Discounted Payback

Learning Outcomes

  • Calculate the payback period using discounted cash flows

Wooden puzzle

We can improve the payback method by using discounted cash flows instead of the nominal (undiscounted, unadjusted) estimates.

Let’s review the example:

JuxtaPos makes interlocking wooden puzzles. One of the machines is old and management is considering either replacing it or refurbishing it. Replacing it will cost $80,000. The new machine would last 10 years and have a residual value of $10,000. Refurbishing the old machine for $56,000 will keep it in service for another 8 years and it will have no residual value at the end of that time.

You have estimated production under both scenarios and used those numbers to compute revenue. You have also estimated operating costs and have created the following table of net cash inflows:

Net cash inflows (additional cash revenues – additional cash expenses)

Refurbish Purchase New
Year 1 $              18,000 $       20,000
Year 2                 16,000           19,000
Year 3                 14,000           18,000
Year 4                 12,000           17,000
Year 5                 10,000           15,000
Year 6                   8,000           13,000
Year 7                   6,000           10,000
Year 8                   4,000             7,000
Year 9             5,000
Year 10 (includes proceeds from sale of machine)           12,000
Single Line$              88,000Double line Single Line$     136,000Double line

 

None of these values are adjusted for the time value of money. For instance, $12,000 10 years from now is not the same as $12,000 today.

Let’s assume JuxtaPos uses a 10% hurdle or discount rate for capital budgeting purposes. If we use a present value table, we find that the factor of 10% and 10 years is 0.386. Multiplying the factor by our $12,000 future value for year 10 gives us a present value of $4,632. That means that if we are to translate all of the future numbers back to Year 0 when we make the initial purchase (beginning of Year 1), the $12,000 future cash flow in Year 10 would be the equivalent of a $4,632 cash flow back at the beginning of Year 1.

Present Value of $1
Periods 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 12% 14% 15% 16% 18% 20%
Period 1 0.990 0.980 0.971 0.962 0.952 0.943 0.935 0.926 0.917 0.909 0.893 0.877 0.870 0.862 0.847 0.833
Period 2 0.980 0.961 0.943 0.925 0.907 0.890 0.873 0.857 0.842 0.826 0.797 0.769 0.756 0.743 0.718 0.694
Period 3 0.971 0.942 0.915 0.889 0.864 0.840 0.816 0.794 0.772 0.751 0.712 0.675 0.658 0.641 0.609 0.579
Period 4 0.961 0.924 0.888 0.855 0.823 0.792 0.763 0.735 0.708 0.683 0.636 0.592 0.572 0.552 0.516 0.482
Period 5 0.951 0.906 0.863 0.822 0.784 0.747 0.713 0.681 0.650 0.621 0.567 0.519 0.497 0.476 0.437 0.402
Period 6 0.942 0.888 0.837 0.790 0.746 0.705 0.666 0.630 0.596 0.564 0.507 0.456 0.432 0.410 0.370 0.335
Period 7 0.933 0.871 0.813 0.760 0.711 0.665 0.623 0.583 0.547 0.513 0.452 0.400 0.376 0.354 0.314 0.279
Period 8 0.923 0.853 0.789 0.731 0.677 0.627 0.582 0.540 0.502 0.467 0.404 0.351 0.327 0.305 0.266 0.233
Period 9 0.914 0.837 0.766 0.703 0.645 0.592 0.544 0.500 0.460 0.424 0.361 0.308 0.284 0.263 0.225 0.194
Period 10 0.905 0.820 0.744 0.676 0.614 0.558 0.508 0.463 0.422 0.386 0.322 0.270 0.247 0.227 0.191 0.162
Period 11 0.896 0.804 0.722 0.650 0.585 0.527 0.475 0.429 0.388 0.350 0.287 0.237 0.215 0.195 0.162 0.135
Period 12 0.887 0.788 0.701 0.625 0.557 0.497 0.444 0.397 0.356 0.319 0.257 0.208 0.187 0.168 0.137 0.112
Period 13 0.879 0.773 0.681 0.601 0.530 0.469 0.415 0.368 0.326 0.290 0.229 0.182 0.163 0.145 0.116 0.093
Period 14 0.870 0.758 0.661 0.577 0.505 0.442 0.388 0.340 0.299 0.263 0.205 0.160 0.141 0.125 0.099 0.078

 

More precisely (because factors in the tables are rounded), the present value of $12,000 at 10% for 10 years is $4,626.5194732… which we could round to $4,626.52.

If we were to invest that amount in a mutual fund that gave us an annual return of 10%, we would see the following results:

BOY interest EOY
Year 1 $      4,626.52 $       462.65 $  5,089.17
Year 2 $      5,089.17 $       508.92 $  5,598.09
Year 3 $      5,598.09 $       559.81 $  6,157.90
Year 4 $      6,157.90 $       615.79 $  6,773.69
Year 5 $      6,773.69 $       677.37 $  7,451.06
Year 6 $      7,451.06 $       745.11 $  8,196.17
Year 7 $      8,196.17 $       819.62 $  9,015.79
Year 8 $      9,015.79 $       901.58 $  9,917.37
Year 9 $      9,917.37 $       991.74 $ 10,909.11
Year 10 $     10,909.11 $     1,090.91 $ 12,000.02

 

In other words, $4,626.52 invested at 10% today will grow to $12,000.02 at the end of the tenth year. So, the present value of $12,000 is $4,626.52. The future value of $4,626.52 invested at 10% for 10 years is $12,000.02.

If we apply this logic to all of the cash flow numbers, using 10% as our discount rate and the appropriate number of years for each cash inflow, we get the following table of discounted cash flows:

Discounted net cash inflows

Refurbish Purchase New
PV Year 1 $               16,364 $        18,182
PV Year 2 $               13,223 $        15,702
PV Year 3 $               10,518 $        13,524
PV Year 4 $                8,196 $        11,611
PV Year 5 $                6,209 $         9,314
PV Year 6 $                4,516 $         7,338
PV Year 7 $                3,079 $         5,132
PV Year 8 $                1,866 $         3,266
PV Year 9 $                    – $         2,120
PV Year 10 $                    – $         4,627
Single Line$               63,971Double line Single Line$        90,816Double line

 

Applying these numbers to our payback analysis, we find that the amount of time it takes for the refurbish option to pay for itself is 5 years 4 months:

Refurbish

% of cash flows months years
Cost to refurbish $        56,000
Recaptured YR 1 $      (16,364) 100% 12.0         1.00
Left to recapture Single Line$        39,636
Recaptured YR 2 $      (13,223) 100% 12.0         1.00
Left to recapture Single Line$        26,413
Recaptured YR 3 $      (10,518) 100% 12.0         1.00
Left to recapture Single Line$        15,895
Recaptured YR 4 $       (8,196) 100% 12.0         1.00
Left to recapture Single Line$         7,699
Recaptured YR 5 $       (6,209) 100% 12.0         1.00
Left to recapture Single Line$         1,489
Recaptured YR 6 $       (1,489) 33% 4.0         0.33
Left to recapture Single Line$             0

 

And the payback period for a new machine is 6 years 10 months.

Purchase

% of cash flows months years
Cost to purchase $       80,000
Recaptured YR 1 $     (18,182) 100% 12.0         1.00
Left to recapture Single Line$       61,818
Recaptured YR 2 $     (15,702) 100% 12.0         1.00
Left to recapture Single Line$       46,116
Recaptured YR 3 $     (13,524) 100% 12.0         1.00
Left to recapture Single Line$       32,592
Recaptured YR 4 $     (11,611) 100% 12.0         1.00
Left to recapture Single Line$       20,981
Recaptured YR 5 $      (9,314) 100% 12.0         1.00
Left to recapture Single Line$       11,667
Recaptured YR 6 $      (7,338) 100% 12.0         1.00
Left to recapture Single Line$        4,329
Recaptured YR 7 $      (4,329) 84% 10.1         0.84
Left to recapture Single Line$          (0)

 

This is an improvement over the simple payback method, but still does not take into account the overall rate of return and it does not address cash flows that occur after the payback period.

Before we address those issues, check your understanding of the payback method using discounted cash flows.

Practice Question