Present Value (Chapter 2)
Present Value (Chapter 2)
Present Value (Chapter 2)
Exercise set 0:
CFt = PV0 ∗ (1 + r )t
Where,
I CFt = the cash-flow in t years.
Exercise set 1:
Exercise set 1:
CFt = PV0 ∗ (1 + r )t
Present Value - flip the question
CFt
PV0 =
(1 + r )t
Exercise Set 4:
I Suppose that the market-appropriate rate of return is 8% per
year. What is the present value if...?
(a) the project makes multiple payouts, 10M in year 1, 10M in
year 2, 10M in year 3, 10M in year 4, 10M in year 5, and no
other payments thereafter.
1
1− (1+r )T
PV0 = CF ∗
r
CF
PV0 =
r
CF
PV0 =
r −g
Exercise Set 5:
I Suppose that the market-appropriate rate of return is 8% per
year. What is the present value if...?
(a) the project makes payments of 10M every year forever, starting
in 1 year.
(b) the project makes a first payment of 10M in year 1, and
payments growing at 5% per year forever.
PV of (In)finite Annuity Starting in “n” Years
I Step 1 gives the ‘project’s value in year n-1’. Step 2 gives the
present value of the ‘project’s value in year n-1’.
PV (In)finite Annuity Starting in t Years - Exercise
Exercise Set 6:
I Suppose that the market-appropriate rate of return is 8% per
year. What is the present value if...?
(a) the project makes five consecutive annual payments of 10M
each starting in year 4.
(b) the project makes five consecutive annual payments growing at
a 10% rate, starting with 10M in year 5.
(c) the project makes infinite annual payments of 10M each
starting in year 6.
(d) the project makes infinite annual payments growing at a 5%
rate, starting with 10M in year 7.
PV of (In)finite “Annuity Due”
I Step 1 gives the ‘project’s value in year -1’. Step 2 gives the
present value of the ‘project’s value in year -1’.
PV (In)finite Annuity Starting in t Years - Exercise
Exercise Set 7:
I Suppose that the market-appropriate rate of return is 8% per
year. What is the present value if...?
(a) the project makes five consecutive annual payments growing at
a 10% rate, starting with 10M today.
(b) the project makes infinite annual payments growing at a 5%
rate starting with 10M today.
Definitions & Exercises
I Opportunity cost of capital: the rate “r” is also called like this
because it’s the return that investors could get by investing in
other financial assets with the same risk as the project.
I Exercise. The return on US bonds is 2%. The average return
on US stocks is 10%. The average return of tech-stocks is
16%. Find the cost of capital if:
(a) Your project has no risk.
(b) You plan to buy a conglomerate firm.
(c) You plan to develop a new software.
Exercise Set 8:
I Which of the following projects increase value for
shareholders?
1. Exercise 4(a) if investment cost is 45M.
2. Exercise 4(b) if investment cost is 45M.
3. Exercise 5(a) if investment cost is 125M.
4. Exercise 5(b) if investment cost is 125M.
5. Exercise 6(a) if investment cost is 45M.
6. Exercise 6(b) if investment cost is 45M.
7. Exercise 6(c) if investment cost is 125M.
8. Exercise 6(d) if investment cost is 125M.
9. Exercise 7(a) if investment cost is 45M
10. Exercise 7(b) if investment cost is 125M.
Useful Applications 2: Mortgage payments
Loan value
CF =
T-year annuity factor
Exercise Set 9:
(a) Verify that you pay the loan after 5 years, and calculate the
annual amortization of the loan.
I Example: you are saving to buy a car. You estimate that once
you graduate you could save 10K a year and earn an 8%
return. How much will you have after five years?
I Long way: calculate the future value of every cash flow (using
future value formula) and sum them.
I Short way: calculate the present value of the cash flows, then
calculate the future value of the present value.
(a) Use the short method to find out how much will you be able
to spend on the car. (hint: T=5, CF=10K)
How Interest is Paid and Quoted
EAR = [1 + (r /m)]m − 1
EARcont = e r − 1