1. Chapter 7
The Timing and Value
of Cash Flows

2. Objectives Valuing Claims to Future Cash Flows: A Comparison Approach
The Basis of Time Value Calculations: The Compounding Process
The Present Value of a Single Cash Flow
The Future Value of a Single Cash Flow
The Present Value of a Multiple Cash Flow Stream
The Future Value of a Multiple Cash Flow Stream
The Rate of Return on an Investment

3. Valuing Claims to Future Cash Flows: A Comparison Approach
A time line shows the point or period in time when a cash flow actually occurs. The periods can be any length but usually are months or years.
Shortime Hotel - will pay $10 in dividends per share each year for the next three years. After three years, we will assume it simply goes out of business and its assets are worthless.
Sametime Hotel’s stock will pay dividends identical to Shortime’s. Sametime’s stock currently sells at $25.50 per share.
25.50 must also represent the price at which Shortime’s stock would sell today.

4. The Compounding Process

5. Present Value CF0 = CFn X 1 = CFn (1 + k)n (1 + k)n
Suppose we wanted to know how much money we would have to invest today to have $ five periods from now when our rate of return per period is 10%. The answer would be:
PV0 = $ X = (1 +.10)5 (1 +.10) = $100.00

6. Present Value of a Single Cash Flow
Table of present value factors (PVn,k) where
PVn,k = 1/(1 + k)n.
Example: the value today (PV0) of the $200,000 to be received in 2 years would be the amount of money we would have to invest today assuming a 20% return in order to have $200,000 two years from now.
PV0 = $200,000 x (1/(1+.20)2)
= $138,880 Or
PV0 = $200,000(PVn=2,k=20)
= $200,000(.6944)
Or – in Excel
= NPV(0.2,0,200000)

7. Present Value of a Multiple Cash Flow Stream
Finding the present value of a multiple cash flow stream where all of the cash flows are different involves simply finding the present value of each cash flow individually and summing those present values.
Example: The worth today of a share of Pleasant Valley’s stock should be equal to the present value of all of the dividends the owner of a share of Pleasant Valley stock will receive:
PV0 = $5(PVn=1,k=12) + $7(PVn=2,k=12) + $9(PVn=3,k=12)
+ $20(PVn=4,k=12)
= $5(.8929) + $7(.7972) + $9(.7118) + $20(.6355)
= $ $ $ $12.71
= $29.16
In Excel: = NPV(.12,5,7,9,20)

8. Future Value of a Single Cash Flow
You put $10,000 in a savings account paying 6% per year. You intend to leave the money in the account for five years and then withdraw and use it for a new car. You would like to know how much money will be in the account at the end of the five years.
FV5 = ?
FVn = CF0(FVn,k)
FV5 = 10,000 (FVn=5, k=6)
= 10,000 (1.3382)
Solving with Excel: =FV(.06,5,0,-10000,1)
OR: =POWER(1+.06, 5)*10,000

9. Future Value of a Multiple Cash Flow Stream
Example: suppose that you will make the following annual deposits into a savings account paying 6% interest per year and would like to know how much money will be in your account at the end of five years
FV5 = $1,000 (FVn=4, k=6)+ $1,500 (FVn=3, k=6)+ $1,200(FVn=2, k=6)
+ 1,800(FVn=1, k=6)+ $600
FV5 = $1,000 (1.2625) + $1,500 (1.1910) + $1,200 (1.1236)
+ 1,800 (1.0600)+ $ = $1, $1, , $1,908 + $600
= $6,905.32

10. Present Value of an Annuity Cash Flow Stream
A cash flow stream with equal cash flows for a finite period of time
Example: Suppose your company had offered to sell a hotel for $5,000,000 cash, but another company counteroffered with an offer of zero cash down and $1,200,000 per year for six years, with the first payment being made one year after the sale date.
If the required rate of return is 10%, one would simply find the factor in the present value of an annuity table for six years at 10% (4.3553) and multiply it by $1,200,000.
This would amount to a present value of $5,226,360 which is $226,360 greater than the cash offer. Hence, the seller should accept the payment plan.
In Excel: = PV(.10,6, ,0,0)

11. Future Value of an Annuity Cash Flow Stream
Example: suppose your hotel puts $50,000 a year into a special account to be used to refurbish the hotel’s rooms and public areas. If the money is invested to earn 6% per year and no money is drawn from the account for four years, how much money will be available for refurbishment at the end of four years?
FV4 = A(FVA n=4,k=6) = $50,000(4.3746) = $218,730
In Excel:
=FV(.06,4,-50000,0,0)

12. Another Annuity Example
Suppose you would like to set aside enough money today to be able to spend $10,000 each year for the next ten years on refurbishing your hotel. You intend to put the money in an account that will earn 7% per year. Thus the amount of money you will have to put in the account today will be equal to:
PV0 = $10,000(PVAn=10,k=7) = $10,000(7.0236) = $70,236
Or solving via Excel: = PV(.07,10,-10000,0,0)

13. Loan payments as Annuities
Example: Suppose the Munch–of–the–Time Restaurant is considering replacing its current dish machine. Of the cost of a new machine, $20,000 will be borrowed from a local bank. The loan will be paid off in five equal annual payments. The bank has said it will lend money to the restaurant at an interest rate of 10%. What will the annual loan payments be?
PV0 = A(PVA n, k) $20, = A(PVA n=5, k=10) = A(3.7908) A = $20,000/ = $5,275.93
Solving using Excel: = PMT(.10,5,-20000,0,0)

14. Loan Amortization Tables

15. Perpetuities
PV0 = CF1 / k
Example: suppose a share of preferred stock pays a dividend of $1.50 per year. Also suppose that the appropriate risk–adjusted discount rate for valuing those dividends is 10% (kE = 10%). The value today of a share of the preferred stock should equal the present value of all dividends received from owning it, or:
PV0 = CF1 / k = DIV1/ k = $1.50/ = $15.00

16. Perpetuities and the Sale Price of a Hotel
Example: suppose the owners of the Infinity Hotel intend to sell the hotel in five years. They estimate that the hotel will generate a constant cash flow stream of $500,000 per year forever from year six onward. They also estimate that the risk associated with those cash flows is equivalent to the risk associated with a 20% rate of return along the security market line. The sales price of the hotel in five years should thus be equal to the present value of $500,000 per year forever, or:
PV5 = CF1 / k = CF6 / k = $500,000/.2
= $2,500,000

17. Rate of Return on an Investment
Example: Maria’s Pizzeria owner Maria would like to borrow $5,000 today to help pay for an expansion of the dining room. A relative of Maria has offered to lend her the $5,000 if she pays him $10,060 back in five years. Maria isn’t sure if this is a good deal or not and would like to determine the annual rate of return she would be paying on this loan.
Solve for the PVn,k factor
PV0 = 5,000
PV0 = 5,000 = $10,060(PVn=5,k=?)
PVn=5,k=? = 5,000/10,060 = .4970
Go to the present value factor table
Find the factor closest to when n = 5 and read up to find the k associated with that factor.
For Maria’s loan, k turns out to be 15%—that is, PVn=5,k=15 = * The 15% is the annual compounded rate of return on the loan
In Excel: = RATE(5,0,5000,-10060,0)

18. Example: suppose that a hotel is considering the purchase of a new computerized front desk system for $200,000. The managers estimate that the net after–tax savings from installing the system will be $29,806 per year for ten years, when the system will need to be replaced. What annual rate of return will the investment in the new system generate?
PV0 = $200,000 = $29,806(PVAn=10,k=?)
PVA n=10,k=? = $200,000/$29,806 =
Reading across the n + 10 row of Exhibit 6.4, we find in the column under 8%
In Excel:
= RATE(10,29806, ,0.0

19. Non-Annual Compounding/Discounting Intervals
If I is the number of compounding intervals within a year (for example, I = 4 for quarterly compounding), then the appropriate future value and present value factors to use when k is the stated annual rate of return will be FVnI,k/I and PVnI,k/I.
Example: suppose a bank offers to pay an annual rate of 12% on an account compounded quarterly. What would be the effective annual return on the account? In this case, k/I = 12%/4 = 3%, and nI = 1 * 4 = 4. Suppose we placed $100 in this account. The future value of the $100 investment will be:
FV1 = $100(FVn=4,k=3) = $100(1.1255) = $112.55

20. Summary
The compounding process serves as the foundation for determining the present or future value of a cash flow and the rate of return on an investment.
When we handle numerical problems involving applications of time value principles, the steps in the solution are to draw a time line, put the relevant cash flows on the time line, locate the reference point, and then draw arrows from the cash flows to the reference point.
We can then set up our numerical calculation of present value, future value, payment, or rate of return.