Nominal Rates and Force of Interest: Formulas and Conversions

Understand Nominal Rates and Force of Interest

Key rules to remember

Nominal to effective (interest)

1 + i = (1 + i(m)/m)^m

i(m) is the nominal annual rate convertible m times a year. Each period earns i(m)/m.

Nominal to effective (discount)

1 - d = (1 - d(m)/m)^m

Also 1 - d = v = (1 + i)^(-1).

Link between nominal interest and discount

(1 + i(m)/m)^m = (1 - d(p)/p)^(-p)

Both sides equal 1 + i. Use this to convert between i(m) and d(p) with different m and p.

Constant force of interest

δ = ln(1 + i), so 1 + i = e^δ and v = e^(-δ)

ln is the natural logarithm.

Force of interest from the accumulation function

δ(t) = A'(t) ÷ A(t) = d/dt ln A(t)

A(t) is the accumulated value at time t of 1 invested at time 0.

Accumulation with varying force

A(t1, t2) = exp(∫ δ(s) ds from t1 to t2)

This is the value at t2 of 1 invested at t1.

Discount factor with varying force

v(t1, t2) = exp(-∫ δ(s) ds from t1 to t2)

This is the present value at t1 of 1 paid at t2.

Limits as m → ∞

lim i(m) = lim d(m) = δ

Also d < d(m) < δ < i(m) < i for m > 1, when i > 0.

Relation of i and d

d = i ÷ (1 + i) = iv, and i - d = id

Useful for quick conversions.

How to solve Nominal Rates and Force of Interest questions

1. 1Write down exactly what you are given: i(m), d(m), δ, or a function δ(t). Note the compounding frequency m.
2. 2Decide what you must find: an effective rate, a different nominal rate, a present value or an accumulated value.
3. 3Convert the given rate to a single common quantity. For constant rates, use 1 + i (the annual accumulation factor) or δ.
4. 4For discount rates, use 1 - d = (1 - d(m)/m)^m, then go to 1 + i = 1 ÷ (1 - d).
5. 5For a varying force, integrate δ(s) over the exact time interval. Split the integral at each point where δ changes form.
6. 6Compute the accumulation factor exp(∫δ) or the discount factor exp(-∫δ).
7. 7Convert the result to the rate asked for, for example i(p) = p[(1 + i)^(1/p) - 1].
8. 8Check the answer is sensible: i(m) is below i for m > 1, and δ lies between d(m) and i(m).

Quickest way: Go through 1 + i every time

Common mistakes in Nominal Rates and Force of Interest

• Treating i(m) as the effective annual rate and using (1 + i(m))^n.

  The word 'rate' makes students forget that i(m) is quoted annually but paid in m parts.

  Fix: Always divide by m inside the bracket and raise to the power m × n. Per-period rate is i(m)/m.

• Using 1 - d(m) instead of 1 - d(m)/m when converting discount rates.

  Students copy the interest formula and forget the discount version needs its own sign and bracket.

  Fix: Write (1 - d(m)/m)^m = 1 - d. Check that d(m) > d for m > 1; if you get d(m) < d, you have made an error.

• Writing δ = i, or using log base 10.

  For small i, δ is close to i, so the difference seems unimportant.

  Fix: Use δ = ln(1 + i) with the natural logarithm. For i = 5%, δ = 4.879%, not 5%.

• Confusing the conversion from d(p) to i(m): inverting the wrong side.

  The negative exponent on the discount side is easy to drop.

  Fix: Write 1 + i on both sides: (1 + i(m)/m)^m = (1 - d(p)/p)^(-p). Then take roots carefully.

• Integrating δ(t) over the wrong limits, or integrating from 0 when the money is invested at time t1 > 0.

  Students memorise exp(∫ from 0 to t) and apply it to every situation.

  Fix: Use the actual start and end times. The limits are the investment time and the valuation time.

• Rounding intermediate values such as 1 + i to 3 or 4 digits.

  Wanting tidy numbers in a long calculation.

  Fix: Store full precision in the calculator memory and round only the final answer.

Worked examples