series of deposits at the end of 5 years.
FV An = A [(1 + r)n - 1] / r
A = $10,000
r = 9% or .09
n = 5
Future value of annuity
= RM10,000 [( )5 - 1] / 0.09
= RM59,847.11

62. Annuity ii) Present worth of an annuity (P/A, i, n)
Question - What is P, given A?
P = A (P/A, i, n)
What is the PW of RM1,000 at the end of each year for over the next 3 years, if the interest rate is 8%?
Equation:
PW = A @ PW = A { [ 1 - (1+i)-n ] / i }
PW = 1,000 { [ ( 1 – (1.08)-3 ] / 0.08 }
PW = 1,000 {2.5771} = RM2,577.10

63. Annuity
What is the PW of RM1,000 at the end of each year for over the next 3 years, if the interest rate is 8% compounded quarterly?
Find the annual effective rate first,
ie = ( /4)4*1-1 = 8.24%
PW = 1,000 { [ ( 1 – (1.0824)-3 ] / }
PW = 1,000 {2.5658} = RM2,565.80
What is the PW of RM1,000 at the end of each semiannual for over the next 3 years, if the interest rate is 8% compounded quarterly? Find the annual effective rate first,
ie = ( /4) 4*1/2-1 = 4.04%
PW = 1,000 { [ ( 1 – (1.0404)-6 ] / }
PW = 1,000 {5.2353} = RM

64. Present Value – Annuity
An annuity is a series of equal annual cash flows.
In general terms, the present value of an annuity may be expressed as follows:
PV An = A A A (1 + r)1 (1 + r)2 (1 + r)n
OR PV An = A { ∑ n      1 / (1 + r)t }          
t = 1
Where
PV An = Present value of an annuity
A = annuity amount (even cash inflows)
r = discount rate
n = number of years of annuity (the last year)

65. Example Years Cash inflows (RM) PV factor at 10%
Let us find out the present value of an annuity of RM10,000 to be received in the next 4 years time, the discount rate being 10%.
Formula: PV An = A [ { 1 - (1/ (1 + r))n } / r ]
Years
Cash inflows (RM)
PV factor at 10%
PV of cash inflows (RM) 1
10,000
0.9091
9,091 2
0.8264
8,264 3
0.7513
7,513 4
0.6830
6,830
Total
31,698.00

66. Annuity iii) Equal-payment series Sinking Fund Factor (A/F, i, n)
Question - What is A, given F?
It is done to calculate an annuity given the future worth.
A = F (A/F, i, n)
(A/F, i, n) = [ i/ (1+i)n -1]
Example:
Firm S is considering to buy an equipment worth RM200,000 in the next four years. Firm S will make annual saving in order to buy that equipment. Find a total annual sum if the saving gives 6% interest rate
A = 200,000 (A/F, 6%, 4)
= 200,000 ( )
= RM45,718

67. Annuity iv) Equal-payment series Capital-Recovery Factor (A/P, i, n)
Question - What is A, given P?
It is done to calculate an annuity given the present worth.
A = P (A/P, i, n)
(A/P, i, n) = [i (1+i)n / (1+i)n -1]
Example:
Firm K is deciding to acquire a new machine worth
RM100,000. If the firm’s MARR is 15% per year, find the total
annual revenue that the firm should get for over 5 years to recover its capital
A = 100,000 (A/P, 15%, 5)
= 100,000 ( )
= RM29,832

68. Uneven Cash Flows Example:
How do we find the PW of a cash flow stream when all of the cash flows are different?
Example:
What is the PW of an investment that yields RM300 to be received in 2 years and RM450 to be received in 8 years if the discount rate is 5%?
PW = (PVIF5%, 2) + 450(PVIF5%, 8)
= (0.907) + 450(0.677) =
= RM576.75 or
PW = ( ) ( )-8
= (0.907) + 450(0.677)

69. Questions & Problems a. FW = PW(1+i)n b. FW = (1+i)/ PW
1. Which of the following is the formula for compound value?
a. FW = PW(1+i)n
b. FW = (1+i)/ PW
c. FW = PW /(1+i)n
d. FW = PW (1+i)-n
2. At 8% compounded annually, how long will it take RM750 to double?
a. 6.5 years
b. 48 months
c. 9 years
d. 12 years
3. At what rate must $400 be compounded annually for it to grow to RM in 10 years?
a. 6%
b. 5%
c. 7%
d. 8%

70. Questions & Problems
4. If you invest RM750 every six months at 8% compounded semi- annually, how much would you accumulate at the end of 10 years?
a. RM10,065
b. RM10,193
c. RM22,334
d. RM21,731
5. A commercial bank will loan you RM7,500 for two years to buy a car. The loan must be repaid in 24 equal monthly payments. The annual interest rate on the loan is 12% of the unpaid balance. What is the amount of the monthly payments?
a. RM282.43
b. RM390.52
c. RM369.82
d. RM353.05

71. Questions & Problems
An investment will pay RM500 in three years, RM700 in five years, and RM1,000 in nine years. If the opportunity rate is 6%, what is the present worth of this investment?
Suppose you are 40 years old and plan to retire in exactly 20 years. 21 years from now you will need to withdraw RM5,000 per year from a retirement fund to supplement your social security payments. You expect to live to the age of 85. How much money should you place in the retirement fund each year for the next 20 years to reach your retirement goal if you can earn 12% interest per year from the fund?
8. In order to send your oldest child to law school when the time comes, you want to accumulate RM40,000 at the end of 18 years. Assuming that your savings account will pay 6% compounded annually, how much would you have to deposit if:
a. you want to deposit an amount annually at the end of each year?
b. you want to deposit one large lump sum today?

72. Questions & Problems
You have been offered the opportunity to invest in a project which will pay RM1,000 per year at the end of years one through 10 and RM2,000 per year at the end of years 21 through 30. If the appropriate discount rate is 8%, what is the present worth of this cash flow pattern?
11. A electric consulting company is estimating its cash flow requirements for the next 7 years. The company expects to replace office equipment at various times over the 7- year planning period. Specifically, the company expects to spend RM8,000 two years from now, RM10,000 three years from now, and RM4,000 five years from now. What is the present worth of the planned expenditures at an interest rate of 9% per year?

73. Questions & Problems
TN Materials Sdn. Bhd. ordered RM7 million worth of seamless tubes for its drill system from the Southern Technology Company. At 9% per year interest, what is the annual worth of the purchase over a 10- year amortization period?
Thermal Products is considering a new annealing-drawing process to reduce costs. If the new process will cost RM2.5 million Ringgit now, how much must be saved each year to recover the investment in 5 years at an interest rate of 12% per year?
Cost is 10%, how If your opportunity much are you willing to pay for an investment promising RM750 per year for the first four years and RM450 for the next six years?

74. Questions & Problems
Consider an investment that has cash flows of RM500 the first year and RM400 for the next four years. If your opportunity cost is 10%, how much is this investment worth to you?
What is the PW of an investment that yields RM500 to be received in 3 years and RM750 to be received in 5 years if the discount rate is 5%?
What is the PW of an investment that yields RM1,000 to be received in 2 years and RM2,500 to be received in 4 years if the discount rate is 6%?
What is the future worth 7 years from now of a present cost of RM175,000 to Company C at an interest rate of 10% per year?

75. Questions & Problems
Western Storage wants to have enough money to purchase a new machine in 5 years at a cost of RM300,000. If the company sets aside RM100,000 in year 2 and RM80,000 in year 3, how much will the company have to set aside in year 4 in order to have the money it needs if the money set aside earns 10% per year?
New simulation technology enables engineers to stimulate complex computer-controlled movements in any direction. If the technology results in cost savings in the design of new roller coasters, determine the future worth in year 6 of savings of RM60,000 now and RM80,000 three years from now at an interest rate of 11% per year.

76. Interest Factors for Discrete Compounding
9.0%
Single Payment
Equal Payment Series
Gradient Series
Compound Amount Factor
(F/P)
Present Worth Factor
(P/F)
(F/A)
Sinking Fund Factor
(A/F)
(P/A)
Capital Recovery Factor
(A/P)
Gradient Uniform Series
(A/G)
Gradient Present Worth
(P/G) n
1.0900
1.9174
1.000
0.9174
0.0000 1
1.1881
0.8417
2.0900
0.4785
1.7591
0.5685 2

77. Thank You