Nominal Rate of Discount

A nominal rate of discount $d^{(m)}$ is convertible m times per year: each sub-period is discounted by $d^{(m)}/m$, and compounding all m of those discounts gives $(1-d^{(m)}/m)^m = v = 1-d$, the annual discount factor.

By the end you'll be able to convert freely among effective interest i, effective discount d, nominal interest $i^{(m)}$, and nominal discount $d^{(m)}$ for any compounding frequency m.

Predict: for the same effective rate and the same m, which is larger — $i^{(m)}$ or $d^{(m)}$? Drag m and check whether the gap shrinks.

The effective annual rate here is fixed at $i = 8\%$ (so $d \approx 7.41\%$ and the force of interest $\delta \approx 7.70\%$ never move). Drag m and watch the nominal interest bar fall and the nominal discount bar rise, both squeezing toward the dashed force-of-interest line — they meet only in the limit, but $i^{(m)}$ never dips below $d^{(m)}$ for any finite m. Hover any bar for its exact value.

Compounding frequency m = 4 times/year

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i⁽⁴⁾ = 7.7706% · d⁽⁴⁾ = 7.6225% · δ = 7.6961%

Check: (1 − d⁽⁴⁾/4)⁴ = 0.925926 — matches v = 1/(1+i) = 0.925926. That equality holds exactly for every m, by construction of d⁽⁴⁾ from v.

Four names for the same growth at i = 8%, current m

i^(m) (nominal interest) d^(m) (nominal discount) δ (force of interest)

Formal

A nominal discount $d^{(m)}$ takes $d^{(m)}/m$ off the balance at the start of each of m sub-periods, so over a year those m discounts compose to the annual discount factor: $(1-d^{(m)}/m)^m = v = 1-d$. Solve either direction: $d = 1-(1-d^{(m)}/m)^m$ (nominal → effective), or $d^{(m)} = m\big[1-(1-d)^{1/m}\big] = m\big[1-v^{1/m}\big]$ (effective → nominal). Because effective interest i, effective discount d, nominal interest $i^{(m)}$, and nominal discount $d^{(m)}$ are all just different names for the same one-year growth factor $1+i = v^{-1}$, you can convert *any* of them to *any* other by routing through $1+i$ (or v) — never mix a nominal rate directly into an effective-only identity like $i-d=id$; that one is for the effective pair only. The cleanest same-m shortcut links the two nominal rates directly: $i^{(m)}/m - d^{(m)}/m = (i^{(m)}/m)(d^{(m)}/m)$, the m-thly echo of $i-d=id$.

Applied

Discount-in-advance shows up whenever a lender deducts interest up front instead of charging it at the end — a bank that sells you a $1000 face-value note for less than $1000 today, or a T-bill quoted at a discount rate rather than a yield. Whenever you see a rate advertised as a "discount rate convertible monthly" or similar, don't plug it straight into a present-value formula built for effective i — convert it to v first, exactly as you would a nominal interest rate.

Worked example

A rate of discount of 8% convertible quarterly means $d^{(4)} = 0.08$, so each quarter discounts by $d^{(4)}/4 = 0.02$. Annual discount factor: $v = (1-0.02)^4 = (0.98)^4 = 0.922368$. Effective annual discount: $d = 1-v = $ 7.7632%. Effective annual interest: $i = 1/v-1 = $ 8.4166%.

Your turn

Same idea, a different frequency: a nominal discount of 6% convertible monthly, $d^{(12)} = 0.06$. Monthly discount: $d^{(12)}/12 = 0.005$. Raising $(1-0.005)$ to the 12th power: $(0.995)^{12} = 0.941623$, so the effective annual discount is $d = 1-0.941623 = $ 5.8377%. Now finish it: $i = 1/v-1 = 1/0.941623 - 1 = $ ____

Reveal the answer

$i = 1/0.941623 - 1 = 1.061996 - 1 = $ 6.1996% — a little above d, exactly as it must be. Set m = 12 on the slider above (between the 4 and 52 ticks) and compare i⁽¹²⁾ and d⁽¹²⁾ against these effective values.

More info — why $i^{(m)}$ and $d^{(m)}$ bracket δ instead of equaling it

At m = 1, $i^{(1)} = i$ and $d^{(1)} = d$ — the widest possible gap, since a single yearly sub-period is the least compounding you can do. Every time you double m, each nominal rate gets one step closer to the continuous limit: $i^{(m)} \downarrow \delta$ and $d^{(m)} \uparrow \delta$ as $m \to \infty$, because $(1+i^{(m)}/m)^m$ and $(1-d^{(m)}/m)^m$ both converge to $e^{\delta} = 1+i$. They never actually cross — $i^{(m)} > \delta > d^{(m)}$ holds for every finite m — because interest-in-arrears can never be cheaper than the same growth charged continuously, and discount-in-advance can never be more expensive than it. See the Binghamton handout in Dive deeper for the full derivation.

Check your understanding

Question 1 of 4

A rate is quoted as $d^{(2)} = 10\%$ convertible semiannually. What are the equivalent effective annual discount rate $d$ and effective annual interest rate $i$?

Question 2 of 4

Holding the effective annual rate fixed, what happens to $i^{(m)}$ and $d^{(m)}$ as the compounding frequency $m$ grows without bound?

Question 3 of 4

A rate is quoted as $i^{(4)} = 8\%$ convertible quarterly (nominal interest). What is the equivalent $d^{(4)}$ (nominal discount), also convertible quarterly?

Question 4 of 4

At a nominal discount rate $d^{(2)} = 9\%$ convertible semiannually, what is the present value of an \$8{,}000 payment due in one year?

Recap

A nominal discount $d^{(m)}$ is applied in advance, m times per year: $(1-d^{(m)}/m)^m = v = 1-d$.
Nominal → effective: $d = 1-(1-d^{(m)}/m)^m$. Effective → nominal: $d^{(m)} = m\big[1-v^{1/m}\big]$.
Effective interest i, effective discount d, nominal interest $i^{(m)}$, and nominal discount $d^{(m)}$ are four names for one growth factor $1+i$; convert through $1+i$ or v, never mix a nominal rate into an effective-only identity.
Same-m shortcut: $i^{(m)}/m - d^{(m)}/m = (i^{(m)}/m)(d^{(m)}/m)$.
For any finite m, $i^{(m)} > \delta > d^{(m)}$; both converge to the force of interest δ as $m \to \infty$, narrowing the gap but never closing it at a finite m.

Dive deeper

Wikibooks — Financial Math FM/Formulas Reference the d^(m)=m(1−(1−d)^(1/m)) nominal-discount conversion.
Binghamton (Arcones) — Manual for SOA Exam FM §1.5: Nominal rates of interest and discount (PDF) Work the full slate of nominal-discount conversion examples.
AnalystPrep — Time Value of Money (SOA Exam FM Study Notes) Cross-check conversions among effective and nominal interest and discount.

Sources

Nominal Rates of Discount and Converting Among All Four Rates