Inflation Answers

1.

This problem gives = 8% and = 10%.

Compute the uninflated (real) interest rate = 0.0185 or 1.85%.

Since all costs and benefits are inflating at the same rate as general inflation, do the analysis use the cash flows given in the problem and the real interest rate,.

x = 1000(P/A, , 5) + 1000(P/G, , 5) = 14,028.62.

Alternative: Inflate dollars and use for the present worth analysis.

x = 1000(F/P, , 1)(P/F, , 1) + 2000(F/P, , 2)(P/F,, 2) + 3000(F/P, , 3)(P/F, , 3) +

4000(F/P, , 4)(P/F, , 4) + 5000(F/P, , 5)(P/F, , 5) =14,028

Note that the factor (F/P, , n) is used to convert the estimate to an actual dollar cash flow at year n.

The alternative computation is definitely harder. When all cash flows are inflating at the same rate as general inflation it is usually much easier to use the cash flows as estimated in today's prices. Then is used in the present worth formula.

1b. Compute the after tax actual dollar cash flow with the specified depreciation. Use trial and error to find the value of the interest rate that makes the present value 0. This is the after tax rate of return including the effect of inflation.

The ROR of the actual-dollar-after-tax cash flow is about 6.5%. This ROR includes inflation since it was computed from after tax cash flows. The real ROR with inflation removed is about -1.3%. This is definitely a bad deal for Gary.

2. Time Cash flow: time 0 is the purchase, times 1 through 8 is the interest of $60 per year, time 8 is the return of the face value of $1,000. These are actual dollar payments since the amounts are fixed by contract and the payments are always made in actual dollars. In the following we use = 25%, the investors MARR when there is no inflation. We compute , the market rate, using the specified general inflation rate for the different parts.  

In each case below, find the purchase price that makes the NPV = 0. We use the market rate of the MARR because the cash flow is in actual dollars.

NPV = - Purchase price + 60(P/A, , 8) + 1000(P/F, , 8) = 0

or

Purchase price = 60(P/A, , 8) + 1000(P/F, , 8)

a. With = 0, use = 25%. Purchase price = $368.

b. With an inflation rate of 16%, her MARR with inflation must be

=0.25 + 0.16 + (0.25)(0.16) = 0.45. Purchase price = $178

c. If the deflation rate is 18% then = -0.18, the market MARR is

=0.25 - 0.18 + (0.25)(-0.18) = 0.025. Purchase price = $1251

Conclusion: Increasing inflation causes a fixed income investment to be reduced in present value.

3. We use the formula

to compute actual dollars. We use in the formula because the cash flow inflates at the rate of general inflation.

The cash flow expressed in actual dollars

b. Since all cash flows increase at the same rate as general inflation, just use the MARR without inflation to compute NPW. Find the NPW.

NPW = -2000 + 1000(P/A, 0.1, 5) - 200(P/G, 0.1, 5)

NPW = -2000 + 1000*3.791 - 200*6.862 = -2000 + 3791 - 1372.4 = 418.6

Since the NPV > 0, accept the investment.

c. For an after tax analyis you must compute the depreciation first

SYD = 1 + 2 + 3 + 4 + 5 = 15. D1 = 5*2000/15 = 666 ... D5 = 2000/15 = 133

For year 1: Tax = (1060 - 666)*.4 = 394*.4 = 157.6, ATCF = 1060 - 157.6 = 902.4

For year 5: Tax = (268 - 133)*.4 = 135*.4 = 54.0, ATCF = 268 - 54.0 = 214.0

Other years are computed similarly. The after-tax cash flows in actual dollars are shown below.

For the actual dollar analysis using the interest rate:

= 0.1 + 0.06 + .1*.06 = 0.16 + .006 = 0.166 or 16.6%.

Computing the net present worth with 16.6%: NPW = 20 > 0, so accept.

4. Since the investment is entirely recovered at the end of the time horizon, the ROR is 200/1000 or 20%. Since the cash flow is in real dollars, this ROR does not include inflation. Including the effects of inflation, the rate of return is

= 0.20 + 0.06 + (0.20)(0.06) = 27.2%

5. Cash Flow: -5000 at year 0, +2500 at years 5, 6, 7, 8. These amounts are in actual dollars because they are payments.

a. NPW at 8% = -5000 + 2500(P/A, 0.08, 4)(P/F, 0.08, 4) =1086

This is an acceptable investment.

b. Deflating the $2500 payments to real dollars.

Year 5: 1958.82; Year 6: 1865.54; Year 7: 1776.7; Year 8 :1692.1

The NPW at 10% is: -5000 + 1958(P/F, .08, 5) + 1865.54(P/F, .08, 6) + 1776.7(P/F, .08, 7) + 1692.1(P/F, .08, 8)

= -540

The investment is not acceptable.

Alternatively, we could use the market MARR with the actual-dollar cash flow..

= .08 + .05 + .004 = 13.4%.

Computing the NPW directly using the formulas NPW = -5000 + 2500(P/A, .134, 4)(P/F, .134, 4) = -540

This is the same value as the real-dollar computation.

6.

a. = 0.2

Compute = (.20 - .05)/(1 + 0.05) = 0.143

NAW = -1500(A/P, , 4) + 500 + 200(A/G, ,4)

b. = .2, = 0.143

First compute the present worth using actual dollars and . Then spread the present worth into a uniform annual equivalent with .

NAW = [-1500 + 900(A/F, , 2)(P/A, , 8)](A/P, , 8)

c. Since the cash flow is in actual dollars and and depreciation is in actual dollars, we work the tax analysis in actual dollars.

Depreciation in each year = 1600/4 = 400 per year

Tax in each year is (900 - 400).4 = 200

The ATCF each year is 900 - 200 = 700

Compute = .1 + .05 + .1*.005 = .155

NPW = -1600 + 700(P/A, i, 4)

7. First alternative: Express all CF as actual dollars and use the MARR with inflation (namely, 0.20 + 0.04 + 0.20*0.04 = 24.8%

We have inflated the costs and salvage using the general rate of inflation..

NPW = -100,000 + 38,400 (P/F, 24.8%, 1) + 36,736 (P/F, 24.8%,2) + ... + 65,198 (P/F, 24.8%, 10) = $24,531

Second alternative: Express all CF as real $ and use the MARR without inflation, namely 20%

NPW = -100,000 + 36,923 (P/F, 20%, 1) + 33,964 (P/F, 20%,2) + ... + 44,045 (P/F, 20%, 10) = $24,531

8.

a. With an inflation rate of 6%, we can find the actual-dollar tuition using the F/P factor for 6%.

Tuition: First year = tuition in actual dollars is 1000(F/P, 0.06, 18)= $28,543

Second year = tuition in actual dollars is 1000(F/P, 0.06, 19) = $30,256

Third year = tuition in actual dollars is 1000(F/P, 0.06, 20)= $32,071

Fourth year = tuition in actual dollars is 1000(F/P, 0.06, 21) = $33,996

b. The amount saved is the present value of the cash requirements at the end of year 18? The present value is computed using the bank interest rate. This is the market rate includes inflation.

$28,453 + $30,256 (P/F, 0.1, 1) + $32,071(P/F, 0.1, 2) + $33,996 (P/F,0.1,3) = $108,095

c. What should the yearly deposit be so that the future worth of the amount in the bank be $108,095? The amounts deposited are in actual dollars, so again we use the market interest rate to compute the deposit amount.

A = $108,095 (A/F, 10%,18) = $2370.55