1. Topic 9
Time Value of Money

2. Topic 9: Time Value of Money
Learning Objectives
Calculate present value and future value of single amounts, annuities, annuities due, uneven cash flows and serial payments.
Calculate NPV and IRR and be able to apply the techniques to financial planning problems.

3. Topic 9: Present Value
The value today of a known amount or series of amounts to be received in the future, given a specified interest rate and number of periods for discounting
Formula for single sum
PV = FV / (1 + i)n

4. Topic 9: Present Value Example
What is the present value of $20,000 in 5 years assuming an 8% rate of return?
It will be worth $13,612
N = 5
I = 8
PMT = 0
FV = 20,000
Solve for PV

5. Topic 9: Future Value
The value that will be available when a present sum or series of periodic sums is invested, with earnings added, most often on a compound basis
Formula for a single sum
FV = PV(1 + i)n

6. Topic 9: Future Value Example
What is the future value of $5,000 invested today in a 4% CD for 3 years?
It will be worth $5,624
N = 3
I = 4
PV = 5,000
PMT = 0
Solve for FV

7. Topic 9: Rule of 72
A tool for estimating the approximate length of time it will take for a single sum of money to double in value at a given compound annual rate of interest
Divide 72 by the interest rate to produce the answer
Example: At a 6% rate of interest it will take approximately 12 years (72/6 = 12) for a lump sum of money to double in value

8. Topic 9: Annuity
A specified amount of money, paid or received at a specified uniform interval, for a specified period of time
Ordinary annuity
Payments or receipts are made at the end of each period
Annuity due
Payments or receipts are made at the beginning of each period

9. Topic 9: Annuity Example
How much should a client pay for an investment that will pay him $1,000 a year for 10 years given a 6% required rate of return?
The client should pay $7,360
N = 10
I = 6
PMT = 1,000
FV = 0
Solve for PV

10. Topic 9: Net Present Value
The present value of all cash outflows and cash inflows
It is how much a client should pay for an asset

11. Topic 9: Net Present Value Example
How much should a client be willing to pay for an investment with the following cash flows assuming a 7% interest rate?
End of Year 1 = $1,000
End of Year 2 = $2,000
End of Year 3 = ($5,000)
End of Year 4 = $10,000
The client should be willing to pay $6,229
Input the cash flows above starting with $0 as the first cash flow
Input 7% as the interest rate
Solve for NPV

12. Topic 9: Internal Rate of Return (IRR)
It is the rate of return on an investment given the cash inflows and outflows
The client wants the IRR to be equal to or greater than the client’s required rate of return

13. Topic 9: Internal Rate of Return Example
If a client pays $6,000 for the investment used in the NPV example, what will be the client’s IRR?
The client’s IRR will be 8.1%
Input the following cash flows
($6,000)
$1,000
$2,000
($5,000)
$10,000
Then solve for IRR which is 8.1%
The IRR will be higher than the 7% since the client paid less than the PV of the future cash flows (NPV>0)

14. Topic 9: Uneven Cash Flows
Typically you will be solving for how much a client should pay for the investment or how much they will have saved at the end of a period of time given the different cash flows from an investment

15. Topic 9: Uneven Cash Flows Example
How much should Peter pay for a bond with a 9% coupon that matures in 7 years if comparable bonds are yielding 10%?
Peter should pay $951
N = 7 x 2 = 14
Due to semi annual coupon payments
I = 10 / 2 = 5
Semi annual interest rate
PMT = 90 / 2 = 45
Semi annual coupon payment
FV = 1000
Solve for PV

16. Topic 9: Serial Payments
The calculation will involve an inflation adjusted rate of return
[(1 + nominal rate)/(1 + inflation rate) – 1] x 100
Example
9% yield in a year with 2.5% inflation
[(1 + 9%) / ( %) –1] x 100
6.34%

17. Topic 9: Annuity Due Serial Payment Example
Susan wishes to accumulate $50,000 in today’s dollars in 4 annual deposits, beginning immediately. The deposits will earn 7% after taxes, and the deposits must also increase each year by 3% to keep up with anticipated inflation. What should be the size of the first deposit?
Susan should deposit $11,354
Set for beginning-of-period payments
N = 4
I = [(1.07 ÷ 1.03) – 1] x 100 = 3.88
PV = 0
FV = 50,000
Solve for PMT

18. Topic 9: Ordinary Annuity Serial Payment Example
Susan wishes to accumulate $50,000 in today’s dollars in 4 annual deposits. The deposits will earn 7% after taxes, and the deposits must also increase each year by 3% to keep up with anticipated inflation. What should be the size of the first deposit?
Susan should deposit $12,149
Set for end-of-period payments
N = 4
I = [(1.07 ÷ 1.03) – 1] x 100 = 3.88
PV = 0
FV = 50,000
Solve for PMT which is 11,795
Add inflation that occurred between today and when the first payment is made by multiplying 11,795 x 1.03 = 12,149

19. Topic 9: Growing Annuity Formulas

20. Topic 9: Loan Amortization Example
Ella took out a $500,000 loan fixed at 6% interest for 30 years? How much is her monthly payment?
N = 30 x 12 = 360, I = 6/12 = 0.50, PV = 500,000, FV = 0, Solve for PMT which is $2,998
How much interest and principal would Ella have paid after 5 years of monthly payments (60 payments) and what is her remaining loan balance?
Without clearing your calculator from the problem above:
2nd, AMORT, 1, set, ↓, 60, set, ↓
$145,137 is the total amount of interest paid
$34,728 is the total amount of principal paid
$465,272 is the remaining loan balance

21. End of Topic 9