Question

1. Find the following values for a lump sum assuming annual compounding:
   1. Consider an uneven cash flow stream:
   2. Find the following values assuming a regular, or ordinary, annuity:
      • The present value of $400 per year for ten years at 10 percent
      • The future value of $400 per year for ten years at 10 percent
      • The present value of $200 per year for five years at 5 percent
      • The future value of $200 per year for five years at 5 percent
   3. The future value of $500 invested at 8 percent for one year
   4. The future value of $500 invested at 8 percent for five years
   5. The present value of $500 to be received in one year when the opportunity cost rate is 8 percent.
   6. The present value of $500 to be received in five years when the opportunity cost rate is 8 percent.

Year	Cash Flow
0	$2,000
1	$2,000
2	$0
3	$1,500
4	$2,500
5	$4,000

1. What is the present (Year 0) value of the cash flow stream if the opportunity cost rate is 10 percent?
   1. What is the value of the cash flow stream at the end of Year 5 if the cash flows are invested in an account that pays 10 percent annually?
   2. What cash flow today (Year 0), in lieu of the $2,000 cash flow, would be needed to accumulate $20,000 at the end of Year 5? (Assume that the cash flows for Years 1 through 5 remain the same.)

Answer 1

Part A:

PV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time.

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r
r - Int rate per period
n - No. of periods

Particulars	Amount
Cash Flow	$         400.00
Int Rate	10.000%
Periods	10

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r

= $ 400 * [ 1 - [(1+0.1)^-10]] /0.1

= $ 400 * [ 1 - [(1.1)^-10]] /0.1

= $ 400 * [ 1 - [0.3855]] /0.1

= $ 400 * [0.6145]] /0.1

$                  2,457.83

Part B:

FV of Annuity:

Annuity is series of cash flows that are deposited at regular intervals for specific period of time.

FV of Annuity = CF [ (1+r)^n - 1 ] / r
r - Int rate per period
n - No. of periods

Particulars	Amount
Cash Flow	400
Int Rate	10.000%
Periods	10

FV of Annuity = Cash Flow * [ [(1+r)^n ] - 1 ] /r

=400 * [ [(1+0.1)^10] - 1 ] / 0.1

=400 * [ [(1.1)^10] - 1 ] /0.1

=400 * [ [2.5937] - 1 ] / 0.1

=400 * [1.5937] /0.1

6374.97

Part C:

Particulars	Amount
Cash Flow	$         200.00
Int Rate	5.000%
Periods	5

PV of Annuity = Cash Flow * [ 1 - [(1+r)^-n]] /r

= $ 200 * [ 1 - [(1+0.05)^-5]] /0.05

= $ 200 * [ 1 - [(1.05)^-5]] /0.05

= $ 200 * [ 1 - [0.7835]] /0.05

= $ 200 * [0.2165]] /0.05

$                      865.90

Part D:

Particulars	Amount
Cash Flow	200
Int Rate	5.000%
Periods	5

FV of Annuity = Cash Flow * [ [(1+r)^n ] - 1 ] /r

=200 * [ [(1+0.05)^5] - 1 ] / 0.05

=200 * [ [(1.05)^5] - 1 ] /0.05

=200 * [ [1.2763] - 1 ] / 0.05

=200 * [0.2763] /0.05

1105.13

Part E:

FUTure Value:

FV = PV (1+r)^n
Where n is Int rate per period
n - No. of periods

= $ 500 ( 1 + 0.08)^1

= $ 500(1.08^1)

= $ 500*1.08

= $ 540

Part F:

FV = PV (1+r)^n
Where n is Int rate per period
n - No. of periods

= $ 500 ( 1 + 0.08)^5

= $ 500(1.08^5)

= $ 500*1.4693

= $ 734.66

Part G:

PV = FV / (1+r)^n
Where n is Int rate per period
n - No. of periods
= 500 / ( 1+0.08)^1

= 500 / (1.08^1)

= 500 / 1.08

= 462.96

Part H:

PV = FV / (1+r)^n
Where n is Int rate per period
n - No. of periods
= 500 / ( 1+0.08)^5

= 500 / (1.08^5)

= 500 / 1.4693

= 340.29

Part I:

PV = Sum [ CF * PVF(r%, n) ]

Year	CF	PVF @10%	Disc CF
0	$ 2,000.00	1.0000	$ 2,000.00
1	$ 2,000.00	0.9091	$ 1,818.18
2	$             -	0.8264	$             -
3	$ 1,500.00	0.7513	$ 1,126.97
4	$ 2,500.00	0.6830	$ 1,707.53
5	$ 4,000.00	0.6209	$ 2,483.69
PV of Cash Flows	$ 9,136.37

Part J:

FV = Sum [ CF * FVF(r%, n) ]

Year	Bal Yrs	CF	FVF @10%	FV of CFs
0	5	$ 2,000.00	1.6105	$   3,221.02
1	4	$ 2,000.00	1.4641	$   2,928.20
2	3	$             -	1.3310	$               -
3	2	$ 1,500.00	1.2100	$   1,815.00
4	1	$ 2,500.00	1.1000	$   2,750.00
5	0	$ 4,000.00	1.0000	$   4,000.00
FV of Cash Flows	$ 14,714.22

Part K:

Let X be the Year 0 CF.

Year	Bal Yrs	CF	FVF @10%	FV of CFs
0	5	X	1.6105	1.6105X
1	4	$ 2,000.00	1.4641	$   2,928.20
2	3	$             -	1.3310	$               -
3	2	$ 1,500.00	1.2100	$   1,815.00
4	1	$ 2,500.00	1.1000	$   2,750.00
5	0	$ 4,000.00	1.0000	$   4,000.00
FV of Cash Flows	11493.2 + 1.6105X

Amount required after 5 Years is 20000

Thus 11493.2 + 1.6105X = 20000

1.6105X = 20000 - 11493.2

= 8506.8

X = 8506.8 / 1.6105

= 5282.09

Pls do rate, if the answer is correct and comment, if any further assistance is required.