# Question about how NPV was used in this 2010 Obama proposal

I'm looking at a 2010 Obama proposal to allow 100% expensing (i.e. allowing a capital good to 100% depreciate in the first year to allow it as a tax expense). Here's the proposal: https://obamawhitehouse.archives.gov/sites/default/files/Expensing_Report.pdf

On Page 3 they have an example blurb that says:

Under current law, the business would only be able to deduct a fraction of its investment each year – about 143,000 in the first year, for example. At a tax rate of 35 percent, that would reduce the business’ taxes in the first year by 50,000. By contrast, under immediate 100 percent expensing, the business could deduct all 1 million in the first year – reducing the business’ taxes by $350,000. Not only does this provide the business with more cash on hand this year – money that can be used to expand and hire new workers – but because businesses value cash today more than cash in the future, immediate expensing also makes the investment more attractive. For example, using a “discount” rate of 6 percent, the value of deducting 1 million in the first year might be worth the full 350,000 for the business – but the value of deducting it over the entire 7-year recovery period would only be about$303,000, reflecting the added incentive the business receives with immediate expensing.

I'm wondering how they calculated that 303,000 number -- if I plug the numbers into Excel and do an NPV(6%, cells) where cells are 7 cells in a row with the tax deduction value of 50,000, I get a value of 279,119.07, about 24,000 shy of the 303K. What am I missing?

• The first year has no need for discounting (it is the same as if you expense immediately, just a smaller sum). The rest is ndone with mid year discounting (1.5 instead of 2 for example). Commented Apr 4, 2023 at 23:12
• @AKdemy interesting, the logic checks out but I still haven't been able to arrive quite at the 303,000 number; did you plug these in and get 303? Commented Apr 5, 2023 at 13:27

As mentioned in my comment, the first year has no need for discounting because it is the same as if you expense immediately, just a smaller sum. Subsequent years are using mid year discounting - the second year will be discounted with 0.5 (not 1.5, because the first year is 0, not 1), the third with 1.5 and so forth.

The text states it is about 303k, because it is not exactly 303k, but USD 134.76 more (see total discounted cashflow below). If you compute it, it looks like this (in Python).

import numpy as np
import pandas as pd

dc_rate = 0.06
total = 350000
years = 7
cf = total / years
val = [50000 for i in np.arange(0,years)]
year_frac = [0]+[i-0.5 for i in np.arange(1,years-0.5,1)]
dcf1 = [f'1/(1+{dc_rate})^{yf}' for yf in year_frac]
dcf2 = [1/(1+dc_rate)**yf for yf in year_frac]
res = [val/(1+dc_rate)**yf for val, yf in zip(val,year_frac)]
df = pd.DataFrame(zip(val, year_frac, dcf1, dcf2,  res), columns=['Cashflows','Year Fraction','Discount Term', 'Discount Factor','Discounted CF'])
print(f'Total discounted cashflow = {np.sum(df["Discounted CF"])}')
df


which yields the following output:

• Thank you! Minor tweak: "The text sates about 300k, because it is not exactly 300k" -> should be changed to "303k", which is what the text states. Commented Apr 10, 2023 at 17:17
• You are right. Sorry about that. I fixed the typo. Commented Apr 10, 2023 at 17:29