You can calculate the lease payments in 3 steps ?
Step A: Calculate the Net Present Value (NPV) of the amount to be amortized.
Step B: Calculate the annual after-tax required lease income. By
taking your answer from Step A as an NPV then use the number of years
and the required rate of return to get the required payment.
Step C: Calculate the lease payment by taking the answer from Step B
and adjusting for taxes. You?ll divide it by (1 - the tax rate).
In Step A it?s very important to note that the salvage value of the
machine ($35,000) will get discounted for net present value (NPV) ?
but not for tax effects.
Why no taxes? Because the leasing company is already showing a
-$40,000 income ? and the salvage simply reduces the loss. In other
words, a company that?s losing money doesn?t have to worry about
paying taxes on any gains that are less than its losses.
The calculations in the three steps are all in this Excel spreadsheet,
which your browser should be able to read, even if you don?t have
Excel itself. By netting up the year 5 salvage and depreciation, we
avoid the potential error mentioned in the previous paragraph. If you
do have Excel, you can download and change the spreadsheet:
NPV Cash Flow
If you have any questions about these calculations, please let us know
via a Clarification Request.
Request for Answer Clarification by
14 Aug 2005 15:48 PDT
The excel sheet is very neat, however I am new in this finance classes
and I was wondering if you could do these 2 things for me. first in
step A.per the teacher's hints, the amount to be amortized needs to be
calculated as the cost of the machine less PV of the after tax salvage
value of the machine and less the PV of the depreciation tax shield.
Second, are you able to show me the finance formulas used to calculate
all these figures, step by step? I need to know the formulas before I
can start using excel. That is all. Thanks for your help.
Clarification of Answer by
15 Aug 2005 03:19 PDT
In Step A, the cash flows should be clear for each year. We use a
?Year 0? convention for payments occurring today. And the NPV of cash
flows are figured as:
NPVi = CFi / (1 + r)^i
NPVi = NPV of cash flow after year i
CFi = cash flow at end of year i
r = discount rate
i = years completed
Cash flow in year 0 is -$200,000.
Cash flow in year 1, 2, 3, 4 is a TAX CREDIT of 0.40 * depreciation.
However, remember that we have the salvage value coming back in the
sale of the equipment in year 5.
Now rather than do something like the following for year 2, typical
cash flow problems set up a special line for the denominator with ?NPV
NPV2 = $16,000/(1.14)^2
This problem is relatively simple, but in more complex cash flow
problems it saves time and allows you to visually check the accuracy
of calculations. It will also help you in step B.
Step B: now we need 5 equal payments ? starting in year 0 because
leases are paid in advance ? that result in an NPV of -$152,342.
Excel actually does this in a trial and error technique but for you it
$X * (1.14)^0
$X * (1.14)^1
$X * (1.4)^2
$X * (1.14)^3
$X * (1.14)^4
That ?NPV factor? line comes in handy here (and I?ve rounded to 4
digits for the NPV factor here) ?
$X * 1
$X * 0.8772
$X * 0.7695
$X * 0.6750
$X * 0.5921
3.9138 * $X = - $152, 342
X = - $38,924.32
The differences with the spreadsheet are less than $1 due to rounding
of the NPV factors here.
Step C: divide that by (1-t), where t = tax rate.
- $38,924.32 / (0.6) = - $64,873.87
Clarification of Answer by
15 Aug 2005 11:31 PDT
The 40% is the tax rate -- so it matches the 40% credit for
depreciation. However, it really has nothing to do with the
depreciation. Imagine a tax code that didn't allow depreciation (or,
like the current U.S. code, used different depreciation allowances).
But the leasing company want AFTER TAX returns on its investment. So
if it gets paid $1, it's shareholders are only receiving $0.60 and the
government gets $0.40. So, we have to price the lease higher to get
I hope that's clear. Taxation issues (and in particular paying
attention to whether taxes are being paid or a credit is coming back)
are one of the hardest part about these investment problems.