Present Value
What is the formula to calculate the Present Value of a future amount to be received in n years, assuming a discount rate r?
PV= FV / ((1+r) ^n)
What is the formula to calculate the future value (FV) of a present amount after n years, assuming a compound return of r?
FV = PV (1+r)^n
When a project’s IRR exactly equals the required return, the project’s NPV is:
(1) Negative (2) Zero (3) Positive (4) Cannot tell
Option 2 is correct: The NPV is zero
How many variables or inputs are there in the Time Value of Money (TVM) equation, and what are they?
The TVM equation has 4 variables or inputs which are:
- PV
- FV
- r (interest rate)
- n (number of periods)
What is the equation to estimate the present value of a constant perpetuity? and of a growing perpetuity, assuming a compound return r, and a growth rate g in the second case
PV of a constant perpetuity
PV = CF1 of the perpetuity / r
PV of a growing perpetuity
PV = CF1 of the perpetuity / (r-g)
Whats is the equation to estimate the Future Value of an investment of $1000 with a maturity of two years, which pays an annual return for the first year of 4% and for the second year of 7%
FV = 1000 x (1+4%) x (1+7%) = $1,112.8
Estimate the IRR for an investment of 1,120 € (today) which will generate the following Cash Flows in subsequent years: Year 1 = 230; Year 2 = 730; Year 3 = 728;
Its IRR is 20% (obtained with Excel formula)
Within the Time Value of Money framework, explain the meaning of:
The compounding process
The discounting process
Compounding: The compounding process is the method of determining the future value (FV) of an amount of money after a certain number of years (nnn), using a given annual interest rate (the compound rate).
Discounting: The discounting process is the method of determining the present value (PV) of an amount of money to be received or paid in the future (nnn years from now), using a given annual interest rate (the discount rate).
What is the present value of $500 to be received in 5 years, assuming an annual interest rate of 5%.
It is $391.76?
What is the future value of $500 invested for 5 years at a compound interest rate of 5 percent.
It is 638.14?
A start-up plans to invest in a machine costing $2,000,000 with a 5-year life and no salvage value. The machine will produce $422,000 at the end of each year. If the project’s discount rate is 8%, determine the correct decision based on the IRR rule:
(1) Accept the investment
(2) Reject the investment
(3) Indifferent
Answer (2) is correct, the IRR is 1.81%, therefore lower than the required return and the NPV is negative -315,076
An investment pays 10% per year (APR), compounded quarterly. Which of the following is the effective annual rate (EAR)?
(1) 10.38% (2) 11.46% (3) 9.31% (4) 8.55%
Optión 1 is correct, 10.38%
obtained with TVM formula
(1+APR/n quarters)^n quarters = (1+EAR)
(1+ 10%/4)^4 = 1.1038, then substracting 1 we obtain the Effective Annual Rate
Estimate the fair price of an apartment using the growing perpetuity formula under the following assumptions:
Annual lease payments (after expenses) of €20,000, collected at the end of each year.
Lease payments are expected to grow at a constant annual rate of 1%.
Required annual return (expected return) is 5%.
Its fair price based on the mentioned assumptions would be 500,000
PV = 20,000 / (5%-1%) = 500,000
How much money will you have in 12 years if you invest €7,000 today in a fund that earns an 18% annual compound return?
(1) €51,013 (2) €36,637 (3) €83,816 (4) €191,751
you will have 51,013 obtained as
FV = 7000 x (1+18%)^12
On October 24, 2021, COMCAST shares were priced at $45. Two years later, on October 24, 2023, the price dropped to $30. What was the compound annual return of investing in COMCAST shares, assuming no dividends?
(1) -18.35% (2) -32.32% (3) +17.7% (4) +23.8%
The correct answer is (1) -18.35%, obtained either with the Excel formula for IRR with these tree values (-45;0;30) or by using the following equation for TVM and solving for r
(1+ Return in two years)= (1+Annual return)^2
(1 + ((45-30)/45)) = (1+ r)^2
What is the present value of a company that generates a cash flow of $100 today, growing at a constant annual rate of 2% indefinitely, if the discount rate is 10%?
It is $1,375 obtained as follows:
Next year Cash Flow = 100 x (1+2%) = 102
PV of all future Cash Flows = 102/(10%-2%) = 1,275
PV of the company = Today Cash Flow + PV future Cash flows = 100 + 1,275 = 1,375
Determine the present value today of an asset that provides annual payments of $500 indefinitely, beginning at the end of year 3. The discount rate is 8%.
It is $5,358
Perpetuity Value in t2 = 500/ 8% = 6250
PV in t0 = 6250 /(1+8%)^2 = 5,358.368
Suppose that starting today, October 29, 2025, you invest €1000 per year for three years in a deposit account that pays 8% compound interest. How much will you have in the deposit account on October 29, 2028?
There will be €3506.11, which I obtain by estimating the FV in Oct 2028 of the amounts invested:
FV1= 1000 x (1+8%)^3 = 1259.71
FV2= 1000 x (1+8%)^2 = 1166.4
FV3= 1000 x (1+8%) = 1080
FV = FV1+FV2+FV3 = 3506.11
Suppose your mortgage has monthly payments and an effective annual rate (EAR) of 6.17%. What is the annual percentage rate (APR)?
We know that
(1+EAR)=(1+APR/n periods)^n periods, so we solve for the APR
(1+6.17%) = (1+ APR/12)^12
(1+6.17%)^(1/12)=(1+APR/12) = 1.005
(1.005 - 1) x12 = APR = 0.0600 = 6.00%
What is the Present Value (PV) of a lottery that pays $1,000 per month for 12 months, starting next month, if the annual percentage rate (APR) is 6% (with monthly compounding)?
It is $11,619
We first estimate the monthly rate which is 0.50% (APR/12) and then we calculate the PV of each monthly payment (tus 12 PVs). Finally we sum all the PVs
1 1000 PV1= 1000/(1+0.5%)^12
2 1000 PV2= 1000/(1+0.5%)^2
... 1000 PVj= 1000/(1+0.5%)^j
12 1000 PV12= 1000/(1+0.5%)^12