Present Value
What is the equation to estimate the present value of a future vaue obtained in n years, assuming a discount rate r.
PV= FV / ((1+r) ^n)
What is the equation to estimate the future value in n years of a present vaue, assuming a compound return r.
FV = PV (1+r)^n
If a project’s IRR equals your required return, its NPV is: (1) negative; (2) zero; (3) positive; (4) We need more information to infer whether is zero, negative or positive.
Option 2 is correct: The NPV will be zero
How many variables does the TVM equation have? and what are they?
The TVM equation has 4 variables which are the PV, the FV, the compound annual interest rate r, and the number of years n.
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 framework of Time Value of Money, what does it mean the compounding process? and the discounting process?
The compounding processe is the process of finding a Future Value in a given number of years (year n), based on an annual interest rate (compound rate)
The discounting process is the process of finding the Present Value or an amount of money obtained or paid in a given number of years (year n), based on an annual interest rate (discount rate)
The present value of $500 to be received in 5 years at an annual interest rate of 5 percent.
What 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?
You have been hired by a start-up that is considering buying a machine. The machine implies an initial investment of $2,000,000, will last for 5 years, and will have no terminal value. The machine will generate an end-of-year positive cash flow of $422,000 each year for the next five years. The appropriate discount rate for this project is 8%. Using the IRR rule, which will be the correct decision? (1) Accept (invest) in the machine; (2) Reject (not invest) in the machine; (3) You will be indifferent to this investment
Answer (2) is correct, the IRR is 1.81%, therefore lower than the required return and the NPV is negative -315,076
An investment at 10 percent APR compounded quarterly has an effective annual rate (EAR) of: (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
Using the growing perpetuity formula, estimate the price of an apartment under the following assumptions: annual lease payment after expenses of €20,000, to be collected at the end of the years, which is estimated to grow at a constant annual rate of 1%; and annual requiered return (or expected return) of 5%.
Its fair price based on the mentioned assumptions would be 500,000
PV = 20,000 / (5%-1%) = 500,000
How much money will I have in 12 years if I invest now €7,000 in a fund that generates a compound annual return of 18%? (1) 51,013; (2) 36,637; (3) 83,816; (4) 191,751
I will have 51,013 obtained as
FV = 7000 x (1+18%)^12
COMCAST shares’ price on October 24, 2021, was 45, if the shares price two years later (on October 24, 2023) is 30, which has been the compound annual return of investing in COMCAST shares assuming no dividents payments? (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 PV of a company which generates cashflows of $100 today, growing at an annual rate of 2% per year to the infinite (or a very very long period), and 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
What is the PV of an asset that pays $500 forever from the end of year 3, with discount rate=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, 2023, you invest €1000 per year for three years in a deposit that pays a compound interest of 8%, how much money will you have in the deposit on October 29, 2026?
There will be €3506.11, which I obtain by estimating the FV in Oct 2026 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 with monthly payments has an EAR of 6.17%, what is the APR of the mortgage?
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 PV of a lottery that pays $1000 for each one of 12 monthly payments starting next month, with annual percentage rate of 6% (monthly annual rate of 6%/12)?
It is $11,619
We first estimate the monthly rate which is 0.50% and then estimate the 12 PVs using this rate and the month as the n periods to be discounted. Finally we sum all the PVs
1 1000 PV1= 1000/(1+0.5%)^1
2 1000 PV2= 1000/(1+0.5%)^2
... 1000 PVj= 1000/(1+0.5%)^j
12 1000 PV12= 1000/(1+0.5%)^12