Real-Value Annuity

How can we evaluate a real-value annuity, and more particularly a composite real-value annuity? An annuity is a contract for a series of periodic constant nominal payments to begin on a given date and end on a subsequent given date or event. A real-value annuity is an annuity with constant real-value payments instead of constant nominal-value payments. Real values are nominal values adjusted for the expected rate of price-level inflation. Thus, real values are expected to have constant purchasing power in terms of some metric such as year-2000 dollars.

Two or more real-value annuities in sequence that increase in their level amount from one annuity to the next, i.e., with each subsequent annuity having an increased level amount of purchasing power, is a composite annuity that is referred to here as a "sequentially-escalating real-value annuity." It is sequential in the sense that the payments from the first annuity are followed by the payments for the second annuity, which in turn are followed by the payments for the third annuity, etc.   Such a composite real-value annuity can be evaluated using the DCF Valuator at Numeraire.com, but the procedure is somewhat tricky and thus requires extra careful attention.

The procedure will be explained with a hypothetical example. It can serve as an interim method for those who can't find a model specifically designed for evaluating composite real-value annuities. The example calculation results are reported with full decimals to enable replication and verification, although the implied precision is spurious due to assumptions about uncertain future inflation rates and investment returns. The calculation is followed by a theoretical explanation.

Investments and returns can be expressed as stocks and flows. Stocks are equivalent to lump sum payments, and flows are equivalent to annuities. Annuities can be either a constant real amount or a constant nominal amount. A rate of increase in price-level inflation (exponential growth rate) translates constant (no growth) real values to non-constant (growing) nominal values.

The DCF Valuator models are designed for exponential cashflow growth patterns. They are not designed for non-exponential increases in cashflow in discrete steps such as sequential annuities. Also, they are not designed for a growth pattern beginning with an initial period of annual cashflows equal to zero. Yet a stream of constant real values is equivalent to a stream of non-constant exponentially growing nominal values. Furthermore, leading zero cashflow years can be handled with additional calculations of present value. In general, two types of transformation are required:

First, decompose the composite annuity into separate component annuities.

Next, transform each constant-level annuity to the equivalent lump sum at a given date based on rate of return and inflation assumptions.

Last, transform each lump sum at a given date to the equivalent lump sum at a prior date based on rate of return and inflation assumptions.

For example, a woman and her husband wish to invest a lump sum now to provide real-value payments after taxes, transactions costs, and trust fund management expenses to their 11-year old granddaughter beginning when she reaches 21 years of age. The payments total $2.2 million over 50 years as follows.

The lump sum is assumed to be invested and generate a return net of taxes and costs equal to an average nominal 8% per year over the 50-year investment horizon, and inflation is assumed to average 4% per year over the same period.

The questions is, How large a lump sum at the beginning of Year 1 would be equivalent to this stream of annual real-value payments?

To answer this question, the DCF Valuator 1-Stage Value model can be used in two sequential steps for each of the four stages of constant annual cashflow: the first step to calculate the present value (PV) of the positive cash flows as of the first year of positive cash flow, and the second step to calculate the present value of this intrinsic value (IV) as of the first year of the investment holding period. With compounding and discounting over continuous time as opposed to discrete-interval time, the present value at the beginning of each stage is equal to the present value at the end of the prior stage. Once all four stages have been evaluated, the four present value sub-totals at the beginning of Year 1 can be added to get the total present value of the lump sum required to generate the stipulated future payments. Each calculation using the 1-Stage Value model is listed in the following table:

The third step is required because the DCF Valuator models are designed with a maximum 20-year investment horizon. Therefore, the 30-year period prior to Stage 4 is divided into two equal 15-year periods.

In the first step of calculating the present value of each stage, the question is, What initial lump sum investment will generate the given constant real-value annuity with zero final value? In step one, therefore, the initial "Base FCFE" is equal to the constant real-value cash payment, the Growth Rate is equal to the rate of inflation, and the Discount Rate is equal to the rate of net return on investment. In addition, the Price/FCFE ratio is equal to zero because the ending value or selling price is equal to zero.

In the second step of calculating the present value of each stage, the question is, What initial lump sum will generate the given ending value or selling price with zero annual cash payments in the interim period? In step two, therefore, the Price/FCFE ratio is set equal to the intrinsic value calculated in the first step, and the initial Base FCFE is set equal to a vanishingly small near-zero unit-value number such as $1. In addition, the Growth Rate is set equal to zero, and the Discount Rate is set equal to the net rate of net return on investment, calculated from the gross rate of net return and the rate of inflation.

For greater precision, the Price/FCFE can be set equal to the intrinsic value with the scale units multiplied by 1,000 or set equal to 1,000 and the initial cash flow divided by 1,000 or set equal to $0.001. The important point is that the product of the Price/FCFE ratio multiplied by the ending year FCFE be equal to the intrinsic value from the first step. The product of either $1.00 or $0.001 FCFE per year times the number of years is insignificant even before discounting and thus can be ignored.

The example is evaluated below to illustrate the method using a scale unit of 1 and total shares outstanding of 1. In step 1, the nominal growth rate (GR) is 4% (set equal to the inflation rate), and the nominal discount rate (DR) is 8% (required rate of nominal net return). The ratio (1+DR)/(1+GR) = 1.08/1.04 = 1.0384615. In steps 2 and 3, to adjust for inflation in terms of the initial reference year for constant money and to adjust for the real initial investment to return the ending real "selling price," the real growth rate is set equal to 0% to maintain a constant unit-value annuity, and therefore the real discount rate is set equal to 3.84615% so that the ratio (1+DR)/(1+GR) has the same value as in step 1. This calculation of the real discount rate is the tricky part.

In Step 2 of Stage 2, multiplying the Price/FCFE ratio by 1,000 to $245,201,180 and simultaneously dividing the Base FCFE by 1,000 to $0.001, decreases the Beginning of Year 1 PV to $168,122.12, a difference of a miniscule $5.73 ($10.00 minus 0.01 or $9.99 before discounting). This has no impact on the rounded final result.

The present value of the stipulated composite real-value annuity is $760,000. Thus, an investment of $760,000 today will generate cash payments of $2,200,000 over 50 years, which is a return-to-investment ratio of 2.90:1. This specific hypothetical example is covered more generally by the following theoretical explanation.

The DCF Valuator models are based on growth patterns. The 1-Stage and 2-Stage models differ in the number of growth stages. Growth stages are not interpreted the same as cashflow stages. In both cases, a stage here refers to a vertical division of the investment time horizon or holding period.

Growth stages assume a stipulated initial or base cashflow in year zero which increases exponentially for a stipulated number of years in the first stage at a stipulated growth rate and then for a stipulated number of years in the second stage at a second stipulated growth rate. The transition between the first and second stage is multiplicative, not additive, in its effect on the cashflow. Therefore, an initial cashflow of zero will result in zero annual cashflows throughout both the first and second stages because zero cashflow in stage one multiplied by any positive growth rate is zero.

In contrast, cashflow stages assume a stipulated constant cashflow per year for a stipulated number of years in the first stage and then at a stipulated constant cashflow per year for a stipulated number of years in the second stage. The transition from stage one to stage two has an addititive, not multiplicative, impact on cashflow. Thus, a first-stage annual cashflow of zero can incrementally increase to a second-stage positive annual cashflow because zero cashflow plus any positive cashflow is positive.

For intrinsic valuation applications where expected future cash flow investment returns are projected in order to calculate a net present value, growth stages are more intuitive to interpret than are cashflow stages. For financial planning applications where expected future cash needs are projected in order to calculate a financially equivalent initial lump sum investment, cashflow stages may be more practicable.

If an initial lump sum investment is projected to earn a compound rate of return throughout a holding period that spans two or more discrete cashflow stages, then it is no longer a straightforward discounting of future values to a present value. There are two factors that simultaneously determine the required initial lump sum: cash returns on the initial lump sum investment, and discounting annual future cash flows to the present.

In such a case, the cash flow stages can be converted to growth rate stages, and the DCF Valuator 1-Stage Value model can be used to estimate the size of the lump sum investment that is monetarily equivalent to stipulated annual cash flows under an assumed expected rate of return on investment. To convert cashflow stages into growth stages, it is necessary to decompose the overall pattern into components.

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