This is a fictional question. I have the opportunity to purchase an aeroplane.
I am 3 years into a lease with Company A. I can continue with them
under the following terms:

I have a deposit with them for $15,686,000.
I have to pay them $311,915 per month for 84 months, or 7 years.
At the end of 7 years, I can buy the plane for $22,000,000.

Company B have come in. I have a termination right with Company A
which means I can buy the aircraft for $39,950,000 (remember that my
deposit with Company A will be freed up). They are saying that they
will give me a loan for $27m (this means that i will have to use
$12,950,000 of my returned deposit). I will need to give them a
deposit for $4m on top. I will repay the loan at $199,326 a month for
120 periods (monthly for 10 years). At the end of the period, i will
pay them $15M.

Using a discount rate of 5%, which is the more favourable deal?
(please show all workings)

Dear leedsutd,

The more favorable deal is the one that costs less after taking into
account all payments and capital costs. We shall first calculate the net
present cost of staying with the lease option, and subsequently that of
switching to the purchase option.

At a discount rate of 5%, the present value of all payments decreases by
5% annually, as does the value of the deposit. We can ignore the effect
of this depreciation over the first three years of the lease, since the
payments already made cannot be recovered regardless of whether we now
choose the lease or the purchase option. We shall compute the value of
all costs as of the present moment, three years into the lease.


From the amount of the monthly payment, we know that annual lease payments
in the upcoming year come to

  12 * $311,915  =  $3,742,980.

Each subsequent year will see the present value of this amount drop
by 5%. Thus, annual lease payments over the next four years are the
following.

  year 1:  $3,742,980
  year 2:  (1 - 5/100) * $3,742,980  =  .95 * $3,742,980  =  $3,555,831
  year 3:  .95 * $3,555,831  =  $3,378,039
  year 4:  .95 * $3,378,039  =  $3,209,137
                                ----------
                       total:  $13,885,987

The lease terminates at the end of these four years, leaving a residual
whose present value is

  .95^4 * $22,000,000  =  $17,919,137.

Thus, the net present cost of staying with the lease option is

  $13,885,987 + $17,919,137  =  $31,805,124.


If we accept Company B's offer, we immediately purchase the airplane,
leaving us with

  $15,686,000 - $12,950,000  =  $2,736,000

of the returned deposit. But we must additionally make a deposit of $4m,
thereby incurring an immediate cost of

  $2,736,000 - $4,000,000  =  $1,264,000.

The annual loan payments nominally amount to

  12 * $199,326  =  $2,391,912.

Over the next ten years, the present value of these payments is the
following.

  year 1:  $2,391,912
  year 2:  .95 * $2,391,912  =  $2,272,316
  year 3:  .95 * $2,272,316  =  $2,158,700
  year 4:  .95 * $2,158,700  =  $2,050,765
  year 5:  .95 * $2,050,765  =  $1,948,227
  year 6:  .95 * $1,948,227  =  $1,850,816
  year 7:  .95 * $1,850,816  =  $1,758,275
  year 8:  .95 * $1,758,275  =  $1,670,361
  year 9:  .95 * $1,670,361  =  $1,586,843
  year 10: .95 * $1,586,843  =  $1,507,501
                               -----------
                       total:  $19,195,716

The present value of the final payment is

  .95^10 * $15,000,000  =  $12,217,594.

Summing the initial cost, the periodic payments, and the final cost,
we obtain a net present cost of

  $1,264,000 + $19,195,716 + $12,217,594  =  $32,677,310.


We conclude that the net present cost of staying with Company A's lease
is more favorable than that of accepting Company B's offer, yielding a
savings of

  $32,677,310 - $31,805,124  =  $872,186

in present dollars.


I have enjoyed addressing this matter on your behalf. If you find fault
with my answer, please advise me through a Clarification Request so that
I may fully meet your needs before you assign a rating.


Regards,

leapinglizard

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Request for Answer Clarification by leedsutd-ga on 18 Apr 2005 02:13 PDT

Also, in the Lease option for company A, you have not taken into
account that we have a deposit with them, and what this would do to
the final analysis.

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Clarification of Answer by leapinglizard-ga on 18 Apr 2005 02:27 PDT

Oh, but I have. The lease deposit is only returned if we choose not to
purchase the plane at the expiry of the lease. If we do choose to buy
out the lease, the deposit is incorporated into the purchase price.
This is made clear by the numbers cited. If the lease purchase price
did not include the deposit, then it would be well below the immediate
purchase price of $39,950,000. This cannot be the case. As for the
purchase plan, the purpose of the deposit again is to act as a surety
against the value of the airplane. If we damage or destroy the plane,
the deposit is lost. But the final payment of $15m already accounts
for the $4m deposit. Otherwise, the final payment would be cited as
$11m.

leapinglizard

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Request for Answer Clarification by leedsutd-ga on 18 Apr 2005 02:58 PDT

No, i meant that Company A (the lease company) has a deposit of $15
million. The calculations done do not take into account that, if we do
exercise the right to purchase the plane for $22m 7 years on, then we
do have the deposit with them. Therefore, the $15m should be NPV'd and
subtracted from the NP cost numbers for the first calculation. Or am i
wrong? It is clear that the deposit does not come into play at all in
your calculations for the lease company.
Please note that the $22m price is lower than the $39,950,000 because
the plane will depreciate over the 7 years of the lease.
I wholeheartedly agree with you calculations for the immediate
purchase and subsequent financing

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Clarification of Answer by leapinglizard-ga on 18 Apr 2005 03:48 PDT

I did consider applying the lease deposit toward the purchase of the
plane, but I quickly concluded that the resulting numbers would be
unrealistic. Consider that even after depreciation, the initial
deposit would be worth over $10m at lease end. The buy-out price of
the plane surely isn't $12m, even after depreciation! If it were,
there would be no need to make any further calculations, since the
purchase plan couldn't possibly be a better deal. For this reason, and
also because the problem statement makes one explicit mention of a
returned deposit and no other, I proceeded with the assumption that
the $22m buy-out incorporates the deposit. Upon further reflection,
I'm still sure this is the right approach.

leapinglizard

I thoguht that the answer was very well given. 
It was easy to understand and has broadened my own knowledge so far as
I managed to do a similar calculation which proved to be correct after
studying ym answer.

Thank you for the rating and the kind words.

leapinglizard