Financial Mathematics: Interest, Annuities, Bonds, and Immunization

Accumulation factor a(t)

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

What is a(0)?

a(0) = 1 by definition.

Effective rate of interest iₜ

iₜ = [a(t) - a(t-1)] / a(t-1).

Meaning of iₜ

Interest earned in period t divided by the amount invested at the beginning of the period.

Effective rate of discount dₜ

dₜ = [a(t) - a(t-1)] / a(t).

Meaning of dₜ

Interest earned in period t divided by the amount invested at the end of the period.

Convert dₜ to iₜ

iₜ = dₜ / (1 - dₜ).

Convert iₜ to dₜ

dₜ = iₜ / (1 + iₜ).

Discount function

a⁻¹(t) = 1 / a(t); it is the present value at time 0 of $1 paid at time t.

Discount factor vₜ

vₜ discounts money from time t to time t-1.

Discount factor formula

vₜ = 1 - dₜ = 1 / (1 + iₜ).

Compound interest accumulation

a(t) = (1 + i)ᵗ.

Compound interest behavior

The effective interest rate is constant each period; the amount of interest varies each period.

Simple interest accumulation

a(t) = 1 + it.

Simple interest behavior

The effective interest rate varies for each period; the amount of interest is constant each period.

Equation of value principle

Time value of inflows = time value of outflows.

Nominal interest rate i⁽ᵐ⁾

i⁽ᵐ⁾/m is the effective rate of interest per m-th of a year.

Nominal discount rate d⁽ⁿ⁾

d⁽ⁿ⁾/n is the effective rate of discount per n-th of a year.

Equivalent nominal interest and discount rates

[1 + i⁽ᵐ⁾/m]ᵐ = [1 - d⁽ⁿ⁾/n]⁻ⁿ. Both sides are 1-year accumulation factors.

Accumulate from t-1 to t: four methods

Multiply by a(t)/a(t-1), multiply by (1+iₜ), divide by (1-dₜ), or divide by vₜ.

Discount from t to t-1: four methods

Multiply by a(t-1)/a(t), divide by (1+iₜ), multiply by (1-dₜ), or multiply by vₜ.

Force of interest δₜ

δₜ = a'(t) / a(t).

Scaling the force of interest

If δ*ₜ = kδₜ, then a(t) = [a(t)]ᵏ.

Accumulation using force of interest

To accumulate from t₁ to t₂, multiply by exp(∫ from t₁ to t₂ of δₜ dt).

Discounting using force of interest

To discount from t₂ to t₁, multiply by exp(-∫ from t₁ to t₂ of δₜ dt).

Force of interest under compound interest

δ = ln(1+i), equivalently i = eᵟ - 1.

Four quantities in the basic interest problem

1) Principal originally invested, 2) length of investment, 3) rate of interest, 4) accumulated value of the principal. Given any three, solve for the fourth.

Yield rate

The rate of interest that establishes an equivalency of value between different points in time.

Time value of money

The value of money at a given time depends on time elapsed since it was paid in the past or time remaining before it is paid in the future.

Equation of value

An equation that accumulates or discounts each payment to a comparison date.

Comparison date under compound interest

For compound interest, you can choose any comparison date.

Sum of a geometric series

If S = a + ar + ar² + ... + arⁿ⁻¹, then S = (a - arⁿ)/(1-r).

Geometric-series memory rule

First term minus the 'next' term, divided by 1 minus what you multiply by each time.

How to count arithmetic-spaced terms

Number of terms = (last term - the term immediately before the first term) / step size. Example: 37,41,...,121 gives (121-33)/4 = 22.

Deferred annuity

Present value of n payments m+1 periods before the first payment; equivalently, an n-payment annuity-immediate deferred for m periods.

Deferred-annuity timing example on the sheet

For four payments at times 3,4,5,6: at time 0 the value is 2|a₄; at time 2 it is a₄; at time 3 it is ä₄; at time 6 it is s₄; at time 7 it is s̈₄.

Grouping payments: a₂ₙ

a₂ₙ = aₙ + vⁿ aₙ.

Grouping payments: s₃ₙ

s₃ₙ = sₙ(1+i)²ⁿ + sₙ(1+i)ⁿ + sₙ.

Fission method

Split payments into smaller payments when that makes the cash-flow pattern easier to value.

Fission example on the sheet

Three payments of 1 at times 5,10,15 can be represented by payments of 1/s₅ over times 1 through 15; PV = (1/s₅)a₁₅.

Annuity-immediate definition

An annuity with payments at the end of each period.

Annuity-immediate present value

aₙ = (1 - vⁿ)/i. This is the PV of n payments one period before the first payment.

Annuity-immediate accumulated value

sₙ = aₙ(1+i)ⁿ = [(1+i)ⁿ - 1]/i. This is the AV at the time of the last payment.

Interest-rate period for annuity formulas

i is the effective interest rate per payment period.

Discount factor period for annuity formulas

v is the discount factor per payment period.

Annuity-due definition

An annuity with payments at the beginning of each period.

Annuity-due present value

äₙ = aₙ(1+i) = (1 - vⁿ)/d. This is the PV at the time of the first payment.

Annuity-due accumulated value

s̈ₙ = äₙ(1+i)ⁿ = [(1+i)ⁿ - 1]/d. This is the AV one period after the last payment.

Perpetuity-immediate

a∞ = lim(n→∞) aₙ = 1/i; PV is one period before the first payment.

Perpetuity-due

ä∞ = 1 + a∞ = 1/d; PV is at the time of the first payment.

TVM worksheet: not in BGN mode

Payments occur at times 1 through N; PV is at time 0 and FV is at time N.

TVM worksheet: BGN mode

Payments occur at times 0 through N-1; PV is at time 0 and FV is at time N.

Annuities payable m-thly

Payments are 1/m every m-th of a year, for a total of 1 per year, for n years.

PV of an annuity payable m-thly

aₙ⁽ᵐ⁾ = (1 - vⁿ) / i⁽ᵐ⁾.

m-thly annuity relation to a level annuity

aₙ⁽ᵐ⁾ = (1/m) aₘₙ, where the per-payment effective rate is i⁽ᵐ⁾/m.

Fusion method example

For semiannual payments, where j is the effective rate per 6 months: a₃⁽²⁾ = 0.5 s₂ at j × a₃ at i.

Increasing arithmetic annuity-immediate

(Ia)ₙ = (äₙ - n vⁿ)/i.

Decreasing arithmetic annuity-immediate

(Da)ₙ = (n - aₙ)/i.

Increasing arithmetic annuity-due

(Iä)ₙ = (äₙ - n vⁿ)/d.

Decreasing arithmetic annuity-due

(Dä)ₙ = (n - aₙ)/d.

Accumulated value of increasing annuity-immediate

(Is)ₙ = (Ia)ₙ(1+i)ⁿ.

Accumulated value of decreasing annuity-immediate

(Ds)ₙ = (Da)ₙ(1+i)ⁿ.

Accumulated value of increasing annuity-due

(Is̈)ₙ = (Iä)ₙ(1+i)ⁿ.

Accumulated value of decreasing annuity-due

(Ds̈)ₙ = (Dä)ₙ(1+i)ⁿ.

Increasing perpetuity-immediate

(Ia)∞ = 1/(id).

Increasing perpetuity-due

(Iä)∞ = 1/d².

Geometric varying annuity payment pattern

Payments at times 1,2,...,n are 1, (1+r), (1+r)², ..., (1+r)ⁿ⁻¹.

Geometric varying annuity-immediate PV

(Ga)ₙ|i,r = [1 - ((1+r)/(1+i))ⁿ] / (i-r).

Geometric varying annuity-due PV

(Gä)ₙ|i,r = äₙ evaluated at the rate (i-r)/(1+r).

Geometric perpetuity-immediate

(Ga)∞|i,r = 1/(i-r) if r

Geometric perpetuity-due

(Gä)∞|i,r = (Ga)∞|i,r(1+i) = (1+i)/(i-r) if r

Geometric annuity special case i=r

(Ga)ₙ|r,r = nv and (Gä)ₙ|r,r = n.

Level continuous annuity present value

PV of 1 per year paid continuously for n years: āₙ = (1-vⁿ)/δ = (i/δ)aₙ.

Continuous perpetuity

ā∞ = 1/δ.

Level continuous annuity accumulated value

s̄ₙ = āₙ(1+i)ⁿ = (i/δ)sₙ.

General varying continuous annuity PV

PV = ∫₀ⁿ f(t) exp[-∫₀ᵗ δₛ ds] dt.

General varying continuous annuity AV

AV = ∫₀ⁿ f(t) exp[∫ₜⁿ δₛ ds] dt.

Continuous increasing annuity PV when f(t)=t and δₜ=δ

(Īā)ₙ = ∫₀ⁿ t vᵗ dt = (āₙ - n vⁿ)/δ.

Continuous increasing annuity AV

(Īs̄)ₙ = ∫₀ⁿ t(1+i)ⁿ⁻ᵗ dt = (Īā)ₙ(1+i)ⁿ.

Continuous increasing perpetuity

(Īā)∞ = 1/δ².

Continuous decreasing annuity PV when f(t)=n-t and δₜ=δ

(D̄ā)ₙ = ∫₀ⁿ (n-t)vᵗ dt = (n-āₙ)/δ.

Continuous decreasing annuity AV

(D̄s̄)ₙ = ∫₀ⁿ (n-t)(1+i)ⁿ⁻ᵗ dt = (D̄ā)ₙ(1+i)ⁿ.

Annuity trick: double-dots cancel

X äₙ = Y äₘ if and only if X aₙ = Y aₘ.

Annuity trick: upper (m)s cancel

A ratio such as ä₂₀⁽¹²⁾ / s̈₅⁽¹²⁾ can be reduced to a₂₀ / s₅.

Annuity trick: a₂ₙ/aₙ

a₂ₙ/aₙ = 1 + vⁿ.

0% test

At 0% interest, the time value of an annuity is the sum of its payments.

Pyramid annuity formulas

PV_pyramid-immediate = aₙ äₙ; PV_pyramid-due = äₙ äₙ.

Pyramid example on the sheet

Payments 1,2,3,2,1 at times 1 through 5 have PV = a₃ ä₃.

Net present value (NPV)

NPV = Σ (net CF)ₜ / (1+r)ᵗ, where r is the required return.

NPV calculator worksheet

Use the CF and NPV worksheets on the calculator.

NPV decision rule

If NPV > 0, the company should do the project.

NPV with limited funds

Choose the project(s) with the greatest NPV.

Internal rate of return (IRR)

IRR solves 0 = NPV = Σ (net CF)ₜ / (1+IRR)ᵗ.

IRR calculator worksheet

Use the CF and IRR worksheets on the calculator.

IRR decision rule

If IRR > r, the company should do the project.

IRR with limited funds

Choose the project(s) with the greatest IRR.

Reinvestment-rate setup

If interest is earned at rate i but interest payments are reinvested at rate j, separate the i-fund and j-fund cash flows and accumulate each appropriately.

Reinvestment-rate example total at time n

For the sheet's example: total at time n = nX + (iX)(Is)ₙ at rate j. The sheet says to understand the process rather than memorize this formula.

Yield-rate equation

Solve: what I have = what I demand.

'What I have' in a yield-rate problem

Cash inflows accumulated at the reinvestment rate.