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![Simple Interest (FV)
• Future Value or terminal value
• The value at some future time of a present amount of
money or a series of payments evaluated at a given
interest rate.
• For any simple interest the future value of an account
at the end of n periods
• FVn= P0 + SI= P0 +P0 (i)(n)…………………………(1)
• Or we can write this as
• FVn= = P0 [1+(i)(n)]………………………………..(2)](https://image.slidesharecdn.com/lec3-170425143546/75/Time-Value-of-Money-Business-Finance-7-2048.jpg)
![Future Value
• What is the Future Value (FV) of the deposit?
• FV = P0 + SI
• = $1,00 + $80
• = $180
• So FV= 100+[100(.08)(10)]=$180
Or
FVn=100[1+(.08)(10)]
=180](https://image.slidesharecdn.com/lec3-170425143546/75/Time-Value-of-Money-Business-Finance-8-2048.jpg)
![Simple Interest (PV)
• Sometimes we need to proceed in opposite direction.
• That is we know the FV of deposit at interest rate deposited for
“n” years but we don’t know the principal originally invested…..so
re arrangement of equation 2 is required to get present value.
• FVn= = P0 [1+(i)(n)]…………..(2)
• Pv0
P0 =FVn/ [1+(i)(n)]
Present Value is the current value of a future amount of money, or
a series of payments, evaluated at a given interest rate.
So PV is the present value is the original amount which was
deposited which is 100.](https://image.slidesharecdn.com/lec3-170425143546/75/Time-Value-of-Money-Business-Finance-9-2048.jpg)
![Example
• Julie Miller wants to know how large her deposit of
$10,000 today will become at a compound annual
interest rate of 10% for 5 years.
• Calculation based on general formula:
• FVn = P0 (1+i)n
• FV5 = $10,000 (1+ 0.10)5
• = $16,105.10
• Calculation based on Table I:
• FV5 = $10,000 (FVIF10%, 5)
= $10,000 (1.611)
= $16,110 [Due to Rounding]](https://image.slidesharecdn.com/lec3-170425143546/75/Time-Value-of-Money-Business-Finance-16-2048.jpg)
![Formula
Lets drive PV from Future value formula
FVn = P0 (1+i)n
Re-arranging it
PV0 = P0 = FVn / (1+i)n or FVn [1/(1+i)n ]
General Present Value Formula:
or PV0 = FVn (PVIFi,n)
Note that the term [1/(1+i)n ] is simply the
inverse of future value of interest factor for “n”
periods.](https://image.slidesharecdn.com/lec3-170425143546/75/Time-Value-of-Money-Business-Finance-19-2048.jpg)
Uploaded byFaHaD .H. NooR
Time Value of Money - Business Finance
The document discusses the time value of money, explaining concepts such as simple and compound interest, future value, and present value. It illustrates how money's worth changes over time due to interest, emphasizing the preference for receiving money today rather than in the future. Examples are provided to show calculations for future values and present values based on different interest rates and time periods.
In this document
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Introduction to the Time Value of Money and the focus of the lecture.
Key concepts include interest rate, simple interest, compound interest, and loan amortization.
Time allows investment to generate interest, highlighting the connection between time and money.
Preference for receiving money today versus in the future illustrates the time value of money.
Simple interest is based on the principal amount, interest rate, and time periods: SI = P0(i)(n).
An example calculating simple interest on a $100 deposit at 8% for 10 years: $80 interest.
Future value formula for simple interest: FVn = P0 + SI, or FVn = P0 [1 + (i)(n)].
FV from a deposit and interest calculation: FV = $1,000 + $80 = $180.
Rearranged future value formula to calculate present value: PV0 = FVn / [1+(i)(n)].
Compound interest involves earning interest on previous interest and principal.
Example calculation for FV of a compound deposit of $1,000 at 8% interest over three years.
Yearly investment growth: First year: $1,000 grows to $1,080 with 8% interest.
Growth after two years: $1,000 to $1,166.64, showing interest earned on previous interest.
Continued growth: $1,000 to $1,259.71 after three years under compound interest.
General future value formula for any investment, including FV interest factors.
Calculation of future value for a $10,000 investment at 10% interest over 5 years: $16,105.
Understanding present value and how future cash flows are evaluated against current dollars.
Example calculating how much to invest today to receive $1,000 in 2 years at 7% interest.
Derivation of present value from future value formula using discounted rates.
Final calculations showing the present value of needing $1,000 in two years at 7%.
Determining the unknown interest rate from known present and future values.
Rearranging formula to solve for compound interest rate when future value is known.
Determining investment duration needed to reach a certain future value at known interest rates.
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Time Value of Money - Business Finance
- 1. Time Value ofMoney Lecture 3
- 2. The Time Valueof Money • The Interest Rate • Simple Interest • Compound Interest • Amortizing a Loan
- 3. Why Time? • Becauseit provides us with an opportunity to put our money to work and start to earn interest from now on. • TIME allows you the opportunity to postpone consumption and earn INTEREST.
- 4. The Interest Rate •Which would you prefer -- $10,000 today or $10,000 in 5 years? Obviously, $10,000 today. You already recognize that there is TIME VALUE TO MONEY Because you can invest it today and start to earn Interest. • In a world where cash flows are certain then rate of return can be used to express the time value of money and rate of interest allows us to to value cash flows.
- 5. Types of Interest •Simple Interest • Interest paid (earned) on only the original amount, or principal borrowed (lent). • Dollar amount of simple interest is a function of three variables. 1:original amount borrowed or lent 2:Interest rate for the time period 3:Number of time periods SI = P0(i)(n)
- 6. Simple Interest Example •Assume that you deposit $1,00 in an account earning 8% simple interest for 10 years. What is the accumulated interest at the end of the 10th year? • SI = P0(i)(n) SI=100(.08)(10)=80
- 7. Simple Interest (FV) •Future Value or terminal value • The value at some future time of a present amount of money or a series of payments evaluated at a given interest rate. • For any simple interest the future value of an account at the end of n periods • FVn= P0 + SI= P0 +P0 (i)(n)…………………………(1) • Or we can write this as • FVn= = P0 [1+(i)(n)]………………………………..(2)
- 8. Future Value • Whatis the Future Value (FV) of the deposit? • FV = P0 + SI • = $1,00 + $80 • = $180 • So FV= 100+[100(.08)(10)]=$180 Or FVn=100[1+(.08)(10)] =180
- 9. Simple Interest (PV) •Sometimes we need to proceed in opposite direction. • That is we know the FV of deposit at interest rate deposited for “n” years but we don’t know the principal originally invested…..so re arrangement of equation 2 is required to get present value. • FVn= = P0 [1+(i)(n)]…………..(2) • Pv0 P0 =FVn/ [1+(i)(n)] Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. So PV is the present value is the original amount which was deposited which is 100.
- 10. Compound Interest • Interestpaid(earned) on any previous interest earned as well as on principal borrowed(lent) • Interest paid on an investment is periodically added to the principal.so this is interest on interest or compounding.
- 11. Future Value Single Deposit •Assume that you deposit $1,00 at a compound interest rate of 8% for 3 years. • So at the end of year 2nd • 0 1 2 3 • $1,00 FV3
- 12. Future Value Single Deposit(Formula) • FV1 = P0 (1+i)1 = $1,00 (1.08) = $1,08 Compound Interest You earned $8 interest on your $1,00 deposit over the first year. This is the same amount of interest you would earn under simple interest.
- 13. Single Deposit • FV2= FV1 (1+i)1 = P0 (1+i)(1+i) = $1,00(1.08)(1.08) = P0 (1+i)2 = $1,00(1.08)2 = $1,16.64 • The initial deposit have grown to 108 at end of year one with 8% return. Going to second year 108 becomes 116.64 as $8 interest earned on initial deposit of $100 and $0.64 is earned on the $8 interest credited to our account at the end of first year.in other words interest is earned on previously earned interest. Hence the name is compound interest
- 14. Single Deposit • P3(1+i)(1+i)(1+i) = $1,00(1.08)(1.08)(1.08) = P0 (1+i)3 = $1,00(1.08)3 = $125.97
- 15. General Future ValueFormula FV1 = P0(1+i)1 FV2 = P0(1+i)2 General Future Value Formula: FVn = P0 (1+i)n or FVn = P0 (FVIFi,n) FVIFi,n (future value interest factor at i% for n Periods or terminal value interest factor) = (1+i)n
- 16. Example • Julie Millerwants to know how large her deposit of $10,000 today will become at a compound annual interest rate of 10% for 5 years. • Calculation based on general formula: • FVn = P0 (1+i)n • FV5 = $10,000 (1+ 0.10)5 • = $16,105.10 • Calculation based on Table I: • FV5 = $10,000 (FVIF10%, 5) = $10,000 (1.611) = $16,110 [Due to Rounding]
- 17. Present Value SingleDeposit (Graphic) • PV or discounted value(interest rate used to convert future values to present values) • We all realize that a dollar today is worth more than a dollar to be received after one, two or three years from now. Calculating the present value of future cash flows allows us to place all cash flows on a current footing so that comparison can be made in terms of today’s dollar.
- 18. Example • Assume thatyou need $1,000 in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount/interest rate of 7% compounded annually? • 0 1 2 7% $1000 PV0 PV1
- 19. Formula Lets drive PVfrom Future value formula FVn = P0 (1+i)n Re-arranging it PV0 = P0 = FVn / (1+i)n or FVn [1/(1+i)n ] General Present Value Formula: or PV0 = FVn (PVIFi,n) Note that the term [1/(1+i)n ] is simply the inverse of future value of interest factor for “n” periods.
- 20. Example Solution • PV0= FV2 / (1+i)2 • = $1,000 / (1.07)2 • = FV2 / (1+i)2 = $873.44 • So finally if we are promised to receive 1000 received today we would prefer to take 1000 today rather than after two years.
- 21. Unknown Interest Rateor discount rate • Some times we are faced with a time value of money in which we know both PV and FV as well as number of time periods involved. What is unknown is interest rate. • Example: • If Cesario invest $1000 today. he will receive $3000 in exactly 8 years. What is the compound rate(discount)?
- 22. Example • Lets startwith future value formula • FV8 = P0 (FVIFi,8) $3000=$1000(FVIFi,8) By re-arranging it FVIFi,8 =$3000/$1000=3 If remember (1+i)8 =3 (1+i) = 30.125 =1.1472 i=0.1472
- 23. Unknown Time periods •Some times we know PV, FV and compound interest rate but what we don’t know is the time period.so again we can drive it using future value formula…lets now jump on example. • For how long will it take for a investment of $1000 to grow to $1900.if we invested it at a compound annual interest rate of 10%? • Lets start with basic future value formula.