Negative Yields

Separate a negative rate from negative cash flows

The sign of a yield belongs to a valuation relationship. The sign of a coupon belongs to the contractual cash-flow schedule. A holder can pay sufficiently more than the scheduled receipts to obtain a negative yield even when every receipt is positive. There is no need to imagine the issuer sending a negative principal payment or collecting an extra fictional coupon from the holder.

Official sources provide historical context without supplying current assumptions for this guide. The ECB's explainer discusses its past negative policy-rate setting, while a BIS review published in March 2015 described negative-yielding debt. A policy rate and a particular bond yield are distinct quantities. The examples below are self-contained arithmetic, not claims about today's policy stance, available securities or executable market prices.

Inspect the discount factor rather than rejecting the sign

For annual nominal yield y compounded m times, a payment at period k has discount factor (1+y/m)^(-k). When y is negative but 1+y/m remains positive, that factor exceeds one. The present value of a positive future payment is therefore larger than the payment itself. The familiar statement that future money is always worth less assumes a positive discount rate and is not a universal identity.

The mathematical domain is important: the compounding base must be strictly positive for this financial interpretation. At or below that boundary, the usual regular-period model is invalid, even if an integer power happens to return a numerical value. A calculation should distinguish an admissible negative yield from an invalid denominator rather than treating every negative input as an error or every finite result as meaningful.

Price a hypothetical negative-yield zero

Assume a single USD 1,000 payment in exactly two years, annual compounding and an annual nominal yield of -1%. With no coupons, calls, default, taxes, costs or accrued interest, price is 1000/(0.99)^2 = USD 1,020.304051. The two-year discount factor is approximately 1.020304051. A holder paying this modeled amount receives less nominal cash at maturity than the initial outlay.

Reverse the calculation using (1000/1020.304051)^(1/2)-1 and the result is approximately -1%, with the tiny rounding difference removed by using the full-precision price. This verifies the internal relationship among price, maturity payment and yield. It does not assess why someone might accept that relationship or whether cash storage, collateral needs, regulation or other alternatives would matter in a particular setting.

Read an increase in yield correctly

Change the assumed annual yield from -1% to -0.5%. This is an increase of 50 basis points even though both rates remain negative. Price becomes 1000/(0.995)^2 = USD 1,010.075503, a decline of approximately USD 10.228547 from the unrounded baseline. No time passes and the USD 1,000 maturity payment remains unchanged. The ordinary inverse price-yield relationship has not reversed at zero.

For this two-year zero, modified duration at -1% is 2/0.99 = approximately 2.020202 years, which exceeds its two-year Macaulay duration. Dividing by a number below one explains that result. A rule asserting that modified duration must always be smaller than Macaulay duration silently assumes positive yield and would incorrectly reject this valid negative-yield arithmetic case.

Distinguish nominal yield from purchasing-power change

A negative nominal yield is different from a negative inflation-adjusted outcome. To isolate that distinction, suppose a separate one-year example has a 2% nominal return and a 3% increase in the relevant price level. Its exact purchasing-power change is 1.02/1.03-1 = approximately -0.970874%. The nominal return is positive; the real result is negative because the comparison uses a different denominator.

That calculation assumes a known realized inflation change over the same horizon, not an inflation forecast or a market inflation-linked yield quotation. Conversely, a negative nominal yield does not by itself determine the sign of purchasing-power change when the price level can also fall. Keep rate type, horizon and inflation assumption visible instead of using the word negative as a complete description of economic performance.

Avoid turning domain validity into an economic endorsement

A positive compounding base establishes that the formula is well defined; it does not make every extreme input plausible. As that base approaches zero from above, long-dated present values can become enormous and numerically difficult. Those values describe the imposed mathematical assumptions. They are not evidence that a corresponding market exists, that an arbitrage is available or that the result can be financed and traded.

Likewise, discount factors above one do not alone prove a contradiction across markets. Such a conclusion would require a specified set of tradable claims, borrowing and lending opportunities, storage or holding costs, and consistent cash-flow timing. The educational example supplies none of that market structure. It is more precise to explain the pricing identity than to infer a trading opportunity from the sign of one yield.

Keep negative-yield review inside the same model contract

The parent model remains a regular noncallable fixed-coupon bullet calculation at a coupon date, with annual nominal yields compounded one, two or four times and maturities of one through thirty whole years. Negative yield does not add support for negative contractual coupons, inflation-linked principal, settlement adjustments or contingent payments. Those are separate structural questions that require their own definitions and cannot be answered by relaxing a sign check.

Useful arithmetic checks are continuity around zero yield, inverse price movement, a positive discount base and agreement between direct discounting and a reverse-yield calculation. State whether a shock is an increase or decrease before describing its sign. Together, these checks allow an unusual-looking but valid calculation to remain understandable without promoting the example into personal advice or an unsupported statement about actual bond conditions.

Primary references

1. ECB: The ECB's Negative Interest Rate, historical explainer
2. BIS: March 2015 Quarterly Review, A Wave of Further Easing