1. A firm can lease a truck for 4 years at a cost of $30,000 annually. It can instead buy a truck at a cost of $80,000, with annual maintenance expenses of $10,000. The truck will be sold at the end of 4 years for $20,000. Calculate the equivalent annual annuity to show the better option if the discount rate is 10%.

The lease option is the better option.

Explanation:

We proceed as follows:

Step 1: Calculation of Lease Option NPV    

Year = n         Details             CF ($)     DF = 1/(1.1)^n   PV ($)

     1     Lease payment   (30,000)        0.9091         (27,273)

    2     Lease payment   (30,000)        0.8264         (24,793)

    3     Lease payment   (30,000)         0.7513         (22,539)

    4     Lease payment   (30,000)         0.6830         (20,490)

                                      Lease option NPV = (95,096)

Step 1: Calculation of Lease Option NPV Buy Option NPV      

Year = n        Details                  CF (CO)     DF = 1/(1.1)^n      PV  

     0  Purchase cost                  (80,000)       1.0000   (80,000)

     1   Maintenance expenses   (10,000)       0.9091      (9,091)

    2   Maintenance expenses   (10,000)       0.8264     (8,264)

    3   Maintenance expenses   (10,000)        0.7513      (7,513)

    4   Maintenance expenses   (10,000)       0.6830     (6,830)

    4   Residual value                   20,000        0.6830      13,660  

                                                     Buy option NPV = (98,038)

Step 3: Calculation of equivalent annual annuity (EAA)

The equivalent annual annuity (EAA) for each option can be calculated as follows:

EAA = (r x NPV) / (1 - (1 + r)^-n )

Where:

EAA = equivalent annuity cash flow

NPV = net present value

r = discount rate per period

n = number of periods

Therefore, we have:

Lease option EAA = (0.1 × -95,096) / (1 - (1 + 0.1)^-4)  = -30,000

Buy option EAA = (0.1 × 98,038) / (1 - (1 + 0.1)^-4)  = -30,928

Since the lease option has a lower EAA of $30,000 in terms of cash outlay than the buy option of higher EAA of $30,928 in terms of cash outlay, the lease option is the better option.

Option 1 is preffered option because:

EAC of Instalment option = $30,000

EAC of Buying option = $37,660

Explanation:

The Equivalent Annual Cost would be calculated using the following formula:

Equivalent Annual Cost = Net Present Value of option / Annuity Factor

Now we will have to find the present value of the each option available.

Option 1

So present value of the installment option is given as under:

Present Value = Annual Cash flow * Annuity Factor

Here the annuity factor can be found from the following formula:

Annuity Factor = (1-(1+r)^-n ) / r

By putting the values we have:

Annuity Factor = (1 - (1.1)^-n) / 10% = 2.487

Present Value = $30000 * 2.487 = $74610

This implies

Equivalent Annual Cost = $74610 / 2.487 = $30000

Option 2

Present Value = $80000 Initial cash outflow +

$20000 cash inflow at Y4 / (1.1)^4 Discount factor at Year 4

Present Value = $80,000 + $20,000 / 1.4641

Present Value = $80,000 + $13,660 = $93,660

And

EAC = $93660 / 2.487 at 10% for 4 Years = $37,660

Decision Rule:

The cheapest option is the one with least value and in this case the cheapest option is option 1.

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