LA/UP 341 Fall 1999

Practical 5 --

The effect of time on the economic
value of land resources: Discounting

In this case I have looked at the travel costs associated with visiting Birder's Paradise as future values that need to be weighed against the capital costs of developing the BP birding and leisure complex.

I chose Birder's Paradise simply because I already had worked through the travel cost evaluation for Practical 4.

The example I have provided below shows how to take a typical value - in this case derived from a travel cost model - project it over the next ten years then bring it back to its Present Value so we can weigh it against the Costs, which are mostly borne at Start-up.

For those costs that occur annually the Present Value is very easy to figure -- it's the same as the current cost would be if we assume that the appropriate discount rate is the same as the inflation rate. Once we accept there will be inflation at a given rate there is good reason to adopt that rate for discounting back to present value for those things like labor etc about which we can be absolutely confident we will need to use, and whose increases will closely follow the inflation rate. See that worked through in the Present Value at Discount rate of 3% portion of the table below.

The Case of Birder's Paradise:

This shows the travel cost model evaluation estimate of $1.049,375 annually being projected out 10 years at 3% inflation per year -- EXCEL formula for inflation:

=SUM($valuetoday*(POWER((1+inflation-rate),time)))

Then those values for each future year brought back to present values using a discount rate of 10% -- EXCEL formula for present value:

=SUM($inflatedvalue/(POWER((1+disc-rate),time)))

Present value of future travel to Birder's Paradise

The inflated costs show the dollars that would have to change hands, and the present value of all those future dollars.

Annual travel cost value:
$1,049,375.00
	year 1	year 2	year 3	year 4	year 5	year 6	year 7	year 8	year 9	year 10
future value at inflation rate of: 3.00%	1,080,856	1,113,282	1,146,680	1,181,081	1,216,513	1,253,009	1,290,599	1,329,317	1,369,196	1,410,272
total inflated cost for next 10yrs. at 3%:	$12,390,806
	year 1	year 2	year 3	year 4	year 5	year 6	year 7	year 8	year 9	year 10
present value at discount rate: 10.00%	982,597	920,068	861,518	806,694	755,359	707,291	662,281	620,136	580,673	543,721
present value of travel at r=10%:	$7,440,337

Next is the peculiar case of using the same discount rate as inflation rate -- if you are very confident about your future values.

Annual travel cost value:
$1,049,375.00
	year 1	year 2	year 3	year 4	year 5	year 6	year 7	year 8	year 9	year 10
future value at inflation rate of: 3.00%	1,080,856	1,113,282	1,146,680	1,181,081	1,216,513	1,253,009	1,290,599	1,329,317	1,369,196	1,410,272
total inflated cost for next 10yrs. at 3%:	$12,390,806
	year 1	year 2	year 3	year 4	year 5	year 6	year 7	year 8	year 9	year 10
present value at discount rate: 3.00%	1,049,375	1,049,375	1,049,375	1,049,375	1,049,375	1,049,375	1,049,375	1,049,375	1,049,375	1,049,375
present value of travel at r=3%:	$10,493,750

The formulae above only give you values for one year at a time. That is practical, maybe, up to about ten years as in the examples but you need another formula to work out the values over long time periods.

This link retrieves an Excel spreadsheet with the correct formulae allowing you to adjust projected benefits, inflation, and discount rate:

Excel table

Discounting over multiple years

Discounted present value of benefits for next
10	years		(using formula vs. summing individual years)
present value of
travel at r=20.00%		4,192,472	*"=SUM(A$4((1+A44)^A$41-1)/(A44(POWER((1+A44),A$41))))"*

10.00%		6,144,567	*"=SUM(A$4((1+A46)^A$41-1)/(A46(POWER((1+A46),A$41))))"*

3.00%		8,530,203	*"=SUM(A$4((1+A48)^A$41-1)/(A48* ((1+A48)^A$41)))"**

3.00%		8,530,203	*"=SUM(A$4((1+A51)^A$41-1)/(A51* (POWER(1+A51,A$41))))"**

Discounted present value of benefits for next
10	years		TAKING INTO ACCOUNT INFLATION
present value of
travel at r=	inflation=
10.00%	3.00%	7,090,256	*"=SUM(A$4((1+((A59-B59)/(1+B59)))^A$41-1)/(((A59-B59)/(1+B59))(POWER((1+((A59-B59)/(1+B59))),A$41))))"*

effective discount rate (the discount rate adjusted for inflation) is calculated by (in the case above) "=((A59-B59)/(1+A59))"

if inflation i =	3.00%
effective discount =
6.80%		7,090,256	*"=SUM(A$4((1+A63)^A$41-1)/(A63(POWER((1+A63),A$41))))"*

Another example - the Tree Farm:

Have a look at these tables, especially those related to the value of trees in a tree farm and see if they make the concept of discounting any clearer.

Effect of a 10% discount rate

These figures represent the value to you of a $100 something you get 100, 50, 40, 30, 20, 10, 5, or 1 year from now, first assuming a fair amount of risk -- 10% -- and next assuming a risk close to the rate of inflation -- 3%.

	discount	rate=10%

present value	year 1	year 5	year 10	year 20	year 30	year 40	year 50	year 100

$ 0.01	$ 0.01	$ 0.01	$ 0.02	$ 0.05	$ 0.13	$ 0.33	$ 0.85	$100.00

$ 0.85	$ 0.94	$ 1.37	$ 2.21	$ 5.73	$ 14.86	$ 38.55	$100.00

$ 2.21	$ 2.43	$ 3.56	$ 5.73	$ 14.86	$ 38.55	$100.00

$ 5.73	$ 6.30	$ 9.23	$ 14.86	$ 38.55	$100.00

$ 14.86	$ 16.35	$ 23.94	$ 38.55	$100.00

$ 38.55	$ 42.41	$ 62.09	$100.00

$ 62.09	$ 68.30	$100.00

$ 90.91	$100.00

$100.00

Effect of a 3% discount rate

This is a little like asking what will my $100 be worth in 100 etc years if I hide it while the world is undergoing 3% inflation. 3% is not much and you could buy $97.09-worth of stuff just one year from now. At 5yrs you will still get $86.26-worth, but at 20yrs it will have dropped to $55.37.

	discount	rate=3%

present value	year 1	year 5	year 10	year 20	year 30	year 40	year 50	year 100

$ 5.20	$ 5.36	$ 6.03	$ 6.99	$ 9.40	$ 12.63	$ 16.97	$ 22.81	$100.00

$ 22.81	$ 23.50	$ 26.44	$ 30.66	$ 41.20	$ 55.37	$ 74.41	$100.00

$ 30.66	$ 31.58	$ 35.54	$ 41.20	$ 55.37	$ 74.41	$100.00

$ 41.20	$ 42.43	$ 47.76	$ 55.37	$ 74.41	$100.00

$ 55.37	$ 57.03	$ 64.19	$ 74.41	$100.00

$ 74.41	$ 76.64	$ 86.26	$100.00

$ 86.26	$ 88.85	$100.00

$ 97.09	$100.00

$100.00

Values of future trees for sale discounted at 10% and 3%

Basically, what this shows is how much those trees are worth now if you expect to get $20, $30, $40, or $50 in five years' time.

SO... if you expect to make $30 each, and you feel the risk is worth 10% discount, then you should invest no more than $18.63 per tree in growing them.

IF... you feel more confident, then it will still be viable at $25.88 of investment.

discount rate=10%

present
value year 1 year 2 year 3 year 4 year 5

$ 12.42 $ 13.66 $ 15.03 $ 16.53 $ 18.18 $20

$ 18.63 $ 20.49 $ 22.54 $ 24.79 $ 27.27 $30

$ 24.84 $ 27.32 $ 30.05 $ 33.06 $ 36.36 $40

$ 31.05 $ 34.15 $ 37.57 $ 41.32 $ 45.45 $50

discount rate=3%

present
value year 1 year 2 year 3 year 4 year 5

$ 17.25 $ 17.77 $ 18.30 $ 18.85 $ 19.42 $20

$ 25.88 $ 26.65 $ 27.45 $ 28.28 $ 29.13 $30

$ 34.50 $ 35.54 $ 36.61 $ 37.70 $ 38.83 $40