Reconciling Discounted Valuation Figures with Rounding and Timing

Summary

The document investigates a discrepancy in a transport appraisal table that values future years of life using a stated monetary amount and a discount rate. The questioner cannot reproduce the table's figure using the displayed rounded quantity. The response observes that successive annual table values are consistent with discounting at the stated rate, then suggests that the underlying years-of-life figure has more precision than the two-decimal value shown.

Using that unrounded quantity and applying a two-period discount factor reproduces the published amount. The example illustrates how displayed rounding and the timing convention for discounting can create small differences when reconstructing a calculation. The explanation is specific to this table; it does not establish the report's full methodology or independently verify the source data and assumptions.

Key ideas

• Successive annual values in the table are consistent with the stated discount rate.
• A displayed quantity rounded to two decimals may conceal precision used in the calculation.
• Using the more precise underlying quantity with the stated discount timing explains the reported value.
• Check rounding and discount-period conventions when reproducing published discounted valuations.

Tags

# Investment Appraisal/Discounting


# Investment Appraisal/Discounting












I would appreciate any consideration given to the question I have with respect to a document published by the UK Dept. of Transport.

https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/639211/research-into-valuing-health-impacts-in-transport-appraisal.pdf

Here, on page 20 (Table 11) of this report, the authors use standard discounted cash flow methods for discounting future costs/future monetisable benefits.

In particular, with an actuarial year of life lived (YLL) set equal to £60,000, they estimate that in 3.92 YLLs in 2012 should be worth £228,476.34.

However, when I use standard discounting formulas in which £60,000 is discounted by 0.015; or discount YLLs by 0.015, I can only seem to get approximately £228,114.

If someone with expertise had 10 mins to sink their teeth into this, would it possible to show how £228,476.34 on Table 11 was obtained please?

## Answer by Bob Jansen (score 2, accepted)

https://quant.stackexchange.com/a/68990

I see that the subsequent years are discounted using 1.5%: \begin{eqnarray} 228,476.34\, /\, 225,099.84 - 1 = 1.5\% \\ 225,099.84\, /\, 221,773.25 - 1 = 1.5\% \end{eqnarray} If you allow that the YLL displayed as 3.92 is actually 3.9230339 but rounded then $$1.015^{-2} \times 60.000 \times YLL = 228,476.34$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.