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INVESTMENT FUNDAMENTALS Bond Math. © 2023 SEI 1 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 . 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)