Valuing a Stream of Cash Flows
- Often time we will have multiple cash flows on our timeline.
- Investment (factory, new store, retirement), debt repayment (mortgage, student loan, interest), salary, bonds, stocks...
- We already have the tools to do this.
- Compute the PV of each cash flow.
- Sum the present values
Present Value of Cash Flows
$PV=C_0+\frac{C_1}{(1+r)}+\frac{C_2}{(1+r)^2}+\frac{C_3}{(1+r)^3}+...+\frac{C_N}{(1+r)^N}$
$FV=C_N+C_{N-1}(1+r)^1+C_{N-2}(1+r)^2+$
$C_{N-3}(1+r)^3+...+C_{N-(N-1)}(1+r)^{N-(N-1)}$
Example 1
You think you will be able to deposit $4,000 at the end of each of the next three years into an account earning 8% interest. You currently have $7,000? How much will you have in three years? Four years? Twenty years?
Solving Example 1: FV at Year 3
You think you will be able to deposit $4,000 at the end of each of the next three years into an account earning 8% interest. You currently have $7,000? How much will you have in three years? Four years? Twenty years?
Future Value (FV)= =FV(rate,nper,pmt,pv)= $21,803.58
Payment(PMT)= -$4000
Rate= 8%
NPER= 3
Present Value(PV)= -$7000
Solving Example 1: FV at Year 4
You think you will be able to deposit $4,000 at the end of each of the next three years into an account earning 8% interest. You currently have $7,000? How much will you have in three years? Four years? Twenty years?
Future Value (FV)= =FV(rate,nper,pmt,pv)= $27,574.87
Payment(PMT)= -$4000
Rate(I%)= 8%
N= 4
Present Value(PV)= -$7000
Solving Example 1: FV at Year 20
You think you will be able to deposit $4,000 at the end of each of the next three years into an account earning 8% interest. You currently have $7,000? How much will you have in three years? Four years? Twenty years?
Future Value (FV)= =FV(rate,nper,pmt,pv)= $215,674.56
Payment(PMT)= -$4000
Rate(I%)= 8%
N= 20
Present Value(PV)= -$7000
Example 2 - Your Turn
You are offered an investment that pays $1,000 at the end of each year for the next 4 years. The discount rate is 4%. What is the maximum you would be willing to pay for this investment.
- Solve using a timeline and formulas
- Solve using Excel function
Answer: $3,629,90
Example 3 Uneven cash flows
You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn 10% per year on your money, how much would you be willing to pay (today) for this investment?
Answer:$4,815.93
Hockey Example
Hockey Contracts
On July 1, 2018 two NHL superstars and Canadian heroes signed long term contracts. Drew Doughty decided to stay in LA for 8 years, $88 million. John Tavares left the failing NYI to join the future Stanley Cup Champion's, the blue and white, pride of Canada, Toronto Maple Leafs. Since he left his originally team he could only sign for 7 years. He signed a $77 million dollar deal. John was represented by CAA (headquartered in LA) and Drew represented himself. Note that each players AAV is $11 Million.
Salary Breakdown
Here is their salary breakdown:
Year |
Doughty |
Tavares |
1 |
$12,000,000 |
$15,900,000 |
2 |
$10,000,000 |
$15,900,000 |
3 |
$11,000,000 |
$12,000,000 |
4 |
$11,000,000 |
$9,350,000 |
5 |
$11,000,000 |
$7,950,000 |
6 |
$11,000,000 |
$7,950,000 |
7 |
$11,000,000 |
$7,950,000 |
8 |
$11,000,000 |
|
Contract PV
Rate |
Doughty |
Tavares |
Difference |
0.00% |
$88,000,000 |
$77,000,000 |
$11,000,000 |
1.00% |
$84,178,258.23 |
$74,430,049.81 |
$9,748,208.42 |
5.00% |
$71,140,691.83 |
$65,434,307.17 |
$5,706,384.65 |
8.00% |
$63,281,615.49 |
$59,811,659.07 |
$3,469,956.41 |
12.00% |
$54,739,700.70 |
$53,490,205.41 |
$1,249,495.29 |
15.03% |
$49,425,668.13 |
$49,425,668.13 |
$0.00 |
20.00% |
$42,347,646.72 |
$43,821,242.18 |
-$1,473,595.45 |
Note: Accounting for signing bonus and assuming salary is paid monthly over 12 months, the break-even occurs at 10.56%.