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Investment fundamentals: Bond math

May 10, 2023
clock 4 MIN READ

Bond prices and interest rates have an inverse relationship. This means that as interest rates rise, bond prices decline (and vice versa). Understanding the math behind this relationship helps investors know what impact interest rate movements have the value of their portfolios.

While bonds are commonly issued at par value (the face value of the bond returned to the bond holder at maturity) and coupon payments are set as a percentage of par value, a bond’s price in the market is determined by its future income stream.

When interest rates (i.e. borrowing costs) rise, the price of existing bonds declines as their coupon payments become less attractive in the market, paying less than new bond issues. Conversely, when interest rates decline, existing bonds become more valuable as their income stream (coupon) is higher than what the market offers.

First, how are bonds priced?

For the sake of simplicity, consider a world where interest rates do not change. Bank A needs to raise cash and decides to issue a $100 bond with a 2% annual coupon, maturing in 10 years. This means that in exchange for $100 today, Bank A will pay the bondholder $2 annually (i.e. 2% of the $100 bond) for 10 years, at which point Bank A will return the $100 to the bond holder. In this example, the price of the bond is its par value $100.1

But what about interest rates?

Having established this example, let’s now introduce interest rates. If the prevailing market interest rates increase—let’s say to 4%—then all things equal, newly issued bonds will pay 4%, or $4, annually for the same $100 par value upfront.

Compare the $4 coupon to the $2, which would a rational investor rather hold? The bond in our first example pays $20 over 10 years, while the newer bond pays $40 over the same time period. Remember, both bonds were issued at the same par value of $100, but one now pays out twice as much. This illustrates that because the price of a bond is a reflection of its future income stream, a newer bond paying 4% is worth more than the older bond paying 2%. Consider: why would an investor pay $100 now for $20 of return, when they could get $40 of return at the same price? As such, if the investor wants to sell Bank A’s bond, they should expect to receive less than what they paid for it; meaning, the bond will trade at a discount (less than $100).2

Conversely, if prevailing interest rates decline, say to 1%, then Bank A’s existing bond paying 2% will return more to the bondholder than a newly issued bond (paying 1%). After all, if prevailing rates suggest that a $100 investment should only pay out 1%, an investment that pays out 2% should be worth more. This would also be reflected in the bond’s price and the bond will trade at a premium (more than $100).3

Bonds in the real world

The above example is for illustrative purposes only and does not capture the full breadth of factors which influence the wide range of debt securities. The intent is to illustrate the inverse relationship between bond prices and interest rates. Indeed, in the real world it is near impossible to consider changes to interest rates without also discussing how such changes affect fixed income markets.

Bond holders may pay close attention to the U.S. Federal Reserve’s (the Fed) monetary policy, and in particular, its stance on interest rates. The Fed may choose to raise interest rates to combat price inflation or lower interest rates to stimulate the economy. Changes in interest rates represent a risk to bonds as fluctuations can either increase or decrease the value of a bond. Indeed, longer term bonds are more exposed to interest rate risk as there is more opportunity (meaning, time) for interest rates to move adversely against the bond holder. Nevertheless, while bonds, like all securities, carry some form of risk, they can also be an effective tool to help diversify your investment portfolio. See SEI’s related pieces on Diversification and Bonds 101 for more information.

1 For those who are more mathematically inclined, to price a bond you must discount the value of future cash flow payments to the present day. PV = present value (or, price), F =Face Value (final payment) C = coupon payment, n = # of remaining payments, r = prevailing interest rate. Thus, PV = C *[(1-(1+r)-n )/r]+[F/(1+r)n ]. Better yet, in Excel, PV=(0.02, 10, -2, -100)

2 Using the calculation above, assuming a 4% discount rate, 10 payment periods, $2 annual payment, and future value of $100: the price of the bond is $83.78. In Excel, PV=(0.04,10,-2,-100)

3 $109.47. In Excel, PV=0.01,10,-2,-100)


Important information

Information provided by SEI Investments Management Corporation (SIMC). This information is for educational purposes only and should not be relied upon by the reader as research or investment advice. Investing involves risk, including possible loss of principal. Bonds and bond funds will decrease in value as interest rates rise. Diversification may not protect against market risk.

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