Summary
This explainer covers the two distinct meanings of “discount rate”: (1) the interest rate the U.S. Federal Reserve charges banks for short-term loans via the discount window, and (2) the rate used in discounted cash‑flow (DCF) analysis to convert future cash flows into present value. I define key terms, summarize how the Fed runs its discount window, show how to choose and compute a discount rate for valuation, give a short checklist, and work a compact numeric DCF example.
Definitions (first use)
– Discount window: the Federal Reserve facility through which the Fed lends short-term funds to depository institutions.
– Fed’s discount rate: the interest rate charged on discount-window loans.
– Discount rate (finance/DCF): the interest rate used to translate future cash flows into present value.
– Present value (PV): today’s equivalent of a future cash flow. Formula: PV = FV / (1 + r)^n, where FV is future value, r is the discount rate, and n is years.
– Net present value (NPV): sum of discounted cash flows less initial investment. Positive NPV implies the project/investment is expected to add value.
– WACC (weighted average cost of capital): a company’s average cost of capital from debt and equity; often used as a discount rate for firm-level cash flows.
– Risk-free rate: the return on a virtually default-free instrument (commonly short-term U.S. Treasury bills) used as a baseline in many models.
How the Federal Reserve’s discount rate and discount window work
– Purpose: provide very short-term liquidity to banks that cannot get funding in the market or face abrupt outflows. It is intended as a backup, not a regular funding source.
– Collateral: discount-window loans are collateralized; the borrower must pledge eligible assets. Emergency lending usually also requires collateral and typically requires Board approval.
– Loan types / tiers: the Fed operates three related types of discount-window credit. The Fed sets the rates for the principal lending categories directly; one category’s rate is tied to prevailing market rates.
– Term and pricing: most discount-window loans are overnight. In crises the Fed can extend loan terms (e.g., during 2007–2009 the Fed extended loans to 30 and 90 days temporarily). Discount-window rates are usually set above interbank market rates to discourage routine use and avoid signaling problems at a borrowing bank.
– Stigma: borrowing at the discount window can create a perception of distress, which is another reason banks typically avoid it unless necessary.
– Historical example (2007–2008): As financial conditions deteriorated, use of the window rose sharply. The Board cut the primary discount rate from 6.25% to 5.75% in August 2007; discount borrowing peaked in October 2008 at about $403.5 billion (compared with a long-run monthly average of around $0.7 billion before the crisis). Many of the temporary term and size changes were reversed after conditions normalized.
How the discount rate is used in DCF (valuation) and how to choose it
– Role: the DCF discount rate expresses the required return given time preference and risk. Higher discount rates reduce present values; lower rates raise them.
– Common choices:
– Risk-free rate: used when valuing cash flows considered essentially risk-free (rare for corporate projects). Usually the short-term Treasury yield is the baseline.
– WACC: often used for firm-level free cash flows because it averages the company’s cost of equity and after-tax cost of debt. Formula: WACC = (E/V)*Re + (D/V)*Rd*(1 − Tc), where E = market value of equity, D = market value of debt, V = E + D, Re = cost of equity, Rd = cost of debt, and Tc = corporate tax rate.
– Cost of equity via CAPM (Capital Asset Pricing Model): Re = Rf + β*(Rm − Rf), where Rf
=CAPM variables and related adjustments=
Re = Rf + β*(Rm − Rf), where:
– Rf = risk-free rate (see note below on nominal vs real).
– β (beta) = sensitivity of the asset’s returns to market returns (systematic risk).
– Rm = expected return on the market portfolio.
– (Rm − Rf) = equity risk premium (ERP): the extra return investors demand for bearing market risk.
Adjusting beta for capital structure
– Unlevered (asset) beta removes financial risk from leverage: beta_unlevered = beta_levered / [1 + (1 − Tc)*(D/E)].
– Re-lever to a new capital structure if you value a project/company with a different D/E: beta_levered_new = beta_unlevered * [1 + (1 − Tc)*(D/E)_new].
These formulas assume debt is relatively safe and taxes are relevant (Tc = corporate tax rate).
Real vs. nominal rates
– Match cash flows and discount rate: use nominal rates with nominal cash flows; use real rates with inflation-adjusted (real) cash flows.
– Fisher relationship: 1 + nominal ≈ (1 + real)*(1 + inflation). More precisely: real = (1 + nominal)/(1 + inflation) − 1.
Project and country risk adjustments
– If a project has risk not captured by the company beta (e.g., small project, regulatory risk), add a specific risk premium to the discount rate or adjust cash flows (project-specific premiums should be conservative and documented).
– For countries with political or currency risk, add a country risk premium (often derived from sovereign CDS spreads or equity-market volatility) to the equity risk premium or to Rd if debt is exposed.
=Step-by-step checklist: choosing a discount rate=
1. Decide whether cash flows you value are nominal or real. Choose matching rate type.
2. Select an appropriate baseline risk-free rate (e.g., Treasury yield with matching maturity).
3. Decide the model: WACC for firm-level free cash flows; CAPM-based cost of equity for equity cash flows; project-specific hurdle rate if justified.
4. If using CAPM, estimate beta carefully (comparable firms, historical regressions, or published betas), and decide on the equity risk premium source (historical vs. implied).
5. If valuing a project with different leverage, unlever and relever beta to the project’s capital structure.
6. Add measurable premiums for country risk, small-firm risk, or other non-market risks if they are not captured by beta.
7. Check consistency: ensure rate, tax assumptions, and cash flow definitions match.
8. Stress-test: run sensitivity analysis for ±100–300 basis points in the discount rate.
=Worked numeric example=
Assumptions:
– Risk-free rate (Rf, nominal): 3.0% (10-year Treasury).
– Expected market return (Rm): 8.0% → ERP = 8 − 3 = 5.0%.
– Observed levered beta for comparables: 1.2.
– Company capital structure: D/E = 0.5 (so D = 0.5, E = 1 → V = 1.5; E/V = 2/3, D/V = 1/3).
– Cost of debt (pre-tax) Rd = 5.0%.
– Corporate tax rate Tc = 21%.
Step A — Cost of equity (CAPM)
Re = 3.0% + 1.2 * 5.0% = 3.0% + 6.0% = 9.0%.
Step B — After-tax cost of debt
Rd_after_tax = Rd * (1 − Tc) = 5.0% * (1 − 0.21) = 5.0% * 0.79 = 3.95%.
Step C — WACC
E/V = 1 / 1.5 = 0.6667; D/V = 0.5 / 1.5 = 0.3333.
WACC = (E/V)*Re + (D/V)*Rd_after_tax
WACC = 0.6667*9.0% + 0.3333*3.95% = 6.000% + 1.3167% = 7.3167% ≈ 7.32%.
Step D — Discounting an illustrative cash flow
If next-year free cash flow = $100, PV = 100 / (1 + 0.073167) = 100 / 1.073167 ≈ $93.22.
Notes: different assumptions (higher ERP, different beta, or different Rd) noticeably change WACC and PV; run sensitivities.
=Common pitfalls and how to avoid them=
– Mixing nominal cash flows with real discount rates (or vice versa). Always match.
– Using a short-term Treasury yield for long-term cash flows without term-structure adjustment. Use a rate whose maturity matches cash flow horizon or adjust for term premium.
– Ignoring project-specific risks that beta doesn’t capture. Document and quantify additions.
– Blindly using published betas without checking comparator suitability. Prefer industry peers or de-lever/re-lever process.
– Forgetting taxes when computing WACC: use after-tax cost of debt.
=Sensitivity and scenario analysis (practical)=
– Create a small table of NPV results for discount rates ±100 and ±300 basis points around base rate.
– Run best-case / base-case / worst-case scenarios combining cash-flow changes with rate changes to see interaction effects.
– Use tornado charts to
to show which inputs (discount rate, volumes, margins, capex) drive the widest NPV swings; focus stress tests on the top 2–4 drivers.
=Worked numerical example — sensitivity table=
Assumptions (example project)
– Initial investment: -$1,000 (Year 0).
– Cash flows: Year1 = $300; Year2 = $400; Year3 = $500.
– Base discount rate = 8.0% (nominal).
We compare NPV at base and at ±100 bps and ±300 bps.
Formula: PV of a cash flow in year t = CFt / (1 + r)^t. NPV = sum(PV of CFs) + initial investment.
Calculations (rounded)
– At 8.0% (base):
PV1 = 300/(1.08)^1 = 277.78
PV2 = 400/(1.08)^2 = 342.94
PV3 = 500/(1.08)^3 = 397.11
NPV = 277.78 + 342.94 + 397.11 − 1000 = 17.83
– At 7.0% (base −100 bps):
PVs = 280.37 + 349.91 + 410.30 → NPV ≈ 40.58
– At 5.0% (base −300 bps):
PVs = 285.71 + 362.81 + 431.92 → NPV ≈ 80.44
– At 9.0% (base +100 bps):
PVs = 275.23 + 335.93 + 386.28 → NPV ≈ −2.56
– At 11.0% (base +300 bps):
PVs = 270.27 + 323.14 + 361.97 → NPV ≈ −43.62
Takeaway: NPV swings from +$80k to −$44k across ±300 bps; sensitivity underscores the importance of choosing a defensible discount rate.
=WACC and cost-of-equity worked example=
Key formulas
– CAPM (cost of equity) r_e = r_f + β × (r_m − r_f), where r_f = risk-free rate, β = beta (systematic risk), r_m − r_f = market risk premium.
– After-tax cost of debt r_d_after = r_d_pre × (1 − TaxRate).
– WACC = (E/(E+D)) × r_e + (D/(E+D)) × r_d_after.
Example inputs
– Market value equity E = $600m; market value debt D = $400m → weights: E/(E+D)=60%, D/(E+D)=40%.
– Risk-free rate r_f = 2.0%.
– Beta = 1.2.
– Market risk premium = 5.0%.
– Pre-tax cost of debt r_d_pre = 4.0%.
– Tax rate = 21%.
Step-by-step
1. Cost of equity: r_e = 2.0% + 1.2 × 5.0% = 8.0%.
2. After-tax cost of debt: r_d_after = 4.0% × (1 − 0.21) = 3.16%.
3. WACC = 0.60 × 8.0% + 0.40 × 3.16% = 4.80% + 1.26% = 6.06%.
Use: apply this WACC to nominal, enterprise-level cash flows that match the firm/project’s risk and currency.
=Sensitivity on WACC (example)=
Vary the market risk premium by +100/300 bps (i.e., 6
.0% and 8.0%) — recompute cost of equity and WACC as follows.
Case A — MRP = 6.0% (+100 bps)
– Cost of equity: r_e = r_f + β × MRP = 2.0% + 1.2 × 6.0% = 9.2%.
– After-tax cost of debt (unchanged):
– After-tax cost of debt (unchanged): 4.0% × (1 − 0.21) = 3.16%.
– WACC = 0.60 × 9.2% + 0.40 × 3.16% = 5.52% + 1.264% = 6.78% (rounded).
Case B — MRP = 8.0% (+300 bps)
1. Cost of equity: r_e = r_f + β × MRP = 2.0% + 1.2 × 8.0% = 11.6%.
2. After-tax cost of debt (unchanged): 3.16%.
3. WACC = 0.60 × 11.6% + 0.40 × 3.16% = 6.96% + 1.264% = 8.22% (rounded).
Quick sensitivity summary (weights: 60% equity / 40% debt)
– Base MRP = 5.0% → WACC = 6.06% (from prior calculation).
– MRP = 6.0% → WACC ≈ 6.78% (+0.72 percentage point).
– MRP = 8.0% → WACC ≈ 8.22% (+2.16 percentage points).
Worked valuation example (constant nominal free cash flow)
Assume enterprise-level FCF = $100.0 (same currency and nominal terms as WACC), zero nominal growth (perpetuity).
– EV at WACC = 6.06%: EV = 100 / 0.0606 ≈ 1,650. (baseline)
– EV at WACC = 6.78%: EV = 100 / 0.0678 ≈ 1,474. (−10.7% vs baseline)
– EV at WACC = 8.22%: EV = 100 / 0.0822 ≈ 1,216. (−26.3% vs baseline)
Interpretation and practical notes
– Mechanism: increasing the market risk premium raises the cost of equity; because equity typically carries the larger weight, WACC rises and valuations fall.
– Duration effect: long-duration cash flows (large weight in years far into the future) are more sensitive to WACC changes. Small WACC moves can materially change present values.
– Consistency required: use nominal WACC with nominal cash flows (or real WACC with real cash flows). Match currency, tax treatment, and risk assumptions.
– Recompute when assumptions change: risk-free rate, β, tax rate, debt yields, or capital structure revisions all require a fresh WACC.
– Sensitivity checklist:
1. Confirm cash-flow basis (nominal vs real) and currency.
2. Choose appropriate r_f, β, and MRP for the firm/project’s risk and market.
3. Calculate after-tax cost of debt using expected debt yields and current tax rate.
4. Compute WACC using current capital structure weights (market-value weights preferred).
5. Run scenario and sensitivity analyses (vary MRP, β, growth, tax rate).
6. Document assumptions and ranges used.
Assumptions made in this example
– Market risk premium is ex ante (expected) and applicable to the firm.
– Capital structure weights (60% equity / 40% debt) are constant and expressed in market values.
– Debt yield and tax rate are constant across scenarios.