1. Chapter 4 Discounted cash flows and valuation

2. Learning objectives
After studying this presentation, you should be able to: 4.1 explain why cash flows occurring at different times must be adjusted to reflect their value as of a common date before they can be compared, and calculate the present value and future value for multiple cash flows 4.2 describe how to calculate the present value and future value of an ordinary annuity and how an ordinary annuity differs from an annuity due
Learning objectives

3. Learning objectives
4.3 explain what perpetuities are, where we see them in business, and calculate the present values of perpetuities 4.4 discuss why the effective annual interest rate (EAR) is the appropriate way to annualise interest rates, and calculate EAR.
Learning objectives

4. Multiple cash flows Future value/present value of multiple cash flows:
Solving future/present value problems with multiple cash flows:
Draw timeline to ascertain each cash flow is placed in correct time period.
Calculate future/present value of each cash flow for its time period.
Add up the future/present values.
Learning objective 4.1

5. Multiple cash flows Multiple cash flows: Future value & Present value
We have many types of cash flows:
1- Mixed stream of cash
2- Annuities
A- Ordinary Annuities
B- Annuity due
C- Perpetuities
Learning objective 4.1

6. Types of cash flows PV=? FV=?
1- Mixed stream of cash
Year 0
Year 2
Year 1
Year 3
$10
$15 $8
PV=?
FV=?
unequal Cash flows occurring at different times

7. A. ordinary annuity: cash flows occur at the end of the period
Types of cash flows
2- Annuities
A. ordinary annuity: cash flows occur at the end of the period
Year 0
Year 2
Year 1
Year 3
PV=?
FV=?
$10
$10
$10
B. Annuity Due: cash flows at the beginning of the period
Year 0
Year 1
Year 3
Year 2
PV=?
FV=?
$10
$10
$10

8. Types of cash flows C. Perpetuities PV=? FV=? Year 2 Year 0 Year 1 $10

9. Multiple cash flows
1- Mixed stream of cash
Future value
Example: Assuming that the relevant interest rote is 5% per annum, value this contract as at 1 August 2017.
Answer:
Deal with each one as single amount cash, the $6000 is in future state, so keep as it is:
Future value Single amount
FV = $10000(1.05)^1.5 + $6000 = $16,759.30

10. Multiple cash flows Answer: Future value: Mixed stream of cash
Example: If the interest rate at the bank is 8 per cent and you can save $6000 now, $8000 at the end of the first year and $10,000 at the end of the second year, how much money will you have to come up with at the end of the third year?
Answer:

11. Multiple cash flows
Present value: Mixed stream of cash
Example: Assuming that the relevant interest rote is 5% per annum, value this contract as at 1 Feb 2014.
$ $
PV= $

12. Multiple cash flows 2- Annuities
An annuity is a stream of equal cash flows, equally spaced in time.
A- Ordinary Annuities
B- Annuity due
C- Perpetuities
Learning objective 4.2

13. Future value of Ordinary annuity:
Multiple cash flows
2- Annuities
A- Ordinary annuities whose cash flow payments occur at the end of the period.
Future value of Ordinary annuity:
Future value annuity calculations usually involve finding what a savings or investment activity is worth at some future point.
For example, saving periodically for vacation, car, house, or retirement.
Find future value Ordinary annuity by finding FV for each individual cash flow using equation 3.1 then add up all together , which will take time or can using Future value of annuity equation 4.2, see next examples
Learning objective 4.2

14. Multiple cash flows Future value of Ordinary Annuities :
Example1: you put $1000 in savings account today and another $1000 a year from now. If the interest is 10 % per year, how much money will you have at the end of 2 years?
Eq. 3.1
Learning objective 4.1

15. Multiple cash flows Future value of Ordinary Annuities :
Example2: you put $1000 in savings account today and another $1000 a year from now. If the interest is 10 % per year, how much money will you have at the end of 2 years?
Learning objective 4.1
Eq. 3.1 not good for long periods, so use Eq.4.1 next

16. Multiple cash flows Equation of Future value Ordinary Annuities

17. Multiple cash flows Future value of Ordinary Annuities :
Suppose that you plan to save $1000 at the end of every year for 4 years. If your bank pays 8 per cent interest a year, how much money you will have at the end of the period?
equation 4.2

18. Time line for Future value of a 4-year Ordinary annuity for last example

19. Multiple cash flows Present value of Ordinary Annuities :
Present value of three cash flows – a time line for a 3-year loan with a payment of $1000 at the end of each year and an annual interest rate of 7 per cent:
This will take time to find each one PV,
so use PV equation 4.1 for PV Ordinary annuities, see next

20. PV Ordinary Annuities:
Multiple cash flows
PV Ordinary Annuities:
PV equation 4.1
Or you can write:

21. Multiple cash flows
Present value of Ordinary Annuities
Example, that a financial contract pays $2000 at the end of each year for 3 years, if the interest rate is 8 per cent, what is the present value of this contact?
Answer
PV equation 4.1
𝑃𝑉𝐴= $ × 1− 1 (1+0.08) 3

22. Multiple cash flows Finding monthly or yearly payment CF
A very common problem in finance is determining the payment CF schedule for a loan on a consumer asset, such as a car or a home that was purchased on credit.
equation 4.1
equation 4.2

23. Multiple cash flows 𝑃𝑉𝐴= CF 0.66667 × 1− 1 (1+0.66667) 300
Finding monthly or yearly payment CF
Example, you have just purchased a $400,000 apartment on the Gold Coast. You were able to obtain a 25-year home loan at 8 per cent. What are your monthly payments?
Answer
300 payments (12 months × 25 years)
interest is % (8%÷ 12 months = % per month)
𝑃𝑉𝐴= CF × 1− 1 ( ) 300
CF= $3,087.28
Loan payment Per month

24. Multiple cash flows B- Annuities due:
Annuity is called an annuity due when there is an annuity with payment being incurred at beginning of each period rather than at end.
Annuity due value = Ordinary annuity value ×(1+i)
Eq. 4.4
×(1+i)
FVA(Annuity due)
Learning objective 4.3
×(1+i)
PVA Annuity due =

25. ×(1+i) Annuities due example
PV Annuity due example: what is the PV of annuities made up of four $1000 cash flows with first payment start immediately and carry an 8 per cent interest rate?
Answer
×(1+i)
PVA Annuity due =
PVA Annuity due = $ × 1− 1 (1+0.08) 4 ×(1+0.08)
= $3,577
What if it was Ordinary annuities? PV then = $3,312

26. Multiple cash flows C- Perpetuity:
contract calling for cash flow payments to continue forever.
- In share markets, preference shares issues are considered to be perpetuities, with issuer paying a constant dividend to holders.
- Equation for present value of a perpetuity can be derived from present value of an annuity equation with n tending to infinity:
Learning objective 4.2
Eq.4.3

27. Perpetuity
Example, suppose you had a great experience during university and decided to create or endow a scholarship fund for finance students.
The goal of the fund is to provide the university with $100,000 of financial support for finance students each year forever.
If the rate of interest is 8 per cent, how much money will you have to give the university to provide the desired level of support?
Answer:
we find that the present value of the perpetuity is:
Thus, a gift of $1,250,000 will provide a constant annual payment of $100,000 to the university forever

28. The effective annual interest rate (EAR)
Interest rates can be quoted in financial markets in variety of ways.
Most common quote, especially for a loan, is annual percentage rate (APR) or the nominal rate.
APR represents simple interest accrued on loan or investment in a single period; annualised over a year by multiplying it by appropriate number of periods in a year.
Learning objective 4.4

29. The effective annual interest rate
Calculating the effective annual rate (EAR):
Correct way to calculate annualised rate is to reflect compounding that occurs; involves calculating effective annual rate (EAR).
Effective annual interest rate (EAR) defined as annual growth rate that takes compounding into account.
Learning objective 4.4

30. The effective annual interest rate
Calculating the effective annual rate (EAR):
EAR =
m is the number of compounding periods during a year.
EAR conversion formula accounts for number of compounding periods, thus effectively adjusts annualised interest rate for time value of money.
EAR is the true cost borrowing and lending.
equation 4.5
Learning objective 4.4

31. Example on the effective annual rate (EAR):
Suppose you borrow $100 on your bank credit card and plan to keep the balance outstanding for 1 year.
The credit card’s stated interest rate is 1 % per month (compounded monthly).
What is the effective annual rate (EAR)?
Answer:
The bank’s annual percentage rate (APR) is 1% per month or 12% annually (nominal rate)
The effective annual rate (EAR)=
= ( /12)12 – 1
0.1268, or 12.68%

32. Summary
Adjusting cash flows that occur at different times to reflect their value as of a common date for comparison.
Present and future value for multiple cash flow and an ordinary annuity.
Difference between an ordinary annuity differs from an annuity due.
Perpetuities in business and calculating the present value.
effective annual interest rate (EAR) is the appropriate way to annualise interest rates and calculate EAR.
Presentation summary