Class 6: Time Value of Money in Real Estate Definitions: PV, FV, annuity.

 Class 6: Time Value of Money in Real Estate

Time Value of Money (TVM) — the essentials, how they fit together, and 

why real estate runs on them.

PV ------------------------>  Int Rate /Time ------------------->  FY

Payment (inflow/Outflow)

Calculator

Core ideas & definitions

• Time Value of Money (TVM)
  A dollar today is worth more than a dollar tomorrow because you can invest it and earn a return (or avoid paying financing costs). TVM turns cash at different dates into comparable values using an interest (discount) rate.

• Interest rate (r)
  The growth (or discount) rate per period that moves money through time. It can be:

  • Compounding: annually, quarterly, monthly, etc.

  • Nominal vs. effective: effective rate reflects compounding (e.g., 6% nominal compounded monthly ≈ 6.168% effective annually).

  • Required return/discount rate when valuing investments (includes risk and opportunity cost).

• Present Value (PV)
  What a future cash flow is worth today given rate r.

  $$PV=\frac{FV}{(1+r)^n}$$

• Future Value (FV)
  What money today grows to in the future after n periods at rate r.

  $$FV=PV(1+r)^n$$

• Periodic Payment (PMT)
  A constant, per-period cash flow (e.g., a mortgage payment or a level lease receipt). Two common patterns:

  • Ordinary annuity (end of period):

    $$PV=\text{PMT}\cdot\frac{1-(1+r)^{-n}}{r},\qquad FV=\text{PMT}\cdot\frac{(1+r)^n-1}{r}$$

  • Annuity due (beginning of period): multiply the above by $(1+r)$.

Tiny numeric feel: $1,000 today at 5% for 3 years → $FV=1000(1.05)^3=1{,}157.63$.
$1,000 received in 3 years, discounted at 5% → $PV=1000/(1.05)^3=863.84$.
$100 each year for 3 years at 5% (ordinary annuity) → $PV\approx \$272.32$.

How they relate

Think of the TVM “family” as one equation with different unknowns:

• Given PV, r, n → find FV (growth).

• Given FV, r, n → find PV (discounting).

• Given PV (or FV), r, n → find PMT (amortizing or saving).

• Given cash flows & PV → solve for r (that rate is the IRR).

• Compounding frequency changes results because it changes how often r is applied.

In practice, you can solve for any one variable if you know the others. That’s why calculators and spreadsheets (e.g., PV, FV, PMT, RATE, NPER, IRR, XIRR in Excel) are so handy.

How TVM is used in real estate

Real estate decisions are streams of cash flows (rents, expenses, loan payments, sale proceeds). TVM is the engine that prices those streams.

1. Mortgage structuring & payments (PMT)

   • Compute the monthly payment on a loan:

     $$\text{PMT}=\frac{L\cdot r_m}{1-(1+r_m)^{-N}}$$

     where $L$ is the loan amount, $r_m$ the monthly rate, and $N$ the number of monthly payments.

   • Split each payment into interest and principal over time (amortization schedule).

2. Investment valuation (PV/discounting)

   • Discounted Cash Flow (DCF): Forecast annual NOI → levered cash flows → reversion (sale price) and discount each at a required return (discount rate) to arrive at present value (value today).

   • Net Present Value (NPV): Sum of discounted cash flows minus the price paid; positive NPV implies value creation at the chosen discount rate.

   • Internal Rate of Return (IRR): The rate that sets NPV = 0 for the projected cash flows (widely used for equity comparisons and waterfalls).

3. Cap rates vs. discount rates

   • A cap rate is NOI / Price in a single-period snapshot.

   • In a stable-growth setup, cap rate ≈ discount rate – long-term NOI growth (a Gordon-growth intuition). TVM underpins how cap rates map to multi-period DCF assumptions.

4. Refinancing & hold/sell decisions

   • Compare the PV (or IRR) of holding vs. selling now (realizing sale proceeds, taxes, and reinvestment options). TVM makes these timelines comparable.

5. Lease analysis & rent escalations

   • Discount stepped rents, free-rent periods, percentage rent, and tenant improvement allowances to a PV for apples-to-apples comparisons across tenants or proposals.

6. Development & construction pro formas

   • Time-phase hard/soft costs, construction draws, and lease-up; discount to PV or compute project IRR. Sensitivities to timing (delays) matter because of compounding.

7. Debt sizing & coverage

   • Use PMT to test DSCR (NOI/Annual Debt Service), size proceeds that satisfy lender constraints, and evaluate LTV at various rates/terms.

8. Waterfalls & promote structures

   • IRR and equity multiples trigger promote tiers; TVM precisely determines when each hurdle is hit.