Perpetuity & Drawdown
About this app
How long does capital last? Balance, interest and withdrawal per period, the perpetuity a = K(q − 1) and the term N = ln(A/(A − S(q − 1)))/ln q with its vertical asymptote, last withdrawal, yearly or monthly, in advance and in arrears.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
Living on the interest
A capital K earns interest K · (q − 1) per interest period, with q = 1 + p/100. Withdraw exactly this amount and the account is back at K after every period: that is a perpetuity. Withdraw more and the capital shrinks, the interest shrinks with it, and the account runs dry after a finite time. A drawdown is the loan repayment calculation with the roles swapped: the capital is the bank's debt S, the withdrawal is the annuity A. Hence the term formula N = ln(A/(A − S(q − 1)))/ln q.
What you see
The top chart shows the balance Rₙ per period as bars with a line at level K. Below it, the interest per period grows as bars against the withdrawal line; the area in between is what comes out of the capital. When the account runs dry, a ring marks the last, smaller withdrawal. The second card shows the term N as a function of the withdrawal a, with a vertical asymptote at the perpetuity. The Working panel puts every number into the lecture formulas.
Try it yourself
Load the drawdown example: €50,000 at 0.5 % per month, €1,000 at the end of each month. It lasts 57.68 months, and €680.68 is left in month 58. Drag the withdrawal line to €250: the bars lie flat on line K. At €260 the line already tips over, and the account is empty after 653 months. In the perpetuity example with €100,000 at 6 % per year and monthly withdrawals in advance, the perpetuity is €484.26.
Frequently asked questions
- What is a perpetuity?
- A perpetuity is a regular payment that equals exactly the interest on a capital. The capital stays unchanged, so the payment can continue without limit. Paid yearly at the end of the year it is K · (q − 1), so €5,000 a year for €100,000 at 5 %. Conversely, K = a/(q − 1) is the present value of the perpetuity.
- How long will my savings last with regular withdrawals?
- The term follows from the repayment formula N = ln(A/(A − K(q − 1)))/ln q, where K is the capital, A the withdrawal per period and q = 1 + p/100. If A is not larger than the interest K(q − 1), the capital lasts forever. For €50,000, 0.5 % interest per month and €1,000 withdrawn, N = 57.68 months.
- How do you compute a perpetuity with monthly payments and an annual rate?
- With m payments per year and pro rata interest, the payments act like a yearly annuity a · (m + (m ± 1)p/200), plus for payments at the start and minus at the end of each sub-period. Setting it equal to the yearly interest K · p/100 gives a = K · p/(100 · (m + (m ± 1)p/200)). For €100,000 at 6 % paid monthly in advance this is €484.26.
- What can I adjust in the app?
- You choose yearly withdrawals, monthly withdrawals with a monthly rate or monthly withdrawals with an annual rate, in advance or in arrears, and set capital, interest rate and withdrawal. You drag the withdrawal directly in the chart; it snaps at the perpetuity. Two examples from the lecture load at the press of a button.
Subject: Finance
More from Finance
- Annuity Loan – Equal Instalments
- Savings Formula – Fixed Instalments
- Arithmetic vs. Geometric
- Loan types compared
- Geometric Series – the Proof
- Sub-annual Payments & Equivalent Annual Amount
- Compounding Frequency & Effective Rate
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