## Discount Rates

### Learning Outcomes

• Understand discount rates and how they are used in capital budgeting

Determining a “hurdle rate,” or the minimum acceptable return on a capital investment based on project risk is a critical step in the capital budgeting process.

Companies often use the weighted average cost of capital (WACC) as the minimum return a company must earn on its projects. It is calculated by weighing the cost of equity and the after-tax cost of debt by their relative weights in the capital structure. WACC is an important input in capital budgeting and business valuation. It is the discount rate used to find out the present value of cash flows in the net present value technique. It is the basic hurdle rate to which the internal rate of returns of different projects are compared to decide whether the projects are feasible.

Projected revenues and accounting earnings of each project are then measured and compared to the hurdle rate to determine if the project is a good investment. Projects that pass this hurdle are then ranked and selected on the basis of factors such as net investment return.

A simpler example of determining a hurdle rate without delving into the complexities of WACC would be to use an opportunity cost model.

For instance, let’s say you can buy a hat-embroidery machine for $40,000 and sell custom hats for$10 each on a website you already own. The hats cost you $4 and we’ll ignore the cost of embroidery thread for this example. Shipping costs$1 per hat. So, you make $5 per hat. If you could invest the$40,000 in a mutual fund and earn 8%, then you would most likely use 8% as your hurdle rate, also known as a discount rate.

Say the embroidery machine is designed to produce 10,000 hats before it breaks down permanently (we are making these assumptions in order to simplify the example.) You can sell 2,500 hats per year. Your time investment is insignificant since it’s all automated. The customer uploads the artwork and the machine does everything, including packaging and applying the shipping label. UPS swings by your house every day to pick up the shipment. In order to further simplify this example, we are also going to temporarily assume that all of the revenue is deposited into your bank account at the end of the fourth year. We’ll change this assumption later to be more realistic.

The future value of a $40,000 investment at 8% per annum compounded annually is$54,419.56. That’s what you would have at the end of four years if you just put your $40,000 into your chosen mutual fund.  Balance at beginning of year + interest at 8% = Principal and interest, end of year 1$ 40,000.00 $3,200.00$ 43,200.00 2 $43,200.00$   3,456.00 $46,656.00 3$ 46,656.00 $3,732.48$ 50,388.48 4 $50,388.48$   4,031.08 $54,419.56 You can also use a present value table to calculate the future value. You multiply the factor times the future value to get the present value, so you can simply divide the present value by the factor to get the future value. Using the table, the factor for four years at 8% is .735. Divide$40,000 by .735 and you get a future value of $54,421.77 (slight difference due to rounding of the factors in the table). Here is the table for calculating the present value, based on the formula PV = FV / (1 + i)n  Present value of$1 5% 6% 7% 8% 9% 10% n = 1 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091 n = 2 0.9070 0.8900 0.8734 0.8573 0.8417 0.8264 n = 3 0.8638 0.8396 0.8163 0.7938 0.7722 0.7513 n = 4 0.8227 0.7921 0.7629 0.7350 0.7084 0.6830 n = 5 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209

In a simple analysis, if you invest your $40,000 into the embroidery machine, at the end of four years, you would have$50,000 (10,000 hats at $5 net cash inflow per hat). That’s less than the$54,419.56 you would have if you just invested in the 8% mutual fund.

Here is another way to look at it: If you discount the $50,000 at 8%, you get$36,751.49, which is less than the $40,000 you are planning to invest. This tells you the same thing, that you are better off investing your$40,000 in a mutual fund that pays 8%.

 Balance at beginning of year + interest at 8% = Principal and interest, end of year 1 $36,751.49$   2,940.12 $39,691.61 2$ 39,691.61 $3,175.33$ 42,866.94 3 $42,866.94$   3,429.36 $46,296.30 4$ 46,296.30 $3,703.70$ 50,000.00

Note: If you know the future value is $50,000 and your chosen discount rate is 8% per year, and your time constraint is four years, then multiply the future value ($50,000) by the factor from the table (0.7350) and you will get the equivalent amount of money in today’s dollars:

50000 * 0.7350 = $36,750 Note that the factor is not as accurate as using the formula PV = FV / (1 + i)n In essence, this is telling you that you are trading your$50,000 for a future cash amount that is the equivalent of $36,750 today. That is not a good trade. In reality, this analysis is flawed because you are not getting the$50,000 in a lump sum at the end of four years—it is coming in as a stream of cash flows. We call a steady stream of fixed cash flows an annuity, and accountants have created a separate table for this:

Present Value of Ordinary Annuity 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 1.970 1.942 1.913 1.886 1.859 1.833 1.808 1.783 1.759 1.736 1.690 1.647 1.626 1.605 1.566 1.528 Period 3 2.941 2.884 2.829 2.775 2.723 2.673 2.624 2.577 2.531 2.487 2.402 2.322 2.283 2.246 2.174 2.106 Period 4 3.902 3.808 3.717 3.630 3.546 3.465 3.387 3.312 3.240 3.170 3.037 2.914 2.855 2.798 2.690 2.589 Period 5 4.853 4.713 4.580 4.452 4.329 4.212 4.100 3.993 3.890 3.791 3.605 3.433 3.352 3.274 3.127 2.991 Period 6 5.795 5.601 5.417 5.242 5.076 4.917 4.767 4.623 4.486 4.355 4.111 3.889 3.784 3.685 3.489 3.326 Period 7 6.728 6.472 6.230 6.002 5.786 5.582 5.389 5.206 5.033 4.868 4.564 4.288 4.160 4.039 3.812 3.605 Period 8 7.652 7.325 7.020 6.733 6.463 6.210 5.971 5.747 5.535 5.335 4.968 4.639 4.487 4.344 4.078 3.837 Period 9 8.566 8.162 7.786 7.435 7.108 6.802 6.515 6.247 5.995 5.759 5.328 4.946 4.772 4.607 4.303 4.031 Period 10 9.471 8.983 8.530 8.111 7.722 7.360 7.024 6.710 6.418 6.145 5.650 5.216 5.019 4.833 4.494 4.192 Period 11 10.368 9.787 9.253 8.760 8.306 7.887 7.499 7.139 6.805 6.495 5.938 5.453 5.234 5.029 4.656 4.327 Period 12 11.255 10.575 9.954 9.385 8.863 8.384 7.943 7.536 7.161 6.814 6.194 5.660 5.421 5.197 4.793 4.439 Period 13 12.134 11.348 10.635 9.986 9.394 8.853 8.358 7.904 7.487 7.103 6.424 5.842 5.583 5.342 4.910 4.533 Period 14 13.004 12.106 11.296 10.563 9.899 9.295 8.745 8.244 7.786 7.367 6.628 6.002 5.724 5.468 5.008 4.611 Let’s change our assumption about when the revenue hits your account from all at once at the end of four years to four equal installments of$12,500 at the end of each year.

Therefore, the cash flow is $12,500 per year (for a total of$50,000) for four years. Using the factor from the table (3.312), we see that the present value of this stream of equal cash flows (annuity) is $41,400. It’s slightly better now than using the lump sum analysis, because it takes into account that you get$12,500 in the first year, $12,500 in the second year, and so on. It’s still not perfect, because an “ordinary” annuity assumes the payment comes in on the last day of the year. Conversely, an annuity “due” assumes the payment comes in on the first day of the year. We could make this more accurate by using a table or formula that computes the present value monthly, using an interest rate of 8%/12 and n=48 months, or we could use a daily discounting (8%/365 and n= 4*365), but there is a limit to how accurate we need to be. We could estimate the rate of return on investment by dividing the initial investment of$40,000 by the $12,500 annual cash flow by the$50,000 we end with to get a factor of 3.2, which is somewhere between 9% and 10% on the n=4 row. In other words, our investment in the hat machine is the equivalent of putting our \$40,000 into a 9% or 10% mutual fund. If 10% was our discount rate, we would probably not invest in the business. If our discount rate (hurdle rate) was 9%, we would invest in the embroidery machine. (The actual rate of return on the machine is 9.5642%. We’ll address ways to figure this out in the next section.)

Here is a quick review of how to calculate the present value of an annuity using a table:

You can view the transcript for “PV of Annuity” here (opens in new window).

You should be able to see now that the time frame and the discount rate are both critical constraints in capital investment analysis, as well as a thorough understanding of the time value of money.

Now, check your understanding of the importance of the discount rate.