Question

b. If the company makes the first deposit one year from now, how much should each deposit be? 3. A start-up internet service provider expects to lose money in each of the first four years. Losses are projected to be $50 million in year one, $40 million in year two, $30 million in year three and $20 million in year four. An interest rate of 10% per year is used. a. What is the present worth of the losses for the first four years? b. What is the equivalent uniform annual worth of the losses through year four? c. In order to recover the losses by the end of year nine, the company's equivalent uniform annual profit in years five through nine must be how much?

Answer 1

1. The company would have to pay $3,200 from end of year 4 to end of year 10

Present value of a payment = Pmt/ (1+r)ⁿ

Pmt is the payment amount, r is the rate of discount, n is the number of periods

Period (n)	Pmt	PV
0		0.00
1		0.00
2		0.00
3		0.00
4	-3200.00	-2185.64
5	-3200.00	-1986.95
6	-3200.00	-1806.32
7	-3200.00	-1642.11
8	-3200.00	-1492.82
9	-3200.00	-1357.11
10	-3200.00	-1233.74
Total		-11704.69

Present worth of the contract = $ (11,704.69)

To find the equivalent annual amount of the contract in years 1 through 10 given the above present value of $ 11,704.69

PVA = A * [((1+r)ⁿ - 1)/r*(1+r)ⁿ] ………….(A)

[((1+r)ⁿ - 1)/r*(1+r)ⁿ] can be found out below first:

((1+0.10)¹⁰ - 1) / 0.10*(1+0.10)¹⁰

(1.10¹⁰ - 1) / 0.10* 1.10¹⁰

(2.5937424601 - 1) / 0.25937424601

1.5937424601 / 0.25937424601

= 6.1446

So -11,704.69 = A*6.1446

A = -11,704.69 / 6.1446 = -1,904.88

Equivalent uniform amount of contract = $ (1,904.88)

If the company were to pre-pay in years 1 -3

PVA = A * [((1+r)ⁿ - 1)/r*(1+r)ⁿ]

[((1+r)ⁿ - 1)/r*(1+r)ⁿ] can be found out below first:

((1+0.10)³ - 1) / 0.10*(1+0.10)³

(1.10³ - 1) / 0.10* 1.10³

0.33 / 0.13

=2.4869

So A = -11,704.69 / 2.4869 = -4,706.63

So the annual payment from years 1 through 3 would be $ (4,706.63)

2. Future value of a deposit is = (1+r)ⁿ where n is the balance periods to the future value date

Period (n)	$ 1 deposit	Balnce periods	Future value at year 10
0		10	0
1		9	0
2		8	0
3	1	7	2.210681
4	1	6	1.973823
5	1	5	1.762342
6	1	4	1.573519
7	1	3	1.404928
8	1	2	1.2544
9	1	1	1.12
10	1	0	1
Total			12.29969

If a $1 deposit according to the above schedule becomes $ 12.2997, then how many $ deposited per year as per above schedule becomes $200,000?

= $200,000 / $ 12.2997 = $ (16,260.56)

If the company makes the first deposit 1 year from now we would need to find an annuity that fits the future value of $200,000 @12% over 10 years

FV = A * ((1+r)ⁿ - 1 )/ r

200000 = A ((1.12)¹⁰ - 1) / 0.10

200000 = A* 2.1058 / 0.10

A = 200000 / 21.05848

A = $ (9,497.36)

3.a Present value of losses = $ 114.71 million

Period	Losses	PV of losses
1	50	($45.45)
2	40	($33.06)
3	30	($22.54)
4	20	($13.66)
Total		($114.71)

3. b Solution is $36.19 million using the PVA formula in eq (A) above (2nd solution of Q 1)

3. c. Solution obtained by dividing the above solution a by the value in below table = $114.71 / 14.7063 = $ 7.80 million in profit per year from year 5 onwards

Period	$ 1 Losses	Balance periods	Future value at year 9
1
2
3
4
5	4	5	6.44204
6	3	4	4.3923
7	2	3	2.662
8	1	2	1.21
9	0	1	0
			14.70634