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Dual Rate Analysis: Foundations and Limitations
Chapter 6: Dual Rate Analysis: Foundations and Limitations
Introduction
This chapter delves into the dual-rate method, a traditional approach to leasehold valuation. We will explore its underlying principles, mathematical formulations, practical applications, and, importantly, its limitations in modern valuation contexts.
6.1 Historical Context and Theoretical Foundations
The dual-rate method emerged in the early to mid-20th century as a response to the specific economic and property market conditions prevalent at the time. Key assumptions underpinning this method include:
- Stable Rental Values: Rental levels were assumed to remain relatively constant throughout the lease term, with limited expectations of rental growth.
- Fixed Profit Rents: Consequently, profit rents (the difference between the rent received and the rent paid) were also expected to remain fixed until the lease expiry.
- Sinking Fund Replacement: The core concept involves the use of a sinking fund, into which a portion of the annual net profit rent is invested. This fund is assumed to accumulate at a low, safe yield, such that at the end of the lease, the fund is sufficient to replace the original purchase price. The low yield assumed for the sinking fund was typically based on rates offered by insurance policies at the time (around 2-3% net of tax).
6.2 Mathematical Formulation of the Dual Rate Method
The dual-rate method utilizes a Years’ Purchase (YP) multiplier calculated using two distinct rates:
- Remunerative Rate (i): This represents the required return on the capital invested, reflecting the risk associated with the leasehold investment. It is typically derived from the freehold all-risks yield but adjusted upwards to account for the inherent disadvantages of a leasehold interest. This is the discount rate applied to the ‘spendable income’.
- Sinking Fund Rate (SFi): This represents the rate of return of capital, the low and supposedly risk-free rate at which the sinking fund accumulates.
6.2.1 Derivation of Key Formulas
To understand the YP dual-rate formula, it is crucial to grasp the concepts of “Amount of £1 per Annum” and “Annual Sinking Fund.”
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Amount of £1 per Annum: This calculates the future value of investing £1 at the end of each year, compounded annually at a rate i for n years.
Formula:
Amount of £1 per annum = ((1 + i)^n - 1) / i -
Annual Sinking Fund (ASF): This calculates the annual amount that needs to be invested to accumulate to £1 at the end of n years, compounding annually at a rate i. It is the reciprocal of “Amount of £1 per Annum”.
Formula:
ASF = i / ((1 + i)^n - 1)
6.2.2 Years’ Purchase Dual Rate Formula
The YP dual rate formula combines these concepts to determine the present value of a leasehold income stream, considering both the return on investment and the return of investment.
*Formula:*
YP dual rate = 1 / (i + (SFi / ((1 + SFi)^n - 1)))
Where:
* i = Remunerative Rate
* SFi = Sinking Fund Rate
* n = Number of years
6.3 Practical Application: An Illustrative Example
Consider a shop leased at a ground rent of £2,000 per annum with 10 years remaining. The lessee sublets the premises for £12,000 per annum. We want to value the head lessee’s interest. Assume a freehold all-risks yield of 6%, a remunerative yield of 7% for the leasehold, and a sinking fund rate of 4%.
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Calculate Profit Rent: £12,000 (Sublease Rent) - £2,000 (Ground Rent) = £10,000
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Calculate the YP Dual Rate: Using the formula with i = 0.07, SFi = 0.04, and n = 10, we get approximately 6.5235.
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Capital Value: £10,000 (Profit Rent) * 6.5235 (YP Dual Rate) = £65,235
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Return On and Return Of Capital Analysis:
- Return on capital = £65,235 * 0.07 = £4,566.45 (spendable income)
- Return of capital = £65,235 * (0.04 / ((1.04)^10 - 1)) = £5,433.42 (sinking fund)
- The sum of the Return on Capital and the Return of Capital equals the Profit Rent (£10,000)
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Verification of Sinking Fund:
- Annual sinking fund * amount of £1 p.a. 10 years at 4% = £5433.42 p.a. × 12.006 = £65,234
- The capital recouped (£65,234) is approximately equal to the original purchase price (£65,235), proving the sinking fund’s effectiveness.
6.4 Limitations and Criticisms of the Dual Rate Method
Despite its historical relevance, the dual-rate method suffers from several limitations that make it less suitable for modern valuation practices, particularly in dynamic property markets:
- Unrealistic Sinking Fund Yields: The assumption of low, risk-free sinking fund yields is often unrealistic. Investors typically seek higher returns on their capital, especially considering loan interest rates are often significantly higher than the assumed sinking fund rate. Borrowers will have a loan with an interest rate in excess of the low accumulative rate and therefore would be extremely unlikely to set up a sinking fund.
- Inflationary Environment: The method fails to adequately account for inflation. The sinking fund only recovers the original purchase price in nominal terms, which may not be sufficient to replace the asset in real terms at the end of the lease. In times of rising inflation the accumulated sinking fund will not be adequate to replace the value in real terms.
- Investment Behavior: Investors often reinvest profit rents directly into other property investments rather than allocating funds to a separate sinking fund. Investors normally purchase a number of property leasehold and freehold investments and a sinking fund is rarely taken out as the whole of the profit rent will be reinvested in other similar property investments.
- Market Inefficiencies and Comparables: The scarcity of comparable leasehold transactions can make it difficult to accurately determine appropriate remunerative yields. The extra 1% or 2% increase on the freehold yield reflects these differences but may not be adequate, as a leasehold interest is a top slice investment with different growth potential and characteristics from a freehold investment (Baum and Crosby, 1995).
- Frequency and Timing of Payments: The method assumes annual payments in arrears, which does not reflect the common practice of quarterly payments in advance in modern leases. The YP dual rate assumes that the profit rent is receivable and the sinking fund payable annually in arrears, whereas most modern leases are let on quarterly in advance terms.
- Occupier-Specific Considerations: The method is not well-suited for valuing leasehold interests acquired by owner-occupiers, as they often view the purchase price as a rental payment made in advance and a deductible expense for income tax purposes.
- Profit Rent Gearing and Growth: The dual rate method is unable to reflect the gearing nature of some leasehold investments. If there are more rent reviews on the sublease than the head lease, then the profit rent has the potential to grow and is highly geared. This complex gearing is not reflected by the use of all risks yield derived from freehold analysis by adopting the all risks yield plus + 1% or 2%.
6.5 DCF as an Alternative
Given the limitations of the dual-rate approach, Discounted Cash Flow (DCF) analysis offers a more robust and flexible framework for leasehold valuation. DCF can explicitly model:
- Expected rental growth patterns
- Rent review cycles
- Varying discount rates reflecting risk profiles
- The terminal value of the asset
Example:
Consider two leasehold investments (Investment A and Investment B), each producing a current profit rent of £45,000 per annum, held on ten-year head leases. Both are sublet at Market Rental Value (MRV) with a rent review in five years. Investment A has a current MRV of £50,000, and Investment B has a current MRV of £250,000.
Using the dual rate method (e.g., 11% remunerative rate and 4% sinking fund), both investments would have the same capital value (£232,808) despite significantly different gearing and growth potential.
However, if rents are growing at 5% per annum:
- In 5 years, Investment A’s profit rent will increase by about 30%
- Investment B’s profit rent will increase by about 154%
A DCF analysis would capture this differential growth potential, assigning a higher value to Investment B. The top slice can be very sensitive to just small changes in the growth of full rental value and the profit rent is therefore very highly geared.
6.6 Conclusion
The dual-rate method provides a historical perspective on leasehold valuation. While it was suitable in a stable, low-inflation environment with predictable rental incomes, its limitations render it less appropriate for modern property markets. DCF analysis provides a more adaptable and comprehensive approach that can account for the complexities of rental growth, risk, and varying lease terms.
Chapter Summary
This chapter, “Dual Rate Analysis: Foundations and Limitations,” within the broader training course “Mastering Leasehold Valuation: From Traditional Methods to Modern DCF Analysis,” examines the scientific underpinnings and practical shortcomings of the dual-rate method for leasehold valuation.
The core scientific principle behind dual-rate analysis lies in the separation of return on capital from the return of capital. It acknowledges that a leasehold is a wasting asset. The method assumes an investor requires a “remunerative yield” (reflecting the risk associated with the investment and comparable to a freehold yield) and the means to recoup the initial investment at lease expiry. This recoupment is achieved through a “sinking fund,” a hypothetical or real fund accumulating at a lower, safe yield (historically based on low-risk insurance policies). The premise is that the annual “profit rent” (rent received minus ground rent) is divided: a portion provides the remunerative yield, and the remainder is allocated to the sinking fund. The Years Purchase (YP) dual-rate formula mathematically combines these two rates to derive a capital value.
The chapter details the mathematical derivation of the “amount of £1 per annum” and “annual sinking fund” formulas, essential components of the dual-rate calculation. It explains how the annual sinking fund formula calculates the small annual sum that should be invested in order to accumulate to £1 at the end of a given period of n years.
However, the chapter critically evaluates the limitations of this approach, particularly in modern property markets. Key shortcomings include:
- Unrealistic Sinking Fund Yields: The assumption of a low, risk-free sinking fund yield is often unrealistic in today’s investment environment. Borrowers are unlikely to accept a lower accumulative yield than the rate they are paying on their loan.
- Inflation and Capital Replacement: The sinking fund only recovers the nominal initial purchase price. In inflationary environments, this amount is insufficient to replace the asset’s real value at lease expiry, undermining the method’s goal of equating a wasting leasehold to a perpetual freehold.
- Investment Behaviour: The assumption that all leasehold investors establish and maintain sinking funds is often inaccurate. Reinvestment in other properties is more common.
- Comparable Evidence: The scarcity of comparable leasehold transactions with identical characteristics complicates the derivation of appropriate remunerative yields.
- Complexity and Opacity: The dual-rate method can obscure the true drivers of value. Different combinations of remunerative and sinking fund rates can produce the same YP multiplier, making it difficult to discern the underlying assumptions.
- Payment Frequency: The formula assumes annual payments in arrears, whereas quarterly payments in advance are now standard.
- Inapplicability to Owner-Occupiers: The concept of a sinking fund is not relevant when valuing an occupier’s interest.
- Gearing and Rental Growth: The dual-rate approach fails to capture the potential for geared rental growth, particularly when sublease reviews are more frequent than head lease reviews. The example provided demonstrates how two leasehold investments with the same current profit rent can have significantly different growth potential, which the traditional approach fails to recognize, while the DCF approach shows a more accurate valuation. The sensitivity of profit rent to changes in full rental value and the resulting complex gearing are also not adequately reflected.
The chapter concludes that the dual-rate method, while historically significant, is often inadequate for modern leasehold valuation. The limitations highlighted underscore the need for more sophisticated techniques, such as Discounted Cash Flow (DCF) analysis, which can explicitly model rental growth, variable yields, and other factors that the dual-rate method ignores. The chapter sets the stage for subsequent sections of the training course that explore these modern approaches.
