Discount Rate

A discount rate is assembled, not observed. The base layer is the real risk-free rate, the compensation demanded for deferring consumption when there is no default or price-level uncertainty. Adding expected inflation gives the nominal risk-free rate; the precise relationship is multiplicative, since one plus the nominal rate equals one plus the real rate multiplied by one plus expected inflation, though the simple sum is an acceptable approximation at low rates.

On top of that sit premia for the specific claim being valued: a default risk premium for the possibility that promised cash is not paid, a liquidity premium where exit is costly or slow, and a maturity premium reflecting greater sensitivity of long-dated instruments to rate changes. Equity claims carry an equity risk premium in place of, or in addition to, these components. Each premium widens when uncertainty about the cash flows rises, which is why the same asset can be discounted at materially different rates across the cycle.

Present value is future value divided by one plus the periodic rate, raised to the number of compounding periods. The two inputs must be stated on the same frequency: a stated annual rate is converted to a periodic rate by dividing by the number of compounding periods per year, and the exponent is then the number of years multiplied by that same frequency. Quoting an annual rate but discounting monthly cash flows without this conversion is the most common arithmetic failure in the topic.

Because compounding frequency alters the effective outcome, the effective annual rate, computed as one plus the periodic rate raised to the number of periods per year, minus one, is the correct basis for comparing rates quoted differently. Under continuous compounding, present value uses the exponential of the negative rate multiplied by time. The relationship also runs backwards: given a price and a known stream of cash flows, the discount rate that equates them is the internal rate of return, and for a bond that solved rate is its yield to maturity.

In fixed income, the discount rate is quoted directly: yield to maturity is a single rate that reprices a bond's coupons and principal to its market price, while an arbitrage-consistent valuation discounts each cash flow at the spot rate matching its own maturity. In corporate valuation, free cash flow to the firm is discounted at the weighted average cost of capital, while free cash flow to equity and dividends are discounted at the cost of equity, often estimated through the capital asset pricing model.

Discount rates also appear inside financial statements. Defined benefit pension obligations are measured by discounting projected benefit payments, typically at high-quality corporate bond yields, so a fall in yields inflates the reported liability. Lease liabilities, provisions and asset impairment tests all embed a rate. Note that the Federal Reserve's discount rate is an unrelated term of art, referring to the rate charged on discount window lending.

Consistency is the governing discipline. Nominal cash flows require a nominal rate and real cash flows a real rate; after-tax flows require an after-tax rate; cash flows denominated in one currency must be discounted at a rate drawn from that currency's market. Mixing conventions produces errors far larger than any refinement in premium estimation.

Candidates should be able to move fluently in both directions: compute present or future value given a rate, and back out the implied rate given prices and cash flows. They should also anticipate sensitivity. Because the discount factor is compounded, an increase in the rate reduces distant cash flows proportionally more than near ones, which is the same intuition that underpins duration in fixed income and terminal value dominance in a discounted cash flow model. Being able to state the direction and rough magnitude of that effect without computation is frequently sufficient for a conceptual item.