Bond Ladder: Method

Bond Ladder: How It Works

What it does

A bond ladder splits one sum across several bonds or certificates of deposit that mature one after another, for example one each year for five years. Each bond is a rung. The page lays the ladder out: how much goes in each rung, what interest it pays, how long the money is tied up on average, how sensitive its market value is to a change in yields, what cash arrives in each year, and what happens to the income as matured rungs are reinvested ("rolled") into new long rungs. Every step is printed as formula, numbers put in, answer.

The method

Symbols. A₀ is the amount invested; n_r the number of rungs; T_k the maturity of rung k in years from today (T_k = first maturity + (k − 1) × spacing); y_k its yield per year; F_k its face amount, the sum repaid at maturity; m_c the number of coupons (interest payments) a year. Yields are used as decimals: 4.65 % is 0.0465.

Par bonds. Each rung is taken as a par bond: its coupon rate equals its yield, so its price equals its face amount. The faces therefore add up to A₀, and rung k pays F_k y_k of interest a year in m_c equal coupons. A newly issued bond or CD is close to this. A bond bought later in the market usually trades above or below its face; the Bond Price and Yield tool handles that case.

Face amounts. Equal face: F_k = A₀ / n_r. Equal income (Advanced): F_k y_k is the same for every rung, which gives F_k = A₀ (1 / y_k) / (1 / y₁ + … + 1 / y_n).

Averages. Yearly interest I_tot = F₁ y₁ + … + F_n y_n. Weighted-average yield = I_tot / A₀. Average maturity T_avg = (F₁ T₁ + … + F_n T_n) / A₀.

Duration. Macaulay duration is the average time to a bond's payments, each weighted by its present value. For a par bond with N = m_c T coupon periods left and a yield per period i = y / m_c,

D = [(1 + i) / i] [1 − (1 + i)^−N] / m_c   (years)

Modified duration MD = D / (1 + i) is the percent fall in price for a one-point rise in yield, to first order. The ladder's durations are the averages of the rungs' durations weighted by the money in each rung, F_k / A₀. The weighted modified duration is exactly the first-order price sensitivity of the ladder when every rung's yield moves by the same amount. The page also reprices each rung exactly at a yield one point higher: price per 1 of face = (c / m_c) [1 − (1 + i′)^−N] / i′ + (1 + i′)^−N, with c the coupon rate and i′ the new yield per period.

Cash-flow calendar. Rung k pays a coupon F_k y_k / m_c at the end of every period until T_k, when it also repays F_k. The calendar adds these payments by year: year 1 is the twelve months from today. The principal column sums to A₀.

Rolling. When rung j matures, at time T_j, its face F_j buys a new par bond with a term of n_r × spacing, which becomes the new longest rung and keeps the spacing even. Its yield Y_new is read from the entered yields at that term (a straight line between rungs; the longest rung's yield when the term is at or beyond it). The yearly interest after the roll-over changes by F_j (Y_new − y_j). In the shifted case every yield has moved by dY, so the new bond yields Y_new + dY while the bonds already held keep their coupons. After j roll-overs the share F₁ + … + F_j of the ladder earns the new yields.

Advanced. After-tax yield = average yield × (1 − t_x). Real yield = (1 + after-tax yield) / (1 + h) − 1, with h the inflation rate. Interest in today's money divides each coupon by (1 + h)^t, t being its date. The comparison table puts all of A₀ in one par bond at the shortest rung's maturity and yield, and at the longest rung's.

What it assumes

Every rung is bought today at par and held to maturity; the issuer pays in full and on time; no bond is called early. Every maturity is a whole number of coupon periods. Coupons are spent, not reinvested. Yields are quoted with compounding m_c times a year. A matured rung is reinvested in full on its maturity date, with no cost and no cash left idle. In the shifted case the whole curve moves by the same amount once, right after today, and then stays. Tax is one rate on interest only; inflation is constant. Default yields are U.S. Treasury par yields of the date shown, a past fact and not a quotation.

When it misleads

Yields do not move in parallel or only once. Short and long yields change by different amounts and keep changing. The shifted case is one clean what-if that shows the mechanism, not a forecast of income.

Duration is a price measure. It describes what the bonds would fetch if sold before maturity. Held to maturity, a rung repays its face whatever yields do; the cost of a rise in yields is then the higher interest not earned in the meantime.

The average yield is not a return. It is the first year's interest over the amount invested. The return actually earned over several years depends on the yields at which rungs are rolled, which are unknown today.

Credit risk and call risk are left out. For bonds that can default or be repaid early, the cash-flow calendar is the promised schedule, not a certain one. Insured CDs and Treasury securities differ from corporate and municipal bonds in this respect.

Real bonds are rarely at par. A bond bought above or below face pays a coupon different from its yield, so its interest and its repaid principal differ from the par figures shown here, although the yield earned is the same.

Whole bonds. Bonds are sold in fixed denominations; the equal-income faces in particular will not be round amounts.

Worked example

The default case: 100,000 in five rungs maturing in 1, 2, 3, 4 and 5 years, at yields of 4.65, 4.86, 4.90, 4.925 and 4.95 % (U.S. Treasury par yields of 12 November 2026; the 4-year figure is midway between the 3- and 5-year yields), coupons every six months, equal face amounts.

Face of each rung: 100,000 / 5 = 20,000. Yearly interest: 20,000 × (0.0465 + 0.0486 + 0.0490 + 0.04925 + 0.0495) = 930 + 972 + 980 + 985 + 990 = 4,857. Weighted-average yield: 4,857 / 100,000 = 4.857 %. Average maturity: (1 + 2 + 3 + 4 + 5) / 5 = 3 years.

Duration of rung 1: i = 0.0465 / 2 = 0.02325, N = 2; D = (1.02325 / 0.02325) × (1 − 1.02325⁻²) / 2 = 0.989 years, MD = 0.989 / 1.02325 = 0.966. The five rungs have D = 0.989, 1.930, 2.826, 3.679 and 4.490 years; their average is 2.783 years, and the average modified duration is 2.716 years. A one-point rise in every yield lowers the ladder's market value by about 2.7 % (2.66 % by exact repricing).

Calendar: year 1 pays 4,857 of interest and returns 20,000; year 2 pays 4,857 − 930 = 3,927; then 2,955, 1,975 and 990. Interest over the five years is 14,704, on top of the 100,000 returned.

Rolling: after 1 year, rung 1 is reinvested for 5 years at 4.95 %. Yearly interest becomes 4,857 + 20,000 × (0.0495 − 0.0465) = 4,917. If all yields had risen one point, the new bond yields 5.95 % and the interest becomes 4,857 + 20,000 × (0.0595 − 0.0465) = 5,117: an average yield of 5.117 %, up 0.26 points, because only one fifth of the ladder has been rolled. After three roll-overs it is 5.545 %.

A calculator for education and analysis, not investment advice.