Selling underwater bonds and reinvesting at today's yield lifts margin at once, but the earn-back arithmetic shows what is gained, what is merely moved, and what it costs in capital.
The earn-back period is the loss realised on the bonds sold divided by the extra interest income the proceeds earn each year. A bank selling 500,000 of 1.80 per cent bonds with six years left, at a market yield of 5.00 per cent, realises a loss of 81,211 and earns 11,939 more a year by reinvesting at 5.00: an earn-back of 6.80 years, longer than the six years the bonds had left. At the same market yield the trade is worth exactly zero; it moves the loss forward in time and the income back.
Worked in full in Bank Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Loss trades, or securities repositionings, became a routine announcement once banks were sitting on bond books bought at low yields. The pitch is always an earn-back period and a margin uplift. The arithmetic below separates the accounting effect from the economic one, and shows where a shorter earn-back actually comes from.
| Input | Value |
|---|---|
| Bonds sold, face and book value | 500,000 |
| Coupon and book yield | 1.80% |
| Years to maturity | 6 |
| Market yield today | 5.00% |
| Tax rate | 21% |
| CET1 capital, unrealised losses filtered out | 650,000 |
| Risk-weighted assets | 5,500,000 |
Price = Σ coupon ÷ (1 + y)t + face ÷ (1 + y)n
Loss = book value − price
Pickup = price × reinvestment yield − face × book yield
Earn-back = loss ÷ pickup
In Excel: =PV(5%,6,-9000,-500000) for the price, 418,789, and =(500000-B1)/(B1*5%-9000) for the earn-back with the price in B1.
Step 1: the price. Six coupons of 9,000 and the 500,000 repayment, discounted at 5.00 per cent, are worth 418,789, 83.76 per cent of par.
Step 2: the loss. 500,000 − 418,789 = 81,211 before tax, 64,157 after.
Step 3: the pickup. The bonds earned 9,000 a year. The proceeds reinvested at 5.00 per cent earn 20,939. The difference is 11,939 before tax, 9,432 after.
Step 4: the earn-back. 81,211 ÷ 11,939 = 6.80 years. After tax it is 64,157 ÷ 9,432, the same 6.80, as long as both sides are taxed at one rate.
6.80 is not a coincidence. It is the future value of a six-year annuity at 5.00 per cent, ((1.05)6 − 1) ÷ 0.05 = 6.80. Over the six years the old bonds had left, the pickup adds up to 71,637, short of the 81,211 loss by 9,574. The gap closes only when each year's pickup is itself reinvested at 5.00 per cent: compounded to the end of year six it is worth exactly 81,211, the principal the bank no longer has, since it holds 418,789 of new bonds at that date instead of 500,000 repaid. Discounted at the market yield, the pickups are worth 60,601 and the lower principal at maturity costs 60,601. The net present value is zero.
So the trade, at the market yield, creates nothing. It brings forward a loss the accounts would otherwise absorb slowly, as below-market coupons, and pays it back slowly through above-book income. Earnings and the margin rise from year one, which is often the point; the economic value of equity does not change.
| Reinvestment yield | New income | Pickup | Pickup after tax | Earn-back, years |
|---|---|---|---|---|
| 5.00% | 20,939 | 11,939 | 9,432 | 6.80 |
| 5.50% | 23,033 | 14,033 | 11,086 | 5.79 |
| 6.00% | 25,127 | 16,127 | 12,741 | 5.04 |
| 7.00% | 29,315 | 20,315 | 16,049 | 4.00 |
Every year cut from the earn-back is bought with risk. A 7.00 per cent yield on a flat 5.00 per cent curve means credit spread, so loans or lower-rated paper, or a longer maturity on an upward-sloping curve. Moving 418,789 from government bonds at a zero risk weight into loans at 100 per cent adds the same amount to risk-weighted assets and, at the post-trade ratio, needs 44,608 of capital.
| Years left | Price | Loss | Pickup | Earn-back, years |
|---|---|---|---|---|
| 2 | 470,249 | 29,751 | 14,512 | 2.05 |
| 4 | 443,265 | 56,735 | 13,163 | 4.31 |
| 6 | 418,789 | 81,211 | 11,939 | 6.80 |
| 8 | 396,589 | 103,411 | 10,829 | 9.55 |
| 10 | 376,452 | 123,548 | 9,823 | 12.58 |
A short earn-back on short bonds is not a better trade, only a smaller one: the two-year bonds would have pulled to par within two years anyway.
For a bank that filters unrealised losses on available-for-sale bonds out of regulatory capital, the sale moves the 64,157 after-tax loss into CET1: the ratio falls from 11.82 to 10.65 per cent, 1.17 points, and takes 6.80 years of after-tax pickup to rebuild. Selling half the position halves the hit, to 32,078 and a ratio of 11.23 per cent, with the same earn-back. Tangible common equity does not move at the sale, because the loss was already in accumulated other comprehensive income, as the TCE ratio article shows. Bonds held to maturity are different: their loss is in neither measure until sold, and selling some can call the classification of the rest into question.
All figures are illustrative. Real repositionings involve bid-ask costs, curve shape, prepayment on mortgage securities and the bank's own tax position.
The book reads one bank's balance sheet four ways, with and without the securities marks; the workbooks are in the free companion files for this case.
There is no market standard, and the figure only makes sense next to the remaining life of the bonds sold. Reinvesting at the same market yield, the earn-back always exceeds that life: 6.80 years on bonds with six years left. A shorter earn-back means the new assets yield more than the old bonds' market yield, so the bank is taking more credit or duration risk, not finding free money.
It reduces CET1 for a bank that filters unrealised losses on available-for-sale securities out of regulatory capital, because the sale realises them. Here 64,157 after tax takes the CET1 ratio from 11.82 to 10.65 per cent. Tangible book value does not fall at the sale for available-for-sale bonds, because the loss was already in accumulated other comprehensive income.
It makes no difference if the loss and the income pickup are taxed at the same rate. The pre-tax loss of 81,211 over the pre-tax pickup of 11,939 is 6.80 years, and the after-tax loss of 64,157 over the after-tax pickup of 9,432 is also 6.80. Mixing the two, an after-tax loss over a pre-tax pickup, understates it.
This article is one calculation from Bank Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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