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The Time Value of Money
       TOPIC 3
Learning Objectives
1.   Define the time value of money
2.   The significance of time value of money in financial
     management
3.   Define and understand the conceptual and
     calculation of future and present value in cash
     flows
4.   Define the meaning of compounding and
     discounting
5.   Work with annuities and perpetuities



                        WRMAS                               2
TIME VALUE OF MONEY
   Basic Principle : A dollar received today is worth more than a
    dollar received in the future.
•   This is due to opportunity costs. The opportunity cost of
    receiving $1 in the future is the interest we could have earned if
    we had received the $1 sooner.

•   Example
    Invest RM1 today at a 6% annual interest rate. At the end of the
    year you will get $1.06.
    SO You can say:
    1. The future value of RM1 today is $1.06 given a 6% interest
        rate a year. OR WE CAN SAY
    2. The present value of the $1.06 you expect to receive in
        one year is only $1 today.
                                WRMAS                              3
   Translate $1 today into its equivalent in the future
    (compounding) – Future Value
     Today                                        Future



                                                    ?
   Translate $1 in the future into its equivalent today
    (discounting)- Present Value
    Today                                       Future



     ?
                               WRMAS                       4
SIGNIFICANCE OF TIME VALUE OF
    MONEY
•    This concept is so important in understanding financial
     management.

•    We must take this time value of money into consideration
     when we are making financial decisions.

•    It can be used to compare investment alternatives and to
     solve problems involving loans, mortgages, leases, savings,
     and annuities.



                                 WRMAS                             5
COMPOUND INTEREST AND FUTURE
            VALUE
• Future value is the value at a given future date of an amount
  placed on deposit today and earning interest at a specified rate.
• Compound interest is interest paid on an investment during the
  first period is added to the principal; then, during the second
  period, interest is earned on the new sum (that includes the
  principal and interest earned so far). (Process of determining FV)
• Principal is the amount of money on which interest is paid.




 WRMAS                           5-6
Simple Interest
• Interest is earned only on principal.
• Example: Compute simple interest on $100 invested
  at 6% per year for 3 years.
      –   1st year interest is $6.00
      –   2nd year interest is $6.00
      –   3rd year interest is $6.00
      –   Total interest earned: $18.00




5-7
Compound Interest and Future
           Value
Example:
Compute compound interest on $100 invested at 6% for 3
years with annual compounding.

   1st year interest is $6.00 Principal is $106.00
   2nd year interest is $6.36 Principal is $112.36
   3rd year interest is $6.74 Principal is $119.11

   Total interest earned: $19.10



                              WRMAS                      8
The Equation for Future Value

• We use the following notation for the various inputs:
  – FVn = future value at the end of period n
  – PV = initial principal, or present value
  – r = annual rate of interest paid. (Note: On financial
    calculators, I is typically used to represent this rate.)
  – n = number of periods (typically years) that the money is
    left on deposit
                             OR


  FVn = PV (1+r)n                   FVn = PV (FVIFr,n)

WRMAS                                                           9
Future Value Example
Example: What will be the FV of $100 in 2 years at interest rate of
  6%?



   Manually                        Table
   FV2= $100 (1+.06)2       FV2= PV(FVIF6%,2)
      = $100 (1.06)2           = $100 (1.1236)
      = $112.36 SAME ANSWER! = $112.36

Exercise
Jane Farber places $800 in a savings account paying 6% interest
compounded annually. She wants to know how much money will be
in the account at the end of 5 years.

                                WRMAS                            10
Future Value
Changing I, N and PV

• Future Value can be increased or
  decreased by changing:
  Increasing number of years of compounding
   (n)
  Increasing the interest or discount rate (i)
  Increasing the original investment (PV)



                       WRMAS                      11
FV- Changing i, n, and PV
Exercise
(a) You deposit $500 in a bank for 2 years. What is the FV at 2%?
   What is the FV if you change interest rate to 6%?
       FV at 2% = 500 (1.0404) = ?
                                               i ( OR ) FV ?
       FV at 6% = 500 (1.1236) = ?
(b) Continue same example but change time to 10 years. What is
   the FV now?
                                               n ( OR ) FV ?
       FV at 6% = 500 (1.7908) = ?
(c) Continue same example but change contribution to $1500.
   What is the FV now?
                                                 PV ( OR ) FV ?
       FV at 6%, year 10 = 1500 (1.7908) = ?
                               WRMAS                                12
Future Value Relationship




We can increase the FV by:
1. Increasing the number of years for which money is invested;
   and/or
2. Investing at a higher interest rate.

WRMAS                                                        13
FV- Finding I and n
Example 1
At what annual rate would the following have to be invested ; $500
to grow to $1183.70 in 10 years.
1183.70       = 500 (FVIFi,,10)
1183.70/500 = FVIFi,10
2.3674        = FVIFi,10 (refer to FVIF table)
2.3674        = 9%

Example 2
How many years will the following take? $100 to grow to $672.75 if
invested at 10% compounded annually.
$672.75       = $100 (FVIF10%,n)
672.75/100 = FVIF10%,n
6.7275        = FVIF10%,n (refer to FVIF table)
6.7275        = 20 years
                               WRMAS                            14
Exercise-Finding i and n
a) How many years will the following take :
   i. $100 to grow to $298.60 if invested at 20% compounded
       annually
   ii. $550 to grow to $1,044.05 if invested at 6% compounded
       annually
b) At what annual rate would the following have to be invested :
    i. $200 to grow to $497.65 in 5 years
    ii. $180 to grow to $485.93 in 6 years




                                WRMAS                              15
DISCOUNT INTEREST AND PRESENT
              VALUE
 Present value reflects the current value of a future payment or
  receipt.
 How much do I have to invest today to have some amount in
  the future?
 Finding Present Values (PVs) = discounting

  Example
  You need RM400 to buy textbook next year. Earn 7% on your
  money. How much do you have to put today?




                              WRMAS                            16
PRESENT VALUE
 Formula of Present Value (PV):

                 FVn
     PV =                     or        PV = FVn (PVIFi,n)
                (1+i   )n
Where;
FVn = the future value of the investment at the end of n years
n     = number of years until payment is received
i     = the interest rate
PV    = the present value of the future sum of money
FVIF = Future value interest factor or the compound sum $1

[ 1/(1+i)n ] is also known as discounting factor
                                WRMAS                            17
Present Value
Example :
  What is the PV of $800 to be received 10 years from
  today if our discount rate is 10%.

Manually
  PV = 800/(1.10)10
        = $308.43
Table
                                       SAME ANSWER!
  PV = $800 (PVIF 10%,10yrs)
      = $800 (0.3855)
      = $308.40
                               WRMAS                    18
Present Value Exercise
Exercise 1 (finding PV)
Pam Valenti wishes to find the present value of $1,700 that will be
received 8 years from now. Pam’s opportunity cost is 8%.

Exercise 2 (changing i)
Find the PV of $10,000 to be received 10 years from today if our
discount rate is:
a) 5%           b) 10%      c) 20% i ( OR ) PV ?

Exercise 3 (finding n)
How many years will it take for your initial investment of RM7,752
to grow to RM20,000 with a 9% interest ?
                                             n ( OR ) PV ?




                               WRMAS                             19
Present Value Relationship




PV is lower if:
1. Time period is longer; and/or
2. Interest rate is higher.

                            WRMAS   20
ANNUITY
An annuity is a series of equal payments for a specified numbers
of years. These cash flows can be inflows of returns earned on
investments or outflows of funds invested to earn future returns.



There are 2 types of annuities*:
 - An ordinary annuity is an annuity for which the cash flow
   occurs at the end of each period (much more frequently in
   finance)
 - An annuity due is an annuity for which the cash flow occurs
   at the beginning of each period.
Note: An annuity due will always be greater than an ordinary
annuity because interest will compound for an additional period.

                              WRMAS                              21
Ordinary Annuity-PV
a) Present Value of Annuity (PVA)
• Pensions, insurance obligations, and interest owed on bonds are
  all annuities. To compare these three types of investments we
  need to know the present value (PV) of each.
                             Formula:
      PVAn = PMT [1-(1+i)-n]           PVAn = PMT (PVIFAi,n)
                     i         or




                              WRMAS                           22
Ordinary Annuity-FV
b) Future Value of Annuity (FVA)
• Depositing or investing an equal sum of money at the end of
   each year for a certain number of years and allowing it to
   grow.
                           Formula
   FVAn = PMT (1+ i)n -1      or    FVAn = PMT (FVIFAi,n)
                     i




                              WRMAS                             23
FV of Annuity: Changing PMT, N & r
1. What will $5,000 deposited annually for 50 years be worth
   at 7%?
   – FV= $2,032,644
   – Contribution = 250,000 (= 5000*50)
2. Change PMT = $6,000 for 50 years at 7%
   – FV = 2,439,173
   – Contribution= $300,000 (= 6000*50)
3. Change time = 60 years, $6,000 at 7%
   – FV = $4,881,122
   – Contribution = 360,000 (= 6000*60)
4. Change r = 9%, 60 years, $6,000
   – FV = $11,668,753
   – Contribution = $360,000 (= 6000*60)



                                                               24
ANNUITY DUE
• Remember-Annuity due is ordinary annuities in which all
  payments have been shifted forward by one time period.
  a) Future Value of Annuity Due (FVAD):

                FVADn = PMT (FVIFAi,n) (1+i)

  b) Present Value of Annuity Due (PVAD) formula:


               PVADn = PMT (PVIFAi,n) (1+i)




                               WRMAS                        25
Earlier, we examined this “ordinary”
                     annuity:
                   500                 500               500


  0                1                   2                 3 …….……5
Using an interest rate of 5%, we find that:
• The FVA (at 3) is $2,818.50
• The PVA (at 0) is $2,106.00
HOW ABOUT ANNUITY DUE?
• FVAD5 (annuity due) = PMT{[(1 + r)n – 1]/r}* (1 + r)
         = 500(5.637)(1.06)
         = $2,987.61
• PVAD0 = $2,818.80
                               WRMAS                           26
Annuity Exercise
Exercise 1
Fran Abrams wishes to determine how much money she will have at
the end of 5 years if he chooses annuity A that earns 7% annually
and deposit $1,000 per year.
Exercise 1
Branden Co., a small producer of plastic toys, wants to determine the
most it should pay to purchase a particular annuity. The annuity
consists a cash flows of $700 at the end of each year for 5 years. The
required return is 8%.
Exercise 3
Determine the answers for exercise 1 and 2 on annuity due.




                                 WRMAS                             27
FV and PV With Non-annual Periods
Non-annual periods : not annual compounding but occur
  semiannually, quarterly, monthly…
   – r = stated rate/# of compounding periods
   – N = # of years * # of compounding periods in a year

• If semiannually compounding :
  FV = PV (1 + i/2)m x 2 or FVn = PV (FVIFi/2,nx2)
• If quarterly compounding :
   FV = PV (1 + i/4)m x 4 or FVn = PV (FVIFi/4,nx4)
• If monthly compounding :
   FV = PV (1 + i/12)m x 12 or FVn = PV (FVIFi/12,nx12)

How about PV?

                                  WRMAS                    28
Compound Interest With Non-annual Periods
Example 1: If you deposit $100 in an account earning 6% with
  semiannually compounding, how much would you have in the
  account after 5 years?
Manually                             Table
FV5    = PV (1 + i/2)m x 2                FV5   = PV (FVIFi/2, nx2 )
       = 100 (1 + 0.03 )10                      = 100 (FVIF 3%,10)
       = 100 (1.3439) = $134.39                 = 100 (1.3439) = $134.39

Example 2: If you deposit $1,000 in an account earning 12% with
  quarterly compounding, how much would you have in the
  account after 5 years?
Manually                           Table
FV5   = PV (1 + i/4)m x 4          FV5 = PV (FVIFi/4, nx4 )
      = 1000 (1 + 0.03)20               = 1000 (FVIF 3%,20)
      = 1000 (1.8061) = $1806.11        = 1000 (1.8061) =$1806.11
                                  WRMAS                               29
Exercise-Non Annual
Exercise 1
How much would you have today, if RM1,000 is being discounted at
18% semiannually for 10 years.
Exercise 2
Calculate the PV of a sum of money, if RM40,000 is discounted back
quarterly at 24% per annum for 10 years.
Exercise 3
Paul makes a single deposit today of $400. The deposit will be invested
at an interest rate of 12% per year compounded monthly. What will be
the value of Paul’s account at the end of 2 years?
Exercise 4
Consider a 10-year mutual fund in which payments of $100 are made at
the beginning of each month. What is the amount today if the annual
rate of interest is 5%?


                                 WRMAS                               30
Quoted Vs. Effective Rate
• We cannot compare rates with different compounding periods.
       5% compounded annually is not the same as 5%
       compounded quarterly.
• To make the rates comparable, we must calculate their
  equivalent rate at some common compounding period by using
  effective annual rate (EAR).




• In general, the effective rate > quoted rate whenever
  compounding occurs more than once per year.


WRMAS                                                      31
Quoted Vs. Effective Rate
Example 1
RM1 invested at 1% per month will grow to RM1.126825
(=RM1.00(1.01)12) in 1 year. Thus even though the interest rate may
be quoted as 12% compounded monthly, the EAR is:
      EAR      = (1 + .12/12)12 – 1 = 12.6825%
Example 2
Fred Moreno wishes to find the effective annual rate associated
with an 8% quoted rate (r = 0.08) when interest is compounded (1)
annually (m = 1); (2) semiannually (m = 2); and (3) quarterly (m = 4).




                                WRMAS                             32
PERPETUITY
• A perpetuity is an annuity that continues forever.
• The present value of a perpetuity is
                                PV = PP
                                      i
  PV = present value of the perpetuity
  PP = constant dollar amount provided by the perpetuity
  i = annuity interest (or discount rate)
Example
What is the present value of $2,000 perpetuity discounted back to
the present at 10% interest rate?
= 2000/.10 = $20,000

                              WRMAS                            33
Perpetuity Exercise
Exercise
What is the Present Value of the following :
- A $100 perpetuity discounted back to the present at
  12%
- A $95 perpetuity discounted back to the present at
  5%
            i ( OR ) P?           $ ( OR ) PV ?




                          WRMAS                    34
WRMAS   35

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3 time value_of_money_slides - Basic Finance

  • 1. The Time Value of Money TOPIC 3
  • 2. Learning Objectives 1. Define the time value of money 2. The significance of time value of money in financial management 3. Define and understand the conceptual and calculation of future and present value in cash flows 4. Define the meaning of compounding and discounting 5. Work with annuities and perpetuities WRMAS 2
  • 3. TIME VALUE OF MONEY  Basic Principle : A dollar received today is worth more than a dollar received in the future. • This is due to opportunity costs. The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner. • Example Invest RM1 today at a 6% annual interest rate. At the end of the year you will get $1.06. SO You can say: 1. The future value of RM1 today is $1.06 given a 6% interest rate a year. OR WE CAN SAY 2. The present value of the $1.06 you expect to receive in one year is only $1 today. WRMAS 3
  • 4. Translate $1 today into its equivalent in the future (compounding) – Future Value Today Future ?  Translate $1 in the future into its equivalent today (discounting)- Present Value Today Future ? WRMAS 4
  • 5. SIGNIFICANCE OF TIME VALUE OF MONEY • This concept is so important in understanding financial management. • We must take this time value of money into consideration when we are making financial decisions. • It can be used to compare investment alternatives and to solve problems involving loans, mortgages, leases, savings, and annuities. WRMAS 5
  • 6. COMPOUND INTEREST AND FUTURE VALUE • Future value is the value at a given future date of an amount placed on deposit today and earning interest at a specified rate. • Compound interest is interest paid on an investment during the first period is added to the principal; then, during the second period, interest is earned on the new sum (that includes the principal and interest earned so far). (Process of determining FV) • Principal is the amount of money on which interest is paid. WRMAS 5-6
  • 7. Simple Interest • Interest is earned only on principal. • Example: Compute simple interest on $100 invested at 6% per year for 3 years. – 1st year interest is $6.00 – 2nd year interest is $6.00 – 3rd year interest is $6.00 – Total interest earned: $18.00 5-7
  • 8. Compound Interest and Future Value Example: Compute compound interest on $100 invested at 6% for 3 years with annual compounding. 1st year interest is $6.00 Principal is $106.00 2nd year interest is $6.36 Principal is $112.36 3rd year interest is $6.74 Principal is $119.11 Total interest earned: $19.10 WRMAS 8
  • 9. The Equation for Future Value • We use the following notation for the various inputs: – FVn = future value at the end of period n – PV = initial principal, or present value – r = annual rate of interest paid. (Note: On financial calculators, I is typically used to represent this rate.) – n = number of periods (typically years) that the money is left on deposit OR FVn = PV (1+r)n FVn = PV (FVIFr,n) WRMAS 9
  • 10. Future Value Example Example: What will be the FV of $100 in 2 years at interest rate of 6%? Manually Table FV2= $100 (1+.06)2 FV2= PV(FVIF6%,2) = $100 (1.06)2 = $100 (1.1236) = $112.36 SAME ANSWER! = $112.36 Exercise Jane Farber places $800 in a savings account paying 6% interest compounded annually. She wants to know how much money will be in the account at the end of 5 years. WRMAS 10
  • 11. Future Value Changing I, N and PV • Future Value can be increased or decreased by changing: Increasing number of years of compounding (n) Increasing the interest or discount rate (i) Increasing the original investment (PV) WRMAS 11
  • 12. FV- Changing i, n, and PV Exercise (a) You deposit $500 in a bank for 2 years. What is the FV at 2%? What is the FV if you change interest rate to 6%? FV at 2% = 500 (1.0404) = ? i ( OR ) FV ? FV at 6% = 500 (1.1236) = ? (b) Continue same example but change time to 10 years. What is the FV now? n ( OR ) FV ? FV at 6% = 500 (1.7908) = ? (c) Continue same example but change contribution to $1500. What is the FV now? PV ( OR ) FV ? FV at 6%, year 10 = 1500 (1.7908) = ? WRMAS 12
  • 13. Future Value Relationship We can increase the FV by: 1. Increasing the number of years for which money is invested; and/or 2. Investing at a higher interest rate. WRMAS 13
  • 14. FV- Finding I and n Example 1 At what annual rate would the following have to be invested ; $500 to grow to $1183.70 in 10 years. 1183.70 = 500 (FVIFi,,10) 1183.70/500 = FVIFi,10 2.3674 = FVIFi,10 (refer to FVIF table) 2.3674 = 9% Example 2 How many years will the following take? $100 to grow to $672.75 if invested at 10% compounded annually. $672.75 = $100 (FVIF10%,n) 672.75/100 = FVIF10%,n 6.7275 = FVIF10%,n (refer to FVIF table) 6.7275 = 20 years WRMAS 14
  • 15. Exercise-Finding i and n a) How many years will the following take : i. $100 to grow to $298.60 if invested at 20% compounded annually ii. $550 to grow to $1,044.05 if invested at 6% compounded annually b) At what annual rate would the following have to be invested : i. $200 to grow to $497.65 in 5 years ii. $180 to grow to $485.93 in 6 years WRMAS 15
  • 16. DISCOUNT INTEREST AND PRESENT VALUE  Present value reflects the current value of a future payment or receipt.  How much do I have to invest today to have some amount in the future?  Finding Present Values (PVs) = discounting Example You need RM400 to buy textbook next year. Earn 7% on your money. How much do you have to put today? WRMAS 16
  • 17. PRESENT VALUE  Formula of Present Value (PV): FVn PV = or PV = FVn (PVIFi,n) (1+i )n Where; FVn = the future value of the investment at the end of n years n = number of years until payment is received i = the interest rate PV = the present value of the future sum of money FVIF = Future value interest factor or the compound sum $1 [ 1/(1+i)n ] is also known as discounting factor WRMAS 17
  • 18. Present Value Example : What is the PV of $800 to be received 10 years from today if our discount rate is 10%. Manually PV = 800/(1.10)10 = $308.43 Table SAME ANSWER! PV = $800 (PVIF 10%,10yrs) = $800 (0.3855) = $308.40 WRMAS 18
  • 19. Present Value Exercise Exercise 1 (finding PV) Pam Valenti wishes to find the present value of $1,700 that will be received 8 years from now. Pam’s opportunity cost is 8%. Exercise 2 (changing i) Find the PV of $10,000 to be received 10 years from today if our discount rate is: a) 5% b) 10% c) 20% i ( OR ) PV ? Exercise 3 (finding n) How many years will it take for your initial investment of RM7,752 to grow to RM20,000 with a 9% interest ? n ( OR ) PV ? WRMAS 19
  • 20. Present Value Relationship PV is lower if: 1. Time period is longer; and/or 2. Interest rate is higher. WRMAS 20
  • 21. ANNUITY An annuity is a series of equal payments for a specified numbers of years. These cash flows can be inflows of returns earned on investments or outflows of funds invested to earn future returns. There are 2 types of annuities*: - An ordinary annuity is an annuity for which the cash flow occurs at the end of each period (much more frequently in finance) - An annuity due is an annuity for which the cash flow occurs at the beginning of each period. Note: An annuity due will always be greater than an ordinary annuity because interest will compound for an additional period. WRMAS 21
  • 22. Ordinary Annuity-PV a) Present Value of Annuity (PVA) • Pensions, insurance obligations, and interest owed on bonds are all annuities. To compare these three types of investments we need to know the present value (PV) of each. Formula: PVAn = PMT [1-(1+i)-n] PVAn = PMT (PVIFAi,n) i or WRMAS 22
  • 23. Ordinary Annuity-FV b) Future Value of Annuity (FVA) • Depositing or investing an equal sum of money at the end of each year for a certain number of years and allowing it to grow. Formula FVAn = PMT (1+ i)n -1 or FVAn = PMT (FVIFAi,n) i WRMAS 23
  • 24. FV of Annuity: Changing PMT, N & r 1. What will $5,000 deposited annually for 50 years be worth at 7%? – FV= $2,032,644 – Contribution = 250,000 (= 5000*50) 2. Change PMT = $6,000 for 50 years at 7% – FV = 2,439,173 – Contribution= $300,000 (= 6000*50) 3. Change time = 60 years, $6,000 at 7% – FV = $4,881,122 – Contribution = 360,000 (= 6000*60) 4. Change r = 9%, 60 years, $6,000 – FV = $11,668,753 – Contribution = $360,000 (= 6000*60) 24
  • 25. ANNUITY DUE • Remember-Annuity due is ordinary annuities in which all payments have been shifted forward by one time period. a) Future Value of Annuity Due (FVAD): FVADn = PMT (FVIFAi,n) (1+i) b) Present Value of Annuity Due (PVAD) formula: PVADn = PMT (PVIFAi,n) (1+i) WRMAS 25
  • 26. Earlier, we examined this “ordinary” annuity: 500 500 500 0 1 2 3 …….……5 Using an interest rate of 5%, we find that: • The FVA (at 3) is $2,818.50 • The PVA (at 0) is $2,106.00 HOW ABOUT ANNUITY DUE? • FVAD5 (annuity due) = PMT{[(1 + r)n – 1]/r}* (1 + r) = 500(5.637)(1.06) = $2,987.61 • PVAD0 = $2,818.80 WRMAS 26
  • 27. Annuity Exercise Exercise 1 Fran Abrams wishes to determine how much money she will have at the end of 5 years if he chooses annuity A that earns 7% annually and deposit $1,000 per year. Exercise 1 Branden Co., a small producer of plastic toys, wants to determine the most it should pay to purchase a particular annuity. The annuity consists a cash flows of $700 at the end of each year for 5 years. The required return is 8%. Exercise 3 Determine the answers for exercise 1 and 2 on annuity due. WRMAS 27
  • 28. FV and PV With Non-annual Periods Non-annual periods : not annual compounding but occur semiannually, quarterly, monthly… – r = stated rate/# of compounding periods – N = # of years * # of compounding periods in a year • If semiannually compounding : FV = PV (1 + i/2)m x 2 or FVn = PV (FVIFi/2,nx2) • If quarterly compounding : FV = PV (1 + i/4)m x 4 or FVn = PV (FVIFi/4,nx4) • If monthly compounding : FV = PV (1 + i/12)m x 12 or FVn = PV (FVIFi/12,nx12) How about PV? WRMAS 28
  • 29. Compound Interest With Non-annual Periods Example 1: If you deposit $100 in an account earning 6% with semiannually compounding, how much would you have in the account after 5 years? Manually Table FV5 = PV (1 + i/2)m x 2 FV5 = PV (FVIFi/2, nx2 ) = 100 (1 + 0.03 )10 = 100 (FVIF 3%,10) = 100 (1.3439) = $134.39 = 100 (1.3439) = $134.39 Example 2: If you deposit $1,000 in an account earning 12% with quarterly compounding, how much would you have in the account after 5 years? Manually Table FV5 = PV (1 + i/4)m x 4 FV5 = PV (FVIFi/4, nx4 ) = 1000 (1 + 0.03)20 = 1000 (FVIF 3%,20) = 1000 (1.8061) = $1806.11 = 1000 (1.8061) =$1806.11 WRMAS 29
  • 30. Exercise-Non Annual Exercise 1 How much would you have today, if RM1,000 is being discounted at 18% semiannually for 10 years. Exercise 2 Calculate the PV of a sum of money, if RM40,000 is discounted back quarterly at 24% per annum for 10 years. Exercise 3 Paul makes a single deposit today of $400. The deposit will be invested at an interest rate of 12% per year compounded monthly. What will be the value of Paul’s account at the end of 2 years? Exercise 4 Consider a 10-year mutual fund in which payments of $100 are made at the beginning of each month. What is the amount today if the annual rate of interest is 5%? WRMAS 30
  • 31. Quoted Vs. Effective Rate • We cannot compare rates with different compounding periods. 5% compounded annually is not the same as 5% compounded quarterly. • To make the rates comparable, we must calculate their equivalent rate at some common compounding period by using effective annual rate (EAR). • In general, the effective rate > quoted rate whenever compounding occurs more than once per year. WRMAS 31
  • 32. Quoted Vs. Effective Rate Example 1 RM1 invested at 1% per month will grow to RM1.126825 (=RM1.00(1.01)12) in 1 year. Thus even though the interest rate may be quoted as 12% compounded monthly, the EAR is: EAR = (1 + .12/12)12 – 1 = 12.6825% Example 2 Fred Moreno wishes to find the effective annual rate associated with an 8% quoted rate (r = 0.08) when interest is compounded (1) annually (m = 1); (2) semiannually (m = 2); and (3) quarterly (m = 4). WRMAS 32
  • 33. PERPETUITY • A perpetuity is an annuity that continues forever. • The present value of a perpetuity is PV = PP i PV = present value of the perpetuity PP = constant dollar amount provided by the perpetuity i = annuity interest (or discount rate) Example What is the present value of $2,000 perpetuity discounted back to the present at 10% interest rate? = 2000/.10 = $20,000 WRMAS 33
  • 34. Perpetuity Exercise Exercise What is the Present Value of the following : - A $100 perpetuity discounted back to the present at 12% - A $95 perpetuity discounted back to the present at 5% i ( OR ) P? $ ( OR ) PV ? WRMAS 34
  • 35. WRMAS 35