Question

Brian is starting a data storage business. He has two sources of servers to store data, used servers from eBay and new servers from Dell. The used servers will cost $1000 and last two years. Additional used servers can be purchased in future years at the same price. The new servers will cost $3,000 and last six years. Each will produce net revenue of $800 per year. The servers are purchased in year 0 and the relevant discount rate is 5 percent (0.05).

a) find the present value of the new and used servers.

b) calculate and compare the equivalent annuity values for both choices.

c) Which type of server should Brian buy?

Answer 1

We will have to create timelines to analyse the cash inflows and outflows in each of the two cases, i.e. used server and new server

Case 1: Used Servers

Purchase in Year 0 (i.e. Start of Year 1) for 1000 $ and will last 2 years;

Therefore next purchase to be made at the end of the Year 2, and then at the end of Year 4 and so on

So, there will be a cash outflow of 1000 $ at the end of Year 0, Year 2, Year 4, Year 6 and so on..

And at the end of each year there is a cash inflow of 800 $ (which is the Net Revenue, as given in the

question)

The structure of Cash Flows will be like the table given below

Year	0	1	2	3	4	5	6
Cash Outflow	-1000		-1000		-1000		-1000
Cash Inflow		800	800	800	800	800	800

These cash flows will keep reoccuring in the future and because of this we have a Perpetual Annuity (or a Perpetuity) here.

To ease our calculations we will consider the above to be two separate Perpetuities:-

(i) In the first one, there is a cash outflow of 1000 every two years

(ii) In the second one, there is a cash inflow of 800 every year

Solving the First Perpetuity

In this case as the cash outflow of 1000 $ is not occuring every year, we cannot use the given discount rate as it is in the general formula of a perpetuity to calculate its present value.

Here, the outflows occur every 2 years and hence our effective Interest Rate will be = ( 1 + r/100)2 - 1

= ( 1 + 0.05)2 - 1

= (1.052 - 1)

= 0.1025

= 10.25%

Our effective interest rate comes out be 10.25% and we will use this in the Formula for Perpetuity as given below to calculate its Present Value

PV1 = C/r (C = The amount of reccurring payment)

PV1 = -1000/10.25%

PV1 = -9756.098

Solving the second Perpetuity

Here, annual payment is 800 and the discount rate for this case will be 0.05 as given in the question because the payments are annual

So,

PV2 = 800/0.05

PV2 = 16000

Now, by adding the values of the two Perpetuities we will get the PV of the original perpetuity as shown in the table above

So, PV of perpetuity in case of used servers = -9756.098 + 16000 = 6243.902

The PV of Used Servers = 6243.902

Case 2: PV of New Servers

Purchase in Year 0 (i.e. Start of Year 1) for 3000 $ and will last 6 years;

Therefore next purchase to be made at the end of the Year 6, and then at the end of Year 12 and so on

So, there will be a cash outflow of 3000 $ at the end of Year 6, Year 12, Year 18, Year 24 and so on..

And at the end of each year there is a cash inflow of 800 $ (which is the Net Revenue, as given in the

question)

The structure of Cash Flows will be like the table given below

Year	0	1	2	3	4	5	6	7	8	9	10	11	12
Cash Outflow	-3000						-3000						-3000
Cash Inflow		800	800	800	800	800	800	800	800	800	800	800	800

Again we have 2 distinct perpetuities and we will solve these two perpetuities separately:-

(i) In the first perpetuity, there is a cash outflow of 3000 every six years

(ii) In the second one, there is a cash inflow of 800 every year

Solving First Perpetuity

Since the payments are not annual therefore, the effective interest rate will be modified as

r = ( 1 + 0.05 )6 - 1 = 0.340096 = 34.01%

PV of perpetuity will be = PV1 = C/r = -3000/34.01% = -8821.048

Solving Second Perpetuity

PV2 = 800/0.05 = 16000

Adding the values of the PV of both the Perpetuities, we will get the value of the series of payments, which is the PV of the New Servers

PV of New Servers = -8821.048 + 16000 = 7178.952

Answers

(a) PV of New Servers = 7178.952 $

PV of Used Servers = 6243.902 $

(b) Formula for PV of an Annuity = P * [1 - (1+r)-n]/r

We will plug in the above formula the values of PV for both the cases (new and old servers) from part (a); r = 0.05; n = 3 (in case of old servers) and n = 6 (in case of new servers); and solve the resulting equations, then we will get the value of the Equivalent Annuity for both the cases, as given below:

(i) Equivalent Annuity for Old Servers = 2292.85 $

(ii) Equivalent Annuity for New Servers = 1414.38 $

(c) Brian should buy New Servers because the PV of the cash flows which he will get in the case of New Servers will be greater than the PV of the cash flows he will get in the case of Old servers.