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