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In Module 3, we defined the following interest rate types/conventions: the effective interest rate, the annual effective interest rate, the n-compounded annual interest rate, the 𝑛-compounded annual discount rate, the simple annual interest rate and the simple annual discount rate.

We have two types of coupon bonds; the annual bonds and semiannual bonds. Annual bonds make coupon payment once a year while semiannual bonds make coupon payments after every six months. Each interest rate describes how the initial loan value (i.e., bond price) increases to its final value (i.e., the par value). Suppose you pay 93 for a 100-nominal zero-coupon bond (i.e., a bond with a par value, or a principal, of 100), which matures in two years. If we work in terms of annual (effective) interest rates (which are the same an once-compounded annual interest rates), the equation that expresses the rate of growth from the price of 93 to the principal of 100 is:

93(1 + 𝑟)! = 100 (1)

If this equation is rearranged, we can calculate the interest rate 𝑟 explicitly:

𝑟 = *100 93 + " ! − 1 = -100 93 − 1 ≈ 0.03695 = 3.695% (2)

This is the return — expressed in annual effective terms — that one gets from investing in the bond and holding it until maturity. It is also the interest rate paid by the borrower on the loan that the bond gives rise to (this assumes they are issuing the bond now — it may have been issued earlier at a different price, as we have not said whether this is a primary or secondary trade). A different convention causes Equation (1) to take on a slightly different form. For example, if using the annual rate compounded quarterly (i.e., the 4-compounded annual interest rate), one would write:

93*1 + 𝑟∗ 4 + $×! = 100, (3)

and, after rearranging, get a slightly different interest rate (which is why we have given it a different symbol (𝑟∗ instead of 𝑟). The various interest-rate formulae given in Module 3 arise from rearranging various versions of equation 1. Every version — every interest-rate convention — is a language for describing how the initial value (93 in the above example) increases to the final par value (here, 100). If we were considering buying two of the above bonds, equation 1 would be rewritten as:

186(1 + 𝑟)! = 200 (4)

Equation (4) shows how both the price (of the total investment) and the (total) par value would double. Notice that this will not change the value for 𝑟 that solved the equation — the multiplication by two (or by any other scaling factor) cancels away. Thus, interest rates are scale-free.

Instead of thinking about the increase from 93, forward in time, to 100, one can think of the decrease, backward in time, from the par value 100 to the current price 93. Mathematically, one could rewrite Equation (1) as:

100(1 + 𝑟)&! = 93 (5)

In this case the multiplicative factor that causes the decrease (namely,100(1+𝑟)"#) which is called a discount factor, as the process of decreasing from a future financial value to a current one is known as discounting (this is not related to a discount rate — one can express the discount factor in terms of an annual interest rate, as in Equation (5), or in terms of, for instance, an annual discount rate). Zero-coupon bonds are sometimes known as discount bonds (or pure discount bonds) — the whole instrument is based on the simple idea of a future value being discounted to some current value. It is customary to price zero-coupon bonds as semiannual bonds.

Before we contrast this with coupon-bearing bonds, let’s summarize by noting that we have created a link between prices and interest rates. In the above example, we have deduced the (annual effective) interest rate associated with a certain price; one could use the link the other way around and calculate a bond price based on a given interest rate. So, we cannot value bonds out of thin air — we can only do so based on given or assumed interest rates. This is still useful, because one can apply the market interest rate (which is usually observable to the whole market) to a particular bond that one is considering. Recalling that interest rates depend on their term, one must find the market interest rate for the appropriate term (by, for example, calculating the return on a government bond of the same term) and then apply it using the equations like the above.

Consider a bond like the above one — par value of 100 and maturity in two years — but one that pays coupons of 3 units at the end of each year. Instead of just discounting the 100 par-value payment like in equation 5, we must include the two coupons (and their corresponding timings):

𝑃𝑟𝑖𝑐𝑒 = 3;1 + 𝑟(") <&" + 103;1 + 𝑟(!)<&! (6)

Notice firstly that each cash flow (the first coupon, paid in a year’s time; and the second coupon combined with the par value, paid in two years’ time) is reflected and discounted separately. Indeed, we are treating the coupon-bearing bond like a portfolio of (combination of) two zero-coupon bonds that collectively give the same cash flows. This hypothetical portfolio behaves the same as the coupon-bearing bond, and so must command the same price.

In summary, coupons are treated additively (they are considered individually, and the individual prices added). Secondly, note how equation 6 accounts for the term structure of interest rates — it uses different rates (denoted, in this case, (𝑟(%)) and (𝑟(#)) to discount cash flows/loans of different terms).

If the coupon bond considered in equation 6 were government-issued (or were issued by an entity with negligible default risk) one could determine the bond’s price by substituting prevailing risk-free interest rate values in for (𝑟(%)) and (𝑟(#)) (using one-year and two-year rates, respectively).

If the issuer had non-negligible default risk, the price reflected by this would be too high — one could then estimate what a suitable price would be, in order to compensate an investor for the possibility of default. For example, if the price suggested by the risk-free interest rate is 98, but the bond is available for 94, one must decide whether the discount of 4 is sufficient to offset the default risk. Another approach would be to add a spread to the risk-free interest rates before substituting them into equation 6 — recall that the spread is the difference between risk-free interest rates and the interest rate implied by a defaultable bond price (so that adding it onto the risk-free rate gives a suitable rate to use in the pricing of the defaultable bond).

Recall that we cannot produce valuations from nothing — we are always using interest rate information (obtained elsewhere) to imply prices, or vice versa. But, again, this can still be useful, because the spread reflects the degree of default risk. Suppose we know that entity A exhibits about the same default risk as entity B (maybe they are companies with a similar standing and financial position), and that we also know that the spreads of entity A’s bond are about 1% (which one could calculate from comparing entity A’s bond prices to government bond prices). It would then make sense to add 1% to risk-free interest rates, and use these adjusted rates to price bonds issued by entity B. If one thought that entity B has more default risk than entity A, then using entity A’s spread would not give a good price estimate for entity B’s bonds, but it would give a good estimate for the upper bound one should pay for these bonds (because an additional discount would be warranted for the greater default risk).

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