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Appendix 3A – Alternative methods of calculating an equivalent yield The first method is by calculating a positive and a negative net present value at two ‘trial’ equivalent yields and then using linear interpolation (through similar triangles) to interpolate the equivalent yield where the net present value (NPV) is zero (i.e. the IRR) (The relationship between the IRR and NPV is actually non-linear.). The net present value is calculated by valuing the property at two trial equivalent yields and then deducting the price paid to calculate one positive and one negative NPV. (1) 8% trial equivalent yield Term rent (£) YP 4 years @ 8% 250 000 3.3121 828 025 Reversion to MR 300 000 YP perpetuity @ 8% 12.5000 Deferred 4 years (PV £1 for 4 years @ 8%) 0.7350 2 756 250 Valuation (£) Less: Term and reversion valuation (£) NPV (£) Property Valuation in an Economic Context © 2007 by Peter Wyatt 3 584 275 –3 189 356 394 919 (2) 9% trial equivalent yield Term rent (£) YP 4 years @ 9% 250 000 3.2397 809 925 Reversion to MR 300 000 YP perpetuity @ 9% 11.1111 Deferred 4 years (PV £1 for 4 years @ 9%) 0.7084 2 361 331 Valuation (£) 3 171 256 Less: Term and reversion valuation (£) -3 189 356 |NPV (£) (3) Estimating an equivalent yield using similar triangles Property Valuation in an Economic Context © 2007 by Peter Wyatt (18 000) NPV (£) A NPV1 IRR estimate x C IRR1 0 NPV2 E B IRR2 IRR (%) D The geometric properties of similar triangles mean that there is a ratio between triangle ABC and triangle ADE. Examining the lengths of the sides of these triangles we can state: x IRR 2 − IRR 1 = NPV1 NPV2 + NPV1 [3.3] Therefore: ⎡ IRR 2 − IRR 1 ⎤ ⎤ ⎡ 1% x = NPV1 × ⎢ ⎥ = 394 919 × ⎢ ⎥ = 0.9564 ⎣ 394 919 + 18 000 ⎦ ⎣ NPV2 + NPV1 ⎦ And the IRR estimate is obtained by adding this increment to the 8% trial yield (the gap between the origin and IRR1 on Figure 3.12), that is, 8% + 0.9564 = 8.96%, which is closer to the 9% trial yield and this can be seen from the fact that the NPV at 9% was much closer to zero that the NPV at 8% yield. As a check, value the property using a term and reversion approach with an equivalent yield: Property Valuation in an Economic Context © 2007 by Peter Wyatt Term rent (£) YP 4 years @ 8.96% 250 000 3.2426 810 650 Reversion to MR (£) 300 000 YP perpetuity @ 8.96% 11.1607 Deferred 4 years (PV £1 for 4 years @ 8.96%) 0.7095 2 375 555 Valuation (£) 3 186 205 There is a small discrepancy in the valuation due to rounding error. And if this equivalent yield is input into a core and top-slice valuation of the same property, an identical valuation should result: Property Valuation in an Economic Context © 2007 by Peter Wyatt Core: Contract rent (£) 250 000 YP in perpetuity @ 8.96% 11.1607 2 790 175 Top-slice: Uplift to MR 50 000 YP perpetuity @ 8.96% 11.1607 Deferred 4 years (PV £1 for 4 years @ 8.96%) 0.7095 395 926 Valuation (£) 3 186 101 The second method, devised back in the days before spreadsheets, was to divide the sum of the term rent and annual equivalent of the gain on reversion by the capital value. So, whereas the initial yield is income divided by capital value, here: Equivalent yield = (term rent + annual equivalent of gain on reversion)/capital value Where annual equivalent of gain on reversion = (gain on reversion x PV £1 for term)/YP for term and the gain on reversion = capital value on reversion – capital value today So, using our term and reversion valuation, the calculation of the equivalent yield would proceed as follows: Gain on reversion = (£300 000 × YP perpetuity @ 11%) – £3 189 356 = £3 333 330 – £3 189 356 = £143 974 Property Valuation in an Economic Context © 2007 by Peter Wyatt Annual equivalent of this gain = (£143 974 × PV £1 @ 8.5%* for 4 years)/YP 4 years @ 8.5% = (£143 974 × 0.7216)/ 3.2756 = £31 717 *First estimate of equivalent yield Equivalent yield = £300 000 + £31 717 £3 189 356 = 0.0883 or 8.83% The annual equivalent of the gain should then be recalculated using this yield in an iterative process until the first and final estimate equate. Property Valuation in an Economic Context © 2007 by Peter Wyatt