Recently increased demand for sovereign bonds has resulted in lower, sometimes negative, yields.^{1}Sovereign bonds are debt securities issued by a national government. These bonds are regarded by investors as having very low risk because governments have the ability, unlike private borrowers, to tax their citizens to pay interest and return principal. This demand comes from central banks attempting to increase their national money supply and reduce interest rates in order to stimulate their economies. Demand also comes from investors seeking the perceived investment safety of government bonds.

Under negative interest rates lenders pay borrowers to take their money. Negative interest rates appear strange because it is expected that borrowers will have to pay lenders for the use of their money or, in other words, to pay a kind of rent for the temporary use of the lender’s funds, usually for the purpose of productive investment. Negative interest rates are also historically very unusual and, in most countries, unprecedented.

#### Interest rates

When people use the term “interest rate” they normally are referring to a nominal, or money, rate of interest. This rate has a real component and an inflation component. Under continuous compounding we can express this as

\begin{equation}y=r+\pi\label{eq:ccfisher}\end{equation}

because

\begin{equation}e^{y}=e^{r}e^{\pi}=e^{r+\pi}\label{eq:fisherexactwhy}\end{equation}

where

\(y\) is the nominal, or money, rate of interest,

\(r\) is the real rate of interest, and

\(\pi\) is the rate of inflation, or the average annual rate of increase of prices for all goods and services. Correspondingly, inflation is the rate of loss of the purchasing power of money (i.e., the rate of decrease of the amount of real goods and services that a unit of money can buy), assumed here to be a constant rate.

The real rate of interest is the rate lenders are most concerned about since it is this rate that measures, all other things remaining constant, the increase in their real wealth (i.e., the capacity to purchase real goods and services) that compensates them for the opportunity cost, and risk, of lending their money. As long as the money rate, \(y\), is greater than the inflation rate, \(\pi\), then the real rate will be positive and this will mean that lenders become richer in real terms for lending their wealth. But, if the money rate is less than the rate of inflation then the real rate is negative and lenders will have their real wealth decrease when they lend. Later, we will discuss why people might still continue to lend even when real rates are negative.

#### Bond pricing

Under continuous compounding^{2}We use continuous compounding since we are interested in the smallest initial principal we would need to deposit in an interest bearing account, having the same risk (and therefore the same return) as our fixed income instrument, such that withdrawals from this account could be used to replicate the cash flows of our bond. This minimum required initial principal is called the present value., the present value, or price, of a bond (disregarding taxes) is:

\begin{align}

P & =(\sum_{i=1}^{kT-m}ce^{-iy_{k}})+Fe^{-(kT-m)y_{k}}\label{eq:bond 1} \\ &=(\sum_{i=1}^{kT-m}\frac{\rho}{k}Fe^{-i\frac{1}{k}y})+Fe^{-(kT-m)\frac{1}{k}y}\label{eq:bond 2}\end{align}

where

\(P\) is the present value, or price, of all remaining income from the bond, evaluated at the time of bond purchase and just after the latest interest income payment (varies with supply and demand for bonds),

\(F\) is the face value, or the sum paid by the issuer at maturity to the current bond holder (fixed at bond issuance),

\(k\) is the number of interest income payments per year (fixed at bond issuance),

\(T\) is the stated term of the bond in years (fixed at bond issuance),

\(m\) is the number of interest income payments already received (fixed at bond purchase),

\(c=\frac{\rho}{k}F\) is the amount of interest income received per payment period (fixed at bond issuance),

\(\rho\) is the coupon, or stated, annual interest rate on the bond (fixed at bond issuance),

\(i\) is the summation index over the number of remaining interest income payments,

\(y\) is the nominal, or money, yield to maturity of the bond, expressed as an annual rate (varies with supply and demand for bonds), and

\(y_{k}=\frac{1}{k}y\) is the nominal, or money, yield to maturity of the bond expressed as a rate over \(\frac{1}{k}\) of a year. Under continuous compounding \(e^{y}=\prod_{i=1}^{k}e^{y_{k}}=e^{\sum_{i=1}^{k}y_{k}}=e^{\sum_{i=1}^{k}\frac{1}{k}y}\).

### Continuously vs. discretely compounded yields

Since bond yields may be quoted on other than a continuously compounded basis, the following can be used to translate discretely compounded yields into continuously compounded yields:

\begin{align}

\delta=k\cdot ln(1+\frac{\Delta}{k})\label{eq:translate 2}\end{align}

or continuously compounded yields into discretely compounded yields:

\begin{alignat}{2}

&&(1+\frac{\Delta}{k})^{k} &=e^{\delta}\notag\\

&\Rightarrow\quad

&1+\frac{\Delta}{k} &=e^{\frac{\delta}{k}}\notag\\

&\Rightarrow\quad

&\Delta &=k(e^{\frac{\delta}{k}}-1)\label{eq:translate}

\end{alignat}

where

\(\delta\) is the continuously compounded yield,

\(\Delta\) is the discretely compounded yield, and

\(k\) is the number of discrete compounding periods per year.

So, for example, if a yield is quoted as a continuously compounded rate (as the European Central Bank does for long-term government bonds) then the equivalent discrete rate with semiannual compounding can be found using the formula

\begin{align}

\Delta=2(e^{\frac{\delta}{2}}-1)\label{eq:translate 3}\end{align}

Now let’s generalize this, and at the same time make our notation simpler, by considering a fixed income security that may, or may not, make an income payment as frequently as \(\frac{1}{k}\) of a year. The formula for the present value of this generalized bond is:

\begin{align}

P & =\sum_{i=1}^{kT-m}C_{i}e^{-iy_{k}}\label{eq:gen bond 1} \\ &=\sum_{i=1}^{n}C_{i}e^{-i\frac{1}{k}y}\label{eq:gen bond 2}\\&=\sum_{i=1}^{n}C_{i}e^{-i\frac{1}{k}(r+\pi)}\label{eq:gen bond 3}\\&=\sum_{i=1}^{n}C_{i}e^{-i\frac{1}{k}\pi}e^{-i\frac{1}{k}r}\label{eq:gen bond 4}\\&=\sum_{i=1}^{n}\theta_{i}e^{-i\frac{1}{k}r}\label{eq:gen bond 5}\\&=\sum_{i=1}^{n}\theta_{i}e^{-i\frac{1}{k}(y-\pi)}\label{eq:gen bond 6}\end{align}

where

\(P\) is the present value, or price, of all remaining fixed income payments \(C_{i}\) (varies with supply and demand for bonds),

\(i\) is the summation index of remaining income payment periods,

\(n\) is the number of remaining income payments (fixed at bond purchase),

\(C_{i}\) is the income payment at end of period \(i\) that may (\(C_{i}>0\)) or may not (\(C_{i}=0\)) occur (fixed at bond issuance),

\(\theta_{i}\) is the real, or inflation adjusted (i.e., expressed in present dollars), income payment at end of period \(i\) that may (\(\theta_{i}>0\)) or may not (\(\theta_{i}=0\)) occur (fixed at bond issuance),

\(y\) is the nominal, or money, yield to maturity of the bond, expressed as an annual rate (varies with supply and demand for bonds),

\(r\) is the real rate of interest or, in this case, the real yield to maturity of the generalized bond, expressed as an annual rate (varies with supply and demand for bonds), and

\(y_{k}=\frac{1}{k}y\) is the nominal, or money, yield to maturity of the bond expressed as a rate over \(\frac{1}{k}\) of a year.

#### Negative yield to maturity

Investors will want to know the bond price above which the real yield to maturity of this bond becomes negative. In order to answer this question we first find the rate of change of bond price with respect to nominal yield to maturity. Using equation \eqref{eq:gen bond 6}:

\begin{align}

\frac{dP}{dy}&=\frac{d}{dy}(\sum_{i=1}^{n}\theta_{i}e^{-i\frac{1}{k}(y-\pi)})\label{eq:der dur 1} \\ &=-\sum_{i=1}^{n}\theta_{i}\frac{i}{k}e^{-i\frac{1}{k}(y-\pi)}<0\label{eq:der dur 2}\end{align}

We see that price is a decreasing function of the yield to maturity. And, by equation \eqref{eq:gen bond 6}:

\begin{equation}P(y=\pi)=\sum_{i=1}^{n}\theta_{i}e^{-i\frac{1}{k}(0)}=\sum_{i=1}^{n}\theta_{i}\label{eq: p at pi}\end{equation}

So, when the continuously compounded nominal, or money, yield to maturity is equal to the rate of inflation the bond price is equal to the sum of all future, inflation adjusted, income payments. This fact, together with the negative derivative of price with respect to yield to maturity, implies that at all bond prices above the sum of future, inflation adjusted, income payments the real yield to maturity, \(r\), is negative.

\begin{equation}P>\sum_{i=1}^{n}\theta_{i}\Leftrightarrow r<0\label{eq: neg r implication}\end{equation}

This means that anyone buying a bond with a negative real yield to maturity (i.e., where \(y<\pi\)) and holding it until maturity is guaranteed to lose real wealth. Why would anyone willingly do this? Either they don’t intend to keep the bond to maturity or they feel that alternative investments would end up being even more costly.

If a buyer does not intend to keep the bond until maturity then they must think they can resell the bond before its maturity date for a price higher than what they bought it for. In other words, they are speculating on bonds. But this is a very risky position to be in. Higher prices mean new buyers are risking even larger real losses if they hold these bonds to maturity. And these new buyers would have to be even more optimistic about future bond prices than the seller. Moreover, increasing bond prices imply interest rates are becoming even more negative when negative interest rates are a very unusual and unstable phenomenon in the first place. Lastly, as we will show below, bond prices become more volatile, or risky, as interest rates decrease.

This speculative scenario is not sustainable, and cannot end well. Negative real interest rates imply that no new bond investors intend to hold their bonds to maturity since to do so would reduce their real stock of purchasing power. And, although early investors (speculators) in these negative real interest rate bonds may profit if they can resell their bonds before prices begin to fall, other not so lucky investors will be forced to sustain substantial losses even if they don’t hold their bonds to maturity.

Investors are willing to pay a premium now for the perceived reliability of interest payments and return of principal of government bonds. But, when the demand for this safety declines so will the temporarily increasing prices of these sovereign bonds.

What if investors in negative real interest rate government bonds know they are a losing bet but feel that to invest elsewhere is likely to be even more costly in the near future? The appearance of negative real interest rate sovereign bonds seems to indicate that many market participants expect desperate economic conditions ahead, the cost of which would be greater than what they risk by buying these types of bonds now.

Both the speculative and safety explanations given above for investor demand for negative real interest rate bonds suggest that an economic or financial crisis may be looming on the horizon. We should assume that the buyers of these bonds are not stupid or irrational. On the contrary, they typically have large sums of money at stake and will for this reason have applied much thought and effort to their purchasing decisions. Moreover, the number of these buyers is large. Therefore, we witness in the price behavior of these bonds a kind of consensus of many thousands of thoughtful minds, strongly motivated by self-interest, that a financial doomsday is imminent.^{3}And, with the advent of securities trading directed by artificial intelligence, prices may also reflect the combined “opinions” of many intelligent machines.

#### Duration

Under continuous compounding the duration of a bond is defined as:

\begin{equation}D=-\frac{1}{P}\frac{dP}{dy}>0\label{eq: duration formula}\end{equation}

Duration is the proportional change in bond price \(P\) resulting from a small change \(\mathop{dy}\) of the yield to maturity. Duration can be considered the interest rate risk of bonds because as market interest rates rise the prices of existing bonds fall proportionally to their duration (for small changes in rates). This is because investors are not willing to purchase a bond at a price that will result in a yield to maturity less than the prevailing market rates of bonds having default risk similar to this bond. And as market interest rates fall, the prices of existing bonds rise. But, so does their price sensitivity to changes in interest rates (i.e., their interest rate risk rises). We can demonstrate this by showing that duration increases with decreasing interest rates.

\begin{align}

\frac{dD}{dy}&=\frac{d}{dy}[-P^{-1}\frac{dP}{dy}]\label{eq:a} \\ &=P^{-2}(\frac{dP}{dy})^{2}-P^{-1}\frac{d^{2}P}{dy^{2}}\label{eq:b}\\&=-P^{-2}[-(\frac{dP}{dy})^{2}+P\frac{d^{2}P}{dy^{2}}]<0\Leftrightarrow P\frac{d^{2}P}{dy^{2}}-(\frac{dP}{dy})^{2}>0\label{eq:c}\end{align}

\begin{align}

P \frac{d^{2}P}{dy^{2}} – (\frac{dP}{dy})^{2}&=(\sum_{i=1}^{n}C_{i}e^{-i\frac{1}{k}y})(\sum_{j=1}^{n}\frac{1}{k^{2}}j^{2}C_{j}e^{-j\frac{1}{k}y})-(-\sum_{i=1}^{n}\frac{i}{k}C_{i}e^{-i\frac{1}{k}y})(-\sum_{j=1}^{n}\frac{j}{k}C_{j}e^{-j\frac{1}{k}y})\label{eq:proof 1} \\ & =\frac{1}{k^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}(j^{2}-ij)C_{i}C_{j}e^{-(i+j)\frac{1}{k}y}\label{eq:proof 2}\\&=\frac{1}{k^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1}{2}[(j^{2}-ij)+(i^{2}-ji)]C_{i}C_{j}e^{-(i+j)\frac{1}{k}y}\label{eq:proof 3}\\ &=\frac{1}{k^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1}{2}(i^{2}-2ij+j^{2})C_{i}C_{j}e^{-(i+j)\frac{1}{k}y}\label{eq:proof 4}\\ &=\frac{1}{k^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1}{2}(i-j)^{2}C_{i}C_{j}e^{-(i+j)\frac{1}{k}y}>0\label{eq:proof 5}\end{align}

Therefore, duration (a measure of interest rate risk for bonds) increases with decreasing yield. So, as interest rates fall, even into negative territory, investors may want to carefully consider the interest rate risk of their bond portfolios.

#### Possible consequences of negative rates

Negative real interest rates on sovereign, or national government, bonds are here now. But what happens when negative real interest rates become commonplace on demand deposit accounts at local banks?

Say your bank, and necessarily other banks as well, suddenly declared that savings accounts would “pay” negative nominal interest rates. This would mean, for example, that if you deposited $100 at the beginning of the year you would end up with only $95 at year’s end if the account had a negative 5 percent interest rate. If this happened (assuming \(y<\pi\)) depositors may simply decide to withdraw their money and take a chance on putting it under their mattress. Then, at least they wouldn’t lose any more of their real wealth. This would occur if the real costs of holding currency outside of the banking system were less than the real costs of negative interest rates on bank deposits.^{4}The costs of being one’s own banker would include, among other things, the cost of preventing theft and of the reduced convenience of accessing the safeguarded money in the event you wanted to spend it.

Even if a bank paid a positive nominal interest rate on deposits but that interest rate was less than the rate of inflation (the situation now for savings accounts) this would still mean that the real rate of interest on your bank deposit would be negative (see equation \eqref{eq:ccfisher}). The amount of real goods and services that you could buy with the balance in your account would then be declining.

If the real costs of hoarding cash were less than the negative real interest charges on bank deposits then banking deposits would dry up and banks would fail. The government would very likely not allow this to happen. As a way to prevent this the government may forbid its citizens to possess or transfer the national currency, or any other type of money (e.g., gold, silver, precious gems), outside of the banking system. In order to ensure compliance, a completely electronic, or cashless, national economic regime would have to be imposed.^{5}See the article by Ross Clark entitled “Politicians want to move us towards a cashless world. It would be a disaster” appearing online in The Spectator on 25 November 2017 at https://www.spectator.co.uk/2017/11/politicians-want-to-move-us-towards-a-cashless-world-it-would-be-a-disaster/ Citizens would then be required to keep their money in banks and to pay the negative real interest charges on deposits.