Pay Off Debt or Invest: Method

Pay Off Debt or Invest: How It Works

What it does

You have a loan and an extra sum each month. The extra can go to the loan (prepaying it) or into an investment. The page follows both choices month by month and reports where each one leaves you after a number of years, the interest and months that prepaying saves, and the investment return at which the two choices end level. Every step is printed as formula, numbers put in, answer. The page describes the two paths on the assumptions you enter; it does not say which to choose.

The method

Symbols. B₀ is the loan balance today; R_L the loan's yearly rate and i_L = R_L / 12 the rate charged each month; n_L the monthly payments left; PMT the loan payment; X_m the extra per month; D_m = PMT + X_m the monthly outlay; r the yearly investment return and j_m = (1 + r)^1/12 − 1 its monthly rate; T_m = 12 × years the months compared. Rates are used as decimals: 6.5 % is 0.065. All payments are made at the end of the month.

The loan payment. A balance that is charged i a month and reduced by a payment P is B (1 + i) − P a month later. The level payment that brings B₀ to zero in n months is

PMT = B₀ i_L (1 + i_L)ⁿ / [(1 + i_L)ⁿ − 1]

and the interest paid over the loan's life is everything paid less the balance, n PMT − B₀.

Same outlay in both paths. To compare like with like, both paths take the same D_m out of the budget every month of the comparison. Whatever does not go to the loan is invested, and once a loan is gone its payment is invested too.

Path A, prepay. The whole D_m goes to the loan. The loan is cleared after n_x = ln[D_m / (D_m − B₀ i_L)] / ln(1 + i_L) months, rounded up to a whole month k_A with a smaller last payment L_k. After that D_m is invested every month. A payment made every month for m months at a rate j grows to the payment times [(1 + j)^m − 1] / j, so with m_A = T_m − k_A

W_A = (D_m − L_k) (1 + j_m)^m_A + D_m [(1 + j_m)^m_A − 1] / j_m

Path B, invest. The loan gets PMT for its n_L months; X_m is invested every month and PMT joins it for the m_B = T_m − n_L months after the loan ends:

W_B = X_m [(1 + j_m)^T − 1] / j_m + PMT [(1 + j_m)^m_B − 1] / j_m

Net worth W is investments less the loan balance still owed. If the comparison ends before a loan is gone, its remaining balance B₀ (1 + i)^T − P [(1 + i)^T − 1] / i is taken off. The result reported is Diff = W_B − W_A.

Break-even return. Write S for the investments and L for the loan balance. In one month, if a part p of the outlay D pays the loan and D − p is invested,

W′ = [S (1 + j) + D − p] − [L (1 + i) − p] = S (1 + j) − L (1 + i) + D

The split p cancels. When j = i this is W′ = W (1 + i) + D for either path, and both paths start at W = −B₀, so they are equal in every month. A unit sent to the loan therefore earns exactly the loan rate. The break-even yearly return is the one whose monthly rate equals i_L:

r_be = (1 + i_L)¹² − 1

This is the loan's rate stated as growth over a full year (its effective annual rate), a little above the quoted rate because the quoted rate is twelve times the monthly one.

Advanced options. Deductible interest: if interest is deductible at a marginal tax rate t_d, each month's interest brings a tax saving of t_d times that interest, which the page invests in the same month in both paths; the loan then costs i_d = i_L (1 − t_d) a month. Tax on returns: a share t_i of each year's return is paid in tax, leaving r (1 − t_i) to compound. With both, r_be = [(1 + i_d)¹² − 1] / (1 − t_i). Employer match: the employer adds a percentage of the money you invest in a month, up to a cap; it is added first, in the month of the investment. The break-even is then found by trial, because the one-line argument above no longer holds. Volatility: each month's growth factor is drawn at random from a lognormal distribution whose yearly average return is r and whose yearly standard deviation of log return is the entered volatility; both paths are followed through several thousand such histories (always the same ones, so equal inputs give equal answers) and the share on which path B ends ahead is reported.

What it assumes

A fixed-rate loan with level monthly payments, interest charged monthly on the balance, no prepayment penalty and no fees. The same extra sum every month, never missed, and the same total outlay in both paths for the whole comparison, including after the loan is gone. A constant investment return (except in the volatility step), with no fees. Amounts are not rounded to cents. Taxes are one flat rate each: the deduction is assumed fully usable every month, and the tax on returns takes the same share of every year's return, which overstates the tax on gains that are not realised each year. The match is paid on invested money only and is not taxed or subject to vesting. Inflation is left out because it affects both paths alike.

When it misleads

The two rates are not the same kind of number. The loan rate is known and the saving from prepaying is certain. The investment return is a guess about an uncertain future, and it can be negative over years. Comparing them as if both were certain overstates the case for investing; the break-even return is the return needed with certainty.

The probability is a model output. It depends entirely on the average return and volatility entered, and the lognormal model understates how often markets fall sharply. It is not a forecast.

Liquidity is not measured. Invested money can usually be withdrawn in an emergency; money sent to a loan cannot be taken back. On the other hand a cleared loan lowers the monthly bills. Neither effect appears in net worth.

The comparison needs the discipline it assumes. Path A ends ahead or level only if the freed payment really is invested after the loan is gone, and path B only if the extra really is invested every month.

Real tax rules are more detailed. Whether interest is deductible, at what rate and up to what limit depends on the loan and the taxpayer; investment gains may be taxed only when realised and at other rates. The two flat rates are an illustration, not tax advice.

Variable-rate loans and loans with penalties are outside the model.

Worked example

The default case: a balance of 25,000 at 6.5 % with 60 months left, 300 extra a month, a 7 % investment return, compared over 10 years.

Loan: i_L = 0.065 / 12 = 0.005417; (1 + i_L)⁶⁰ = 1.38282; PMT = 25,000 × 0.005417 × 1.38282 / 0.38282 = 489.15. Interest if the loan runs its course: 60 × 489.15 − 25,000 = 4,349.22. Outlay: D_m = 489.15 + 300 = 789.15. Investment rate: j_m = 1.07^1/12 − 1 = 0.005654.

Path A: n_x = ln[789.15 / (789.15 − 135.42)] / ln(1.005417) = 34.85, so the loan ends in month 35 with a last payment of 670.25. Interest: 34 × 789.15 + 670.25 − 25,000 = 2,501.48, a saving of 1,847.75, and 25 months sooner. The outlay is then invested for 120 − 35 = 85 months: (1 + j_m)⁸⁵ = 1.61486, so W_A = 118.90 × 1.61486 + 789.15 × 0.61486 / 0.005654 = 192.01 + 85,816.63 = 86,008.64.

Path B: 300 a month for 120 months and 489.15 a month for the last 60: (1 + j_m)¹²⁰ = 1.96715 and (1 + j_m)⁶⁰ = 1.40255, so W_B = 300 × 0.96715 / 0.005654 + 489.15 × 0.40255 / 0.005654 = 51,315.52 + 34,825.72 = 86,141.24.

Difference: 86,141.24 − 86,008.64 = 132.60 in favour of path B if the return is 7 % every year. Break-even: r_be = 1.005417¹² − 1 = 6.697 %; at that return both paths end at 85,091.38.

The page's results carry full precision; the hand figures above are rounded, so a last digit can differ.

A calculator for education and analysis, not investment advice.