Effective and nominal rates, discounting, annuity-immediate and annuity-due formulas, perpetuities.
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- What is the effective annual rate (EAR)?
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The actual annual rate of interest earned, accounting for all compounding within the year.- Define a nominal interest rate.
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The stated annual interest rate that does not account for intra-year compounding.- Convert nominal rate i^(m) compounded m times per year to effective annual rate i.
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i = (1 + i^(m)/m)^m - 1- What is the force of interest (delta, δ)?
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The instantaneous rate of interest per unit time under continuous compounding.- Relationship between force of interest δ and effective annual rate i.
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e^δ = 1 + i, or equivalently δ = ln(1 + i)- Define an annuity-immediate (ordinary annuity).
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A series of equal periodic payments made at the end of each period for a fixed number of periods.- Define an annuity-due.
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A series of equal periodic payments made at the beginning of each period for a fixed number of periods.- Present value of an annuity-immediate (n periods, unit payment, rate i): formula.
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a_n⌉i = (1 - v^n) / i, where v = 1 / (1 + i)- Accumulated value of an annuity-immediate (n periods, unit payment, rate i): formula.
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s_n⌉i = ((1 + i)^n - 1) / i- Present value of an annuity-due (n periods, unit payment, rate i): formula.
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ä_n⌉i = (1 + i) * a_n⌉i, or equivalently (1 - v^n) / d- Accumulated value of an annuity-due (n periods, unit payment, rate i): formula.
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s̈_n⌉i = (1 + i) * s_n⌉i- What is a perpetuity-immediate?
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A series of equal payments made at the end of each period indefinitely.- What is a perpetuity-due?
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A series of equal payments made at the beginning of each period indefinitely.- Present value of perpetuity-immediate (unit payment, rate i): formula.
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a_∞⌉i = 1 / i- Present value of perpetuity-due (unit payment, rate i): formula.
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ä_∞⌉i = 1 / d, or equivalently (1 + i) / i
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