1. Chapter 3 - Support
The Time Value of Money

2. Remember? Future Value Single Deposit (Graphic)
Assume that you deposit $1,000 at a compound interest rate of 7% for 2 years. 7%
$1,000
FV2

3. Future Value Excel Formula [Calculates a single value in the future based on current expectations]
Excel function is =FV(rate,nper,pmt,pv,type)
rate: the interest rate per period
nper: the total number of compounding periods
pmt: the payment made each period and cannot change over the life of the annuity ($0 in a single cash flow)
pv: the present value you begin with
type: is the number 0 (normal period-end) or 1 (beginning of the period) and indicates when payments are due/occur

4. Future Value Example
Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other present value problems on the “Future Value” tab. Students may find this much easier!

5. Story Problem Revisited
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.
10%
$10,000
FV5

6. Future Value Example
Refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) on the “Future Value” tab. This is the identical answer as our other methods in one formula shown in the boxed area above!

7. Another Future Value Story Problem
John and Mary are saving for retirement and currently have $127, as a nest egg.
John indicates that they plan to retire 25 years from today while Mary expects that a 6% rate of return is appropriate for their risk level given historical returns.
Calculate how large the account is expected to grow.

8. “New” FV Story Problem
Refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) on the “Future Value” tab.

9. “New” FV Story Problem
John and Mary will have their $ 127, investment grow to $ 548, in 25 years if they earn exactly 6% each year.
Note that the Excel answer is a ‘positive’ amount. This indicates that John and Mary DEPOSITED $ 127, (the negative amount as they have less cash) to receive the positive $548, (when they receive cash at retirement when they WITHDRAW the funds).

10. Remember? Present Value Single Deposit (Graphic)
Assume that you need $1,000 in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 7%
$1,000
PV0
PV1

11. Present Value Excel Formula [Calculates a single current value based on future expectations]
Excel function is =PV(rate,nper,pmt,fv,type)
rate: the interest rate per period
nper: the total number of discounted periods
pmt: the payment made each period and cannot change over the life of the annuity ($0 in a single cash flow)
fv: the future value you expect to attain
type: is the number 0 (normal period-end) or 1 (beginning of the period) and indicates when payments are due/occur

12. Present Value Example
Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other present value problems on the “Present Value” tab. Students may find this much easier!

13. Story Problem Revisited
Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000 in 5 years at a discount rate of 10%.
10%
$10,000
PV0

14. Present Value Example
Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other present value problems on the “Present Value” tab. Students may find this much easier!

15. Another Present Value Story Problem
John and Mary are expecting to build a $100,000 nest egg to use to travel the world upon retirement. They would like to know how much they need to set aside today to reach this goal.
John indicates that they will retire 20 years from today while Mary thinks that a 6% rate of return is appropriate for their risk level. Calculate how much they need to set aside today.

16. “New” PV Story Problem
Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xls) to use in developing their own solutions to other present value problems on the “Present Value” tab.

17. “New” PV Story Problem
John and Mary need to set aside $31, today if they earn exactly 6% each year for the next 20 years to reach their goal.
Note that the Excel answer is a ‘negative’ amount. This indicates that John and Mary will need to DEPOSIT this amount of money (they have less cash) to receive the positive $100,000 (when they receive cash) they would WITHDRAW monies.

18. Remember? Double Your Money!!!
Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?
We will use the “Rule-of-72”.

19. NOTE: The same answer! Solved in only one cell in Excel!!
Output: The Periods!
Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other interest rate problems on the “Periods” tab. Students may find this much easier!
NOTE: The same answer! Solved in only one cell in Excel!!

20. Remember? Double Your Money!!!
What if you were given the periods (years) and wanted to solve for the interest rate? Wouldn’t it be the same concept for the rule of 72?
Yes! We simply need to solve using a different function in Excel called “rate”.

21. Output: The Interest Rate!
Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other interest rate problems on the “Rate” tab.
Note that we “guessed 12% as 12 x 6 = 72. The answer was very close!

22. Remember? Example of an Ordinary Annuity -- PVA
Cash flows occur at the end of the period 7%
$1, $1, $1,000
$934.58
$873.44
$816.30
$2, = PVA3
PVA3 = $1,000/(1.07) $1,000/(1.07) $1,000/(1.07)3
= $ $ $ = $2,624.32

23. Present Value of an Annuity [Calculates a single current value based on future expectations]
Excel function is =PV(rate,nper,pmt,fv,type)
rate: the interest rate per period
nper: the total number of payments or periods
pmt: the payment that is made/received each period and cannot change over the life of the annuity
fv: a single future value you expect to receive (can be $0)
type: is the number 0 (normal period-end) or 1 (beginning of the period) and indicates when payments are due/occur

24. PV Annuity Example (Ordinary Annuity)
Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other present value problems on the “PV Annuity” tab. Students will find this much easier!

25. “New” PV Annuity Example (Ordinary Annuity)
John and Mary are trying to build a nest egg to use in the future. They would like to know how much they need to set aside in a single lump sum today to be equivalent to investing $10,000 each year starting one year from today to reach this goal.
John indicates that they will use the money 20 years from today while Mary thinks that a 6% rate of return is appropriate for their risk level.
Calculate the equivalent present value of this ordinary annuity stream.

26. “New” PV Annuity Example (Ordinary Annuity)
Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other present value problems on the “PV Annuity” tab. Students will find this much easier!

27. “New” PV Annuity Example (Ordinary Annuity)
John and Mary need to set aside $127, today to be equivalent to setting aside $10,000 per year at exactly 6% each year for the next 25 years.
In this case, John and Mary need to decide which is their preference. This sum will grow to exactly the same as the future value of an ordinary annuity (see slides 7 to 9).

28. Remember? Example of an Ordinary Annuity -- FVA
Cash flows occur at the end of the period 7%
$1, $1, $1,000
$1,070
$1,145
FVA3 = $1,000(1.07) $1,000(1.07)1 + $1,000(1.07)0
= $1,145 + $1,070 + $1, = $3,215
$3,215 = FVA3

29. Future Value of an Annuity [Calculates a single current value based on future expectations]
Excel function is = FV(rate,nper,pmt,pv,type)
rate: the interest rate per period
nper: the total number of payments or periods
pmt: the payment that is made/received each period and cannot change over the life of the annuity
pv: a single present amount you begin with (can be $0)
type: is the number 0 (normal period-end) or 1 (beginning of the period) and indicates when payments are due/occur

30. Future Value Annuity Example (Ordinary Annuity)
Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other present value problems on the “FV Annuity” tab. The same answer as before is arrived at in a single step in Excel - one formula ‘=FV(.07,3,-1000,0,0)’!

31. “New” Future Value Annuity Example (Ordinary Annuity)
John and Mary are trying to build a nest egg to use in the future. They would like to know how much they need to set aside in a single lump sum today to be equivalent to investing $10,000 each year starting one year from today to reach this goal. (See slides 21 to 23 and also 7 to 9)
John indicates that they will use the money 20 years from today while Mary thinks that a 6% rate of return is appropriate for their risk level.
Calculate the equivalent present value of this ordinary annuity stream.

32. “New” Future Value Annuity Example (Ordinary Annuity)
Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other present value problems on the “FV Annuity” tab. Students will find this is the exact same answer as those derived using PV of an annuity and then Future Value!

33. “New” Future Value Annuity Example (Ordinary Annuity)
John and Mary will accumulate nearly $550,000 by investing $10,000 per year at exactly 6% each year for the next 25 years.
In this case, note that this result is equivalent to the future value of a single sum where John and Mary needed to set aside over $127,000 to generate this sum.

34. Remember? Example of an Annuity Due – PVAD
Cash flows occur at the beginning of the period 7%
$1, $1, $1,000 $ $
$2, = PVADn
PVADn = $1,000/(1.07)0 + $1,000/(1.07) $1,000/(1.07)2 = $2,808.02

35. PV Annuity Example (Annuity Due)
Students may refer to the supporting Excel file for Chapter 3 to use in developing their own solutions to other present value annuity problems.

36. “New” PV Annuity Example (Annuity Due)
John and Mary are trying to build a nest egg to use in the future. They would like to know how much they need to set aside in a single lump sum today to be equivalent to investing $10,000 each year starting today to reach this goal.
John indicates that they will use the money 25 years from today while Mary thinks that a 6% rate of return is appropriate for their risk level.
Calculate the equivalent present value of this annuity due stream.

37. “New” PV Annuity Example (Annuity Due)
Students may refer to the supporting Excel file for Chapter 3 to use in developing their own solutions to other present value annuity problems.

38. “New” PV Annuity Example (Annuity Due)
John and Mary need to set aside $135, today to be equivalent to setting aside $10,000 per year at exactly 6% each year for the next 25 years.
In this case, John and Mary need to decide which is their preference. This sum will grow to exactly the same as the future value of an ordinary annuity (see slides 24 to 26) plus EXTRA interest.
EXTRA INTEREST: So the amount is one years interest higher for each payment or $127,833.56*6% = $7, higher PV!

39. Annuity Due
An annuity due is used when the cash flow occurs at the beginning of the period
As before, you see the value is higher by an amount equal to i% times the ordinary annuity value.
Present values of annuities will be larger because each cash flow is “discounted” one less period. See previous examples.
Future values of annuities will be larger because each cash flow gets compounded one “extra” period. No examples shown here.

40. Mixed Flows Example
Julie Miller will receive the set of cash flows below. What is the Present Value at a discount rate of 10%.
10%
$ $600 $400 $400 $100
PV0

41. Solve a “mixed flows” problem using the NPV function
Period
Cash Flows
Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other present value problems on the “Mixed Flows” tab.
Simply type in the cash flows in the green column and the correct interest rate in yellow and you get your answer! 1 $
600.00
Interest Rate: 2 $
600.00
10%
(discount rate) 3 $
400.00 4 $
400.00
Present Value*:
$1,677.15 5 $
100.00 6
=NPV(F3,C3:C22) 7 8
* Do NOT include cash 9
flows in period 0. Please 10
add or subtract these flows 11
from the final answer above. 12
(Cell F6) 13 14 15 16 17 18 19 20

42. Solve a “mixed flows” problem using the NPV function
Period
Cash Flows
Students can also solve this in one formula step again!
=npv(.10, 600, 600, 400, 400, 100)
This will generate the same and correct output. The model on the right is designed to be slightly more flexible in solving these types of problems.
Remember the NPV function, because we will use it later! 1 $
600.00
Interest Rate: 2 $
600.00
10%
(discount rate) 3 $
400.00 4 $
400.00
Present Value*:
$1,677.15 5 $
100.00 6
=NPV(F3,C3:C22) 7 8
* Do NOT include cash 9
flows in period 0. Please 10
add or subtract these flows 11
from the final answer above. 12
(Cell F6) 13 14 15 16 17 18 19 20

43. Remember? BWs Effective Annual Interest Rate
Basket Wonders (BW) has a $1,000 CD at the bank. The interest rate is 6% compounded quarterly for 1 year. What is the Effective Annual Interest Rate (EAR)?
EAR = ( 1 + 6% / 4 ) = = or 6.14%!

44. BWs Effective Annual Interest Rate
We can use a single function in Excel again to solve for the Effective Annual Interest Rate (EAR)
Nominal
Periods
Effective
Formula Used 6% 1
6.0000%
=EFFECT(B3,C3) 6% 2
6.0900%
=EFFECT(B4,C4) 6% 4
6.1364%
=EFFECT(B5,C5) 6% 6
6.1520%
=EFFECT(B6,C6) 6% 12
6.1678%
=EFFECT(B7,C7) 6%
365
6.1831%
=EFFECT(B8,C8) 6%
1,000,000
6.1837%
=EFFECT(B9,C9)
So it is very simple to solve for the effective rate using Excel. Note that a large number of periods per year can be used to approximate continuous compounding!

45. Remember? Amortizing a Loan Example
Julie Miller is borrowing $10,000 at a compound annual interest rate of 12%. Amortize the loan if annual payments are made for 5 years.
Step 1: Payment
PV0 = R (PVIFA i%,n)
$10,000 = R (PVIFA 12%,5)
$10,000 = R (3.605)
R = $10,000 / = $2,774

46. Remember? Amortizing a Loan Example
The first step is to use the “PMT” function to determine the yearly (in this case) payment on the loan
Now you can use Excel to easily create the table you see below! Refer to ‘VW13E-03.xlsx’ on the ‘Effect and Loan’ tab.
Amortizing a loan
Step 1: Calculating the loan amount
Rate:
12.00% ï
Interest rate per period (year in this case)
nper: 5 ï
Number of periods (5 years in this case) pv $
10,000.00 ï
Beginning loan balance today (positive) fv $ - ï
Ending loan balance at end of periods
payment:
($2,774.10) ï
Payment needed (negative)
Step 2: Create a table
Period
Beginning Bal
Payment
Interest in Period
Principal in Period
Ending Balance $
10,000.00 1 $
10,000.00
$2,774.10 $
1,200.00
$1,574.10 $
8,425.90 2 $
8,425.90
$2,774.10 $
1,011.11
$1,762.99 $
6,662.91 3 $
6,662.91
$2,774.10 $
799.55
$1,974.55 $
4,688.37 4 $
4,688.37
$2,774.10 $
562.60
$2,211.49 $
2,476.87 5 $
2,476.87
$2,774.10 $
297.22
$2,476.87 $
0.00