8.5 Yield Capitalization Property Models and Valuing Leased Fee and Leasehold Interests

8.5 Yield Capitalization Property Models and Valuing Leased Fee and Leasehold Interests

1. Property Models: Matching the Rate to the Income Pattern

Here $\text{SFF}$ is the sinking fund factor, $\text{SFF}_{Y,n} = Y / [(1+Y)^n - 1]$, and $\text{CR}$ is the annual compound rate of change (the $g$ in the Gordon model of Section 8.3).

Comparing the Recapture Premises

A building with a 25-year remaining economic life and a 9% yield rate:

• Ring (straight-line): $R = 0.09 + 1/25 = 0.09 + 0.04 = 0.13$, or 13.0%.
• Inwood (level annuity): $\text{SFF}_{9\%,25} = 0.09 / (1.09^{25} - 1) \approx 0.0118$, so $R \approx 10.18\%$.
• Hoskold at a 4% safe rate: $\text{SFF}_{4\%,25} \approx 0.0240$, so $R \approx 11.40\%$.

Ring produces the highest rate (lowest value) because it assumes income declines; Inwood produces the lowest because it assumes level income with recapture reinvested at the full yield rate.

Value Change Models

An investor requires a 10% yield and expects property value to rise 20% over a 10-year holding period:

• Level income with value change: $\text{SFF}_{10\%,10} \approx 0.0627$, so $R = 0.10 - 0.20 \times 0.0627 \approx 8.75\%$.
• Straight-line change in income and value: $R = 0.10 - 0.20 \times 0.10 = 8.0\%$.
• Exponential change at 2% a year: $R = 0.10 - 0.02 = 8.0\%$.

The Mortgage-Equity (Ellwood) Formula

The Ellwood formula extends the band of investment (Section 8.2) to a holding period with loan amortization and value change:

$$R_o = Y_e - M \left( Y_e + P \times \text{SFF}_{Y_e,n} - R_m \right) - \Delta_o \times \text{SFF}_{Y_e,n}$$

Where $Y_e$ is the equity yield rate, $M$ the loan-to-value ratio, $P$ the fraction of the loan paid off during the holding period, $R_m$ the mortgage constant, and $\Delta_o$ the total change in property value.

Example: A 70% loan at 6.5% interest, amortized monthly over 25 years ($R_m \approx 0.08102$), is held for 10 years. After 120 payments the balance is about 77.51% of the original loan, so $P \approx 0.2249$. With a 12% equity yield ($\text{SFF}_{12\%,10} \approx 0.0570$) and no change in value:

$$R_o = 0.12 - 0.70 \times (0.12 + 0.2249 \times 0.0570 - 0.08102) \approx 0.12 - 0.0362 \approx 0.0837$$

The indicated overall rate is about 8.37%. If value were expected to fall 10% over the decade, adding $0.10 \times 0.0570$ raises the rate to about 8.94%.

2. Mortgage Calculations for Reversions

The loan balance at the end of a holding period equals the payment times the present value annuity factor for the remaining term. For the loan above, the balance after 10 years is about 77.51% of the original principal: on an $8,361,403 loan from Section 8.2, about $6,481,000. If the property's net resale proceeds are $13,000,000, the equity reversion is $13,000,000 - $6,481,000 = $6,519,000.

3. Valuing Fee Simple, Leased Fee, and Leasehold Interests

Yield Capitalization of a Below-Market Lease

A single-tenant building is leased for 5 more years at a flat net rent of $400,000. Market net rent is $500,000, and market conditions are stable. The property is expected to be re-leased at market at the end of the term, when it should be worth $500,000 ÷ 0.08 = $6,250,000. The market discount rate for the leased fee is 8%, and the tenant's position (a leasehold) carries more risk, warranting 12%.

The sum falls about $39,000 short of the fee simple value because the leasehold is discounted at the higher 12% rate. Discounted at 8%, the leasehold would be worth $399,271, and the two interests would add back to $6,250,000. This is why the guide's earlier rule of thumb says fee simple is only approximately the sum of the leased fee and the leasehold.

Direct Capitalization of a Long-Term Ground Lease

A tenant built an office building on ground-leased land with 60 years remaining. The building produces $1,150,000 of NOI, and the tenant pays $250,000 a year in ground rent.

• Leasehold (tenant's position): ($1,150,000 - $250,000) ÷ 9.0% leasehold rate = $900,000 ÷ 0.09 = $10,000,000.
• Leased fee in the land (landowner's position): $250,000 ÷ 6.25% ground lease rate = $4,000,000.
• Fee simple of the whole, if unencumbered: $1,150,000 ÷ 8.0% = $14,375,000.

Direct capitalization is reasonable here because the term is long and the income is stable; with short or changing terms, yield capitalization is the better tool.