Interest and Time Value

Sources: Wikipedia (CC BY-SA 4.0) · Adam Smith, Wealth of Nations (1776, public domain)

A dollar today is worth more than a dollar tomorrow. Compound interest turns this intuition into arithmetic: future value grows exponentially, present value shrinks by discounting. Every investment decision reduces to comparing present values.

Compound interest

If you invest principal P at annual rate r, compounded once per year, after t years you have FV = P(1 + r)^t. Compounding more frequently (n times per year) gives FV = P(1 + r/n)^nt. As n approaches infinity, you get continuous compounding: FV = Pe^rt.

Scheme

; Compound interest: FV = P * (1 + r)^t
; P = principal, r = annual rate, t = years

(define (future-value P r t)
  (* P (expt (+ 1 r) t)))

; $1000 at 7% for 10, 20, 30 years
(display "10 years: $")
(display (round (future-value 1000 0.07 10))) (newline)
(display "20 years: $")
(display (round (future-value 1000 0.07 20))) (newline)
(display "30 years: $")
(display (round (future-value 1000 0.07 30))) (newline)

; Rule of 72: doubling time ~ 72/r
(display "Rule of 72 doubling time at 7%: ")
(display (/ 72 7))
(display " years")

Present value and discounting

Present value reverses the direction: PV = FV / (1 + r)^t. The discount rate r encodes your opportunity cost. A dollar arriving in 10 years at 7% discount is worth only $0.51 today. This is the foundation of all valuation.

Scheme

; Present value: PV = FV / (1 + r)^t

(define (present-value FV r t)
  (/ FV (expt (+ 1 r) t)))

; What is $1000 received in 10 years worth today?
(display "PV of $1000 in 10 years at 7%: $")
(display (round (present-value 1000 0.07 10))) (newline)

(display "PV of $1000 in 20 years at 7%: $")
(display (round (present-value 1000 0.07 20))) (newline)

; Higher discount rate = lower present value
(display "PV of $1000 in 10 years at 12%: $")
(display (round (present-value 1000 0.12 10)))

Net present value

NPV sums the present values of all future cash flows, minus the initial investment. If NPV is greater than zero, the project creates value. NPV is additive: two independent projects can be evaluated separately.

Scheme

; NPV = sum of discounted cash flows - initial cost
; Cash flows: list of (year . amount) pairs

(define (present-value FV r t)
  (/ FV (expt (+ 1 r) t)))

(define (npv rate initial-cost cash-flows)
  (define (sum-pv flows)
    (if (null? flows)
        0
        (+ (present-value (cdar flows) rate (caar flows))
           (sum-pv (cdr flows)))))
  (- (sum-pv cash-flows) initial-cost))

; Project: costs $5000 upfront, returns $2000/year for 3 years
(define flows (list (cons 1 2000) (cons 2 2000) (cons 3 2000)))

(display "NPV at 5%: $")
(display (round (npv 0.05 5000 flows))) (newline)
(display "NPV at 15%: $")
(display (round (npv 0.15 5000 flows)))

Annuities

An annuity is a fixed payment C repeated for n periods. The present value of an annuity has a closed form: PV = C * (1 - (1+r)^-n) / r. Mortgages, bonds, and pensions are all annuities in disguise. The connection to integrals: a continuous annuity is the integral of Ce^-rt from 0 to T.

Scheme

; Annuity PV = C * (1 - (1+r)^(-n)) / r
; C = payment per period, r = rate, n = periods

(define (annuity-pv C r n)
  (* C (/ (- 1 (expt (+ 1 r) (- n))) r)))

; $500/month for 30 years at 6% annual (0.5% monthly)
(define monthly-pv (annuity-pv 500 0.005 360))
(display "PV of $500/month for 30 years: $")
(display (round monthly-pv)) (newline)

; Mortgage: what monthly payment for $300,000 loan?
; C = PV * r / (1 - (1+r)^(-n))
(define (mortgage-payment PV r n)
  (* PV (/ r (- 1 (expt (+ 1 r) (- n))))))

(define pmt (mortgage-payment 300000 0.005 360))
(display "Monthly payment on $300k mortgage: $")
(display (round pmt))

Neighbors

Cross-references

Calculus Ch.7 — integrals: continuous discounting as integration

Foundations (Wikipedia)

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