Relating Real Estate Cap Rates to Discount Rates and Income Growth

# The link between discount rates and cap rates?


# The link between discount rates and cap rates?












I’m reading an article found here: https://www.gwlrealtyadvisors.com/research_report/yielding-perspective-cap-rates-discount-rates-and-relative-value-for-real-estate/.

The article mentions the relationship between the discount rate/cap rate spread, and NOI. The author shows that the discount rate/cap rate spread moves with changes in NOI.

He mentions that “Generally, the higher the spread between discount rates and cap rates, the higher the income growth required to justify current property values.” How does this explain the correlation between the spread and NOI? Does that statement not assume stable property values which must be held?

## Answer by Sergei Rodionov (score 1, accepted)

https://quant.stackexchange.com/a/63032

Discount rate is a rate at which future cash flows are deflated to current dollars.

Lets assume we have a real estate asset producing a series of payments in perpetuity:

$$ C_0, C_1, C_2, ... $$

If the payment amount is fixed ($C_0$ = $C_i$), the present value of these payments can be calculated as follows:

$$ { PV = \frac{C_0}{R_f + R_p} \qquad (1) }$$

Where $R_f$ is the nominal risk free rate, and $R_p$ is the risk premium, and $R_f + R_p$ is the discount rate.

If the payment amount is growing at a rate $R_g$, the formula is:

$$ { PV = \frac{C_0}{R_f + R_p - R_g} \qquad (2) }$$

The cap rate $R_c$ is the ratio of the most recent payment (first payment in our sequence) to price:

$$ { R_c = \frac{C_0}{Price} \qquad (3) }$$

Assuming the cash flow is fairly valued by the market, we can substitute $Price$ for $PV$ in (3):

$$ { PV = \frac{C_0}{R_c} \qquad (4) }$$

From (2) and (4) we can observe that $R_f + R_p - R_g = R_c$. Re-arranging:

$$ { (R_f + R_p) - R_c = R_g \qquad (5) }$$

We get the "spread between discount rates and cap rates" on the left side. It also means that this spread must be equal to the growth rate for a fairly priced asset.

> Does that statement not assume stable property values which must be held?

Property prices can move in any direction as long as the linear relationship (5) is maintained.

For example, if $R_g$ in (5) remains unchanged (landlords not able to extract more rental income) whereas $R_f$ increases due to inflation, the prices will have to fall in order for $R_c$ to "absorb the hit". In the below example, nominal risk free rate increases from 2% to 4%, prices decrease by 22%:

$$ \begin{matrix} R_f & R_p & R_g & PV\\ 0.02 & 0.06 & 0.01 & 1/(0.02+0.06-0.01)=14.29 \\ 0.04 & 0.06 & 0.01 & 1/(0.04+0.06-0.01)=11.11 \\ \end{matrix} $$

You can calculate other scenarios from this, but keep the assumptions in mind.

As a minor note, the article states that:

> "discount rates are typically based on their (pension funds) required liabilities and future funding levels".

This is most likely incorrect. The fund's capital structure has no bearing on the discount rate of a real estate asset. Perhaps the author was referring to IRR.