Time value of money calculations

¶ … Value of Money

For this example, we will assume a $1,000 face value on the CD. The formula for continuous compounding is:

FV = (Pe)^rt

Thus, FV = 1000 * ((2.71828)^(.0358*5)) = $1,196.02

The formula for monthly compounding is:

FV = P (1+r/12)^

Thus, FV = 1000 (1+.003)^60 = $1,196.89

Therefore, the first CD, the one with the 3.6% rate that compounds monthly, is the better deal of the two, because it has a slightly higher final value.

To calculate this question in the year 2010, one would use Excel, specifically the Solver function. What we learn is that the equal payments at 6% and at 4% will depend on what the inputs are. With two unknowns, the inputs can be set at any level where the second is higher than the first. For example, if the first five years the input is $2,266.48, the last seven years will require payments of $2,700. If the first five years see an input of $2,162.02, then the final seven years will require a payment of $2,800. Thus, with two unknown variables, it is impossible to select only one correct answer, but a multitude of answers can be generated with a little spreadsheet knowledge.

3. To calculate this, again the solver is the best approach. The solver delivers an equal payment of $2,487.97 to deliver a final value of $40,000, with the first five years at 6% and the final four years at 4%.

4. To calculate when 60% of the mortgage would be paid down, one would need to know what the principle is. The point at which this occurs is not going to change depending on the principle, so any principle value can be used. In this example, P = $200,000. The monthly payment on this would be

Pmt = P * ((r/12) / (1 -- ((1 + (r/12))^-t)))

Therefore Pmt = 200,000 * ((.00325) / (1 -- ((1+.00325)^-360

Pmt = $943.34

60% of the principle is $120,000, so this will be paid down when P = $80,000. Using Excel, this point is determined to be after month 261, when P = $79,746.

This answer makes sense because during the early months of the mortgage, most of each payment goes towards interest. The principle portion of the payment is relatively small, so that it will take much longer than 60% of the mortgage's length to eliminate the mortgage.

5. The total remaining after 12 years was $135,146.44. The interest rate is 6.6% and the loan was originally for 30 years. This implies that the principle was not too far from the remaining after 12 years. The monthly payments are unknown at this time, so they must be calculated. The formula is

Pmt = P * ((r/12) / (1 -- ((1 + (r/12))^-t)))

However, without knowing the principle, there are two unknown variables in this equation. The only known value in the repayment schedule is that after month 144, P = $135,146.44. This can therefore be solved, because of the relationship between the original principle at the monthly payment. The original principle was therefore $167,660 and the monthly payments were $1,070.77.

This makes sense because the P. after 12 years (40% of the loan time) was still 80% of the loan's original value. The principle payments are small at the beginning of the loan relative to the interest payments.

6. Angelo's contract paid him $1,200,000 on Sept 1, 2008 and then on Sept 1, 2009 it would have paid him $1,248,000. The net present value of his remaining payments would be calculated by adding the present value of each future payment. The PV of a given payment is calculated as follows:

P / (1 + r) ^ t where P. is the payment, r is the discount rate and t is the number of periods. Thus, the NPV table for the remainder of Angelo's contract is as follows:

Question 6

Year

0

1

2

3

Payment

1297920

1349837

1403830

1459983

PV

1297920

1249849

1203558

1158982

NPV=

4,910,309.04

d

0.08

A ten percent discount rate would be worse for Angelo, because the higher the discount rate, the more value the future payment is reduced. As illustrated:

Year

0

1

2

3

Payment

1297920

1349837

1403830

1459983

PV

1297920

1227124

1160190

1096907

NPV=

4,782,141.87

d

0.1

This makes sense because the 10% rate is higher, which diminishes the present value of the future cash flows further, as it increases the size of the denominator.

7.a) the value of the flows would be a perpetuity.

PV = C / I

$45,300,000 * .18 = $8,154,000

This makes sense because the high discount rate makes the future payments worth relatively little. The front-end payments therefore need to be high.

b) the growing perpetuity formula solves this:

PV = C / i-g

Thus I -- g = 5/45.3

.18 -- g = .11 or -- g = -.07, giving us g = 7%

This makes sense because the starting cash flows are lower than for the no-growth example. The growth rate of 7% is therefore required to make up for this over time.

8. To calculate this, a table can be drawn in Excel. The salary increases 2.5% every year. The percentage of the salary donated each year is 5% and this is added to the previous years' balance. Thus,

Year

0

1

2

29

30

Salary

70000

71750

73543.75

143248.5

146829.7

Begin Principle

0

11433.28

792652.9

862964.8

Payment

Interest

16168.19

855802.4

931227.7

End Principle

11433.28

19845.38

862964.8

938569.2

The end balance would therefore be $938,569.20. This makes sense because the interest on the first few payments will have compounded over the course of decades. For example, the first payment will have seen compounding of:

$3,500 ( 1.07)^30 = 3500 * 7.6122, making it worth $26,642.89 at the end of the period. There are also the increases in the payments to consider. While the higher payments will receive less compounding, all of the payments will have a higher starting point than the first one.

9. To calculate this question, an NPV comparison will need to be made. Apparently, there is no difference in the cash flow generated by either machine, so the tax writeoff and initial cost are the only two cash flow components. The cost of Machine a does not count in the calculation, as that is a sunk cost.

d

0.15

Machine a

Year

0

1

2

3

4

5

Tax Reduction

PV

NPV

$24,135.52

Machine D

Purchase

-75000

Tax Reduction

Sale of Machine a

0

PV

-75000

NPV

$ (59,915.30)

The sale price of Machine a to make buying Machine D. justifiable would be the point at which the NPV is positive. Thus,

59,915.30 + 24,135.52 = $84,051

This makes sense because the sunk cost of Machine a does not count. The cost of Machine D. does count, so there needs to be a positive cash flow that offsets this purchase, plus the difference in the tax savings that accrues from the different machines. The proof is here:

Question 9

d

0.15

Machine a

Year

0

1

2

3

4

5

Tax Reduction

PV

NPV

$24,135.52

Machine D

Purchase

-75000

Tax Reduction

Sale of Machine a

84051

PV

NPV

$24,135.70

This gives Machine D. A higher NPV, which is the point at which the project would theoretically be chosen over Machine a.

10. The two options would be compared using an NPV analysis. Thus:

Year

0

1

2

3

4

5

Machine 1

Cost

-420,000

Operating Costs

-68,000

-68,000

-68,000

Salvage Value

42,000

Flows

-420,000

-68000

-68000

-26000

PV

-420,000

-59649.1

-52323.79

-17549.3

NPV

-549,522

Machine 2

Cost

-640000

Operating Costs

-46,000

-46,000

-46,000

-46,000

-46,000

Salvage Value

64,000

Flows

-640000

-46000

-46000

-46000

-46000

18000

PV

-640000

-40350.9

-35395.51

-31048.7

-27235.7

NPV

-764682.13

a) the calculation implies that Machine a is the better purchase. It is worth noting that after three years, another machine will need to be purchased. This cost should be included (i.e. The costs for years 4 and 5) in order to adequately assess the full cost difference between the two machines. After three years, Machine 2 still has a worse NPV than does Machine a, which implies to that point that Machine a is still better. The future decision after Year 3, because it is unknown at this point, cannot be included in the calculation.

b) NPV analysis supports the answer to question a. Indeed, NPV analysis is how the answer to question a was derived. With an interest rate to work with, NPV is the most appropriate means of determining the value of each of these projects, so it is what was used. Any other method of calculating question a would be inferior, so would not make sense.

11. For Kerry's purposes, current costs and sunk costs are not included in the NPV calculation. Thus, the $6,000 marketing expense is not included; nor is the $18,000 in taxes and insurance. Income tax will only be considered on the incremental cash flows from the project. Thus, the most Kerry will pay for this machine is the net present value of the machine. This should be calculated on Excel using Solver, to have an NPV of 0. The reason this is the best method is because the purchase cost impacts the tax expense, as a result of the depreciation tax credit. That means that six total elements of the equation are unknown, but all are related to the original purchase cost.