Let us consider the meaning of present value (P V ) for a future stream of losses (P V F L ), as opposed to income (P V ). This discussion comes out of the long and vigorous discount debate in the context of climate economics. This debate can be exemplified by the discussion between William Nordhaus [3], and Nicholas Stern [5] [2]. Here we will show that both sides of this debate have missed a key matter. For our presentation here, we accept the nominal use of discounting a future loss stream, but note the ethics debate on inter-generational equity that questions the morality of discounting climate losses at all [6, p 24]. Discounting of a future loss stream, like discounting of a future income stream is based on two aspects of discounting, the time value of money (d TVM ) and the risk discount factor (r rar ) that is applied to d TVM To examine this matter of discounting PV FL for risk, we will start with income, which is the usual case that PV was created for, step through that logic, then apply the same logic to a stream of future losses. We will begin with an expositional discussion, and then proceed to the mathematical proof. When an evaluation is made of risk relative to a future stream of income, what is the risk an investor worries about? The risk is that real future income will be lower than expected based on current discount rates. So, to address this, a projection is made using a higher discount rate to show what the effect on PV is. The higher the discount rate, the lower the PV becomes. For the purposes of this discussion, we will nominally assign the mean inflation adjusted federal reserve discount rate (1.43%) to d TVM . The precise value of d TVM can be varied. The PV curve for d TVM is visible in fig. 1(a). To adjust a future projection of income for risk, this baseline rate, d TVM , will be modified by the risk discount factor, r rar , giving us a discount net of risk: d NR = d TVM × r rar eq. 1. William Nordhaus used up to 5.1% risk accounted discount rate [4]. Using a 5.1% DICE discount rate, r DF = d DICE d T V M = 3.57. This 3.57 risk adjustment factor we apply to d T V M to obtain a risk curve, for d Risk . in fig. 1(a). (a) Calculation of r rar using DICE rate. Where:

3 Present value of future losses (PV FL )

Now, let us ask the question, "What does risk mean for a stream of future losses?" Risk in this case means that the losses, in real terms, could be larger than expected, rather than smaller. Note that this is the opposite of the meaning of risk for future income, although the direction of net income after accounting for risk is the same, as both curves move in a negative direction. So how can we represent losses being greater than expected using a discount rate? To represent larger losses using a discount rate, the rate must be lower than than the baseline d TVM rate. So, instead of multiplying to increase the discount rate by r rar as is correct for PV of future income, one must divide by r rar . (Alternatively, a risk factor could be subtracted from the discount rate.) The curve for the PV FL version of d NR is shown in eq. 2 and fig. 1(b). 4 A proof that risk adjustment for losses must have the opposite sign to the discount rate for the time value of money

Let us begin with a project that has an expected positive revenue stream E(t), and an actual positive revenue stream A(t). At an arbitrary value of t, E(t) is compared with A(t). We will assume that both E(t) and A(t) start at the same initial value of income I, and grow exponentially at different rates g E and g A respectively. The actual and expected returns are:

Where: g E and g A > 0

The discounted value of those cash flows subtracts a positive term for the time value of money, d tvm , from both exponents, to account for the lower present value of future income versus income today. Where:

Let us next consider the relevant risk in the ordinary present value case, that actual returns will be below expected returns. It is possible for a project with a positive future income stream to be net present value positive on a time value of money basis, but lower-or even negative-on a risk adjusted basis. Thus, a risk adjustment rate, d rar , is applied to E tvm (t) to cover the case where returns are lower than expected. There is no need to apply d rar to actual returns, because actual returns are the basis for determining what the correct fit value of d rar is. The correct fit value for d rar (which cannot be known in advance) equates the expected stream to the actual stream. Call this d rarF . To calculate it, we only care about the growth terms in eq. 5. Thus:

Therefore, because we are risk adjusting for g A being lower than g E , d rarF is positive, as expected, and as accords with standard practice. Now let us consider a calamity, with an expected negative stream of damages D E and an actual negative stream of damages D A . The loss streams (L) start at the same initial value L < 0. These damages grow over time at the rates g E and g A . And as before, a time value of money discount d tvm is applied to both loss streams, since losses at a future date are less concerning than losses in the present.
In the case of future losses, the risk is that these will be larger than expected. We therefore seek to find a value of d rar to account for true losses of a larger magnitude than expected losses. As before, d rar is applied only to expected losses, since D Atvm (t) is the benchmark we seek to match:

As in the positive income stream case, we are only interested in the growth terms of eq. 11. We are again interested in finding the fit (or actual) value for d rar , which is d rarF . g E -d tvm -d rarF = g A -d tvm Where: g A < g E < 0 (12) g E -d rarF = g A (13)

Therefore, because the growth rate of actual damages, g A , exceeds those of expected damages g E , d rarF will be a negative quantity. Since this term is subtracted from the exponent, the effect is to make the total discount smaller, not larger, when risk is taken into account. Stern is correct [5] in his contention that after taking risk into account, a lower discount rate must be used.