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In the previous set of notes, we looked at an equation (number 6) that gave the price of a certain coupon bond. In that case, there were two cash flows. Consider the following expression for the price of a general fixed-income portfolio that involves 𝑛𝑛 cash flows:

Price = ∑𝑖𝑖=1 𝑛𝑛𝑃𝑃𝑖𝑖(1 + 𝑟𝑟(𝑡𝑡𝑖𝑖))−𝑡𝑡𝑖𝑖, (7)

where 𝑃𝑃1, 𝑃𝑃2, . . . , 𝑃𝑃𝑛𝑛 are the cash flow amounts, 𝑡𝑡1, 𝑡𝑡2, . . . , 𝑡𝑡𝑛𝑛 are the respective terms of these cash flows (i.e., the time until maturity, given in years), and 𝑟𝑟(𝑡𝑡1), 𝑟𝑟(𝑡𝑡2), . . . , 𝑟𝑟(𝑡𝑡𝑛𝑛) are annual effective interest rates corresponding to these terms.

Yield to Maturity is the interest rate that equates the present value of cash flows received from a debt instrument with its value today. If the bonds are semiannual bonds. We need to do three changes; the annual coupon is divided by 2, the number of years is multiplied by 2 for number of coupon payments and the YTM is divided by 2.

The price of the bond has a direct relationship with coupon rate and yield to maturity rates: 1. When the coupon rate is less than the yield to maturity, the bond sells for a discount against its par value. That is, the price of the bond is less than the par value. We call this kind of bond a discount bond.2. When the coupon rate is more than the yield to maturity, the bond sells for a premium above its par value. We call this kind of bond a premium bond.3. When the yield to maturity and coupon rate are the same, the bond sells for its par value. We call this kind of bond a par value bond.

If we consider a fixed-income portfolio — the right to receive a set of fixed cash flows at future dates — that is comprised of a cash flow of 3 in a year’s time, and of 103 in two years’ time, then we have the coupon bond considered in Equation (6) (we would have 𝑃𝑃1 = 3, 𝑃𝑃2 = 103, 𝑡𝑡1 = 1 and 𝑡𝑡2 = 2). Equation (7) can thus handle coupon-bearing bonds (where the cash flows are small and constant until the final one, 𝑃𝑃𝑛𝑛, is large, as it gives the par value payment as well as the final coupon), or it can handle more complicated sets of cash flows (such as the cash flows that arise for the combination of a number of coupon bonds). The fixed cash flows at regular intervals of time is known as annuity. The present value of annuity can be calculated by the formula

𝑃𝑃𝑃𝑃 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃1 −1/(1 + 𝑟𝑟)𝑡𝑡 𝑟𝑟 (8)

Whereas PMT shows the fixed cash flows, r is the interest rate and t is the time period. Just as we treated a coupon bond like a portfolio of zero-coupon bonds (by separating the individual cash flows and them adding them together), we can treat a whole fixed-income portfolio by individually treating each cash flow involved.

In order to value a fixed-income portfolio (such as a coupon bond), we need to have interest rates with which to discount each cash flow ((1 + 𝑟𝑟(𝑡𝑡𝑖𝑖))−𝑡𝑡𝑖𝑖 is the discount factor for the 𝑃𝑃th cash flow 𝑃𝑃𝑖𝑖). These rates (𝑟𝑟(𝑡𝑡1), 𝑟𝑟(𝑡𝑡2), . . . , 𝑟𝑟(𝑡𝑡𝑛𝑛) in Equation (7)) can be read off a yield curve, which plots interest rates against terms (to find 𝑟𝑟(𝑡𝑡1), we look at the vertical-axis value corresponding to 𝑡𝑡1 on the horizontal axis). This assumes the yield curve is given in the interest-rate convention that is used in the valuation equation — just like Equation (7) can be written in any convention, a yield curve can be given in any convention.

The yield curve is a very powerful idea. It summaries the interest-rate information of the whole market in one simple mathematical object. Referring to that object in the correct mathematical way allows you to value any fixed-income portfolio (with Equation (7), or some suitable variant). The yield curve is the most convenient and useful way to exploit the link we have created between prices and interest rates, because it is applicable to any portfolio (although credit risk can be a factor, and will be briefly discussed in the next section). The yield curve can be determined once, and then used over and over again for different portfolios.

The fact that we use different interest rates for different terms accounts for the term structure of interest rates; we must remember to use interest rates with a suitable spread included in order to account for any default risk present. So, beginning with a risk-free yield curve (one based on government bonds), one can add spreads to attain risky yield curves. Although it is not obvious how large a spread is needed for a particular entity, these ideas can still be used to make informative comparisons. If you price a defaultable coupon-bearing bond with the risk-free yield curve, you get an upper bound for the price — you can then decide what additional discount is needed for the possibility of default.

In the previous video, we supposed that government zero-coupon bonds of many maturities were observable, in which case one simply needs to calculate the corresponding yields to form a yield curve. This process — known as bootstrapping the yield curve — can be more difficult if only coupon bonds are observable, because one coupon bond price does not imply a single interest rate. One coupon bond price implies one equation, involving many interest rates — bootstrapping the yield curve involves writing many such equations and solving for the unknown rates simultaneously.

To calculate a bond’s yield-to-maturity, we must ignore the term structure of interest rates by solving the following equation, which involves just one rate, 𝑦𝑦:

𝑃𝑃𝑟𝑟𝑃𝑃𝑃𝑃𝑃𝑃= ∑𝑖𝑖=1 𝑛𝑛𝑃𝑃𝑖𝑖(1 + 𝑦𝑦)−𝑡𝑡𝑖𝑖. (9)

The equation links the yield-to-maturity 𝑦𝑦 with the bond’s (or bond portfolio’s) price. If we input the price into Equation (9), and solve for 𝑦𝑦, we find the yield-to-maturity (the total return, averaging over all cash flows) that investing in the bond/bond portfolio offers (assuming we buy at this inputted price). If we input 𝑦𝑦 and solve for the price, we are determining the price that gives the total return of 𝑦𝑦.

Equation (9) might look like a naive valuation equation; one written by someone who doesn’t know about the term structure of interest rates or who is simplifying by ignoring it. However, we are using Equation (8) after a proper valuation has taken place (or we are linking hypothetical valuations to yields-to-maturity). This allows us to find the average return over all aspects of a fixed-income portfolio, given a certain price for the portfolio (or to find the price that gives rise to a certain average, total return).

In the case of a zero-coupon bond, we find the price to be lower than the par value (as it is just the discounted par value). A coupon bond includes other cash flows (the coupons), which can increase the price relative to the par value. If the price is still less than the par value, the bond is said to be trading at a discount; if it is greater, the bond is said to be trading at a premium. There is an important intuition here: if a bond is trading at a premium, then the coupons are more than sufficient to cover the interest accruing, and so the buyer has to pay an extra premium to enter the bond. The loan induced by a zero-coupon bond has all the interest paid at the end; the loan induced by a coupon bond involves regular interest payments. If a bond price is equal to its par value (is trading at par or is priced at par), the coupon payments are exactly covering the interest due on the loan at each period, so that no additional payment is made at maturity (the initial loan value, the price, is just returned to the lender).

Finally, let us see how longer terms are riskier. A 100-nominal, one-year zero-coupon bond price is given by:

100�1 + 𝑟𝑟(1)�−1, (10)

whereas a two-year version of the same bond has a price:

100�1 + 𝑟𝑟(2)�−2. (11)

The exponent of -2 makes this second price more sensitive to changes in the interest rate than Equation (9). In fact, the second price is doubly sensitive — we can see this by writing Equation (10) as:

100�1 + 𝑟𝑟(2)�−1�1 + 𝑟𝑟(2)�−1. (12)

The discount factor (1 + 𝑟𝑟(1))−1 in Equation (10) makes the first bond price risky (because it might change), the second bond price involves two such discount factors. In general, bond prices involving longer terms are more sensitive to changes in the interest rates pertaining to their cash flows. In other words, longer terms result in greater sensitivity to the yield curve. This is why interest-rate risk is not so relevant in the context of money-market debt instruments, which are characterized by short terms.

Duration is a notion of average term of a coupon bond or bond portfolio. The term of a zero-coupon bond is obvious and can be easily compared to other zero-coupon bonds; in more complicated cases, duration helps us summarize the many terms involved. It is defined with:

𝐷𝐷𝐷𝐷𝑟𝑟𝐷𝐷𝑡𝑡𝑃𝑃𝐷𝐷𝑛𝑛 = � 𝑡𝑡𝑖𝑖𝑃𝑃𝑖𝑖(1 + 𝑦𝑦)−𝑡𝑡𝑖𝑖𝑛𝑛 𝑖𝑖=1 � 𝑃𝑃𝑖𝑖(1 + 𝑦𝑦)−𝑡𝑡𝑖𝑖𝑛𝑛 𝑖𝑖=1 . (13)

This is a weighted average: it takes each term 𝑡𝑡𝑖𝑖, weighs it by the discounted cash flow corresponding to that term 𝑃𝑃𝑖𝑖(1 + 𝑦𝑦)−𝑡𝑡𝑖𝑖, adds up the weighted sum, and then (in the denominator) divided by the sum of the weights used in the averaging. This is the standard procedure for taking a weighted average. For the weighted sum used by duration, the sum of the weights is the price of the portfolio. The idea is that we want an average of all of the terms, but we want to place more importance on the terms with larger cash flows (and more valuable discounted values), as these are more financially relevant. Because longer terms involve more interest-rate risk, and because duration is a measure of the terms involved in a portfolio, duration is a risk measure of a portfolio: it is a quantitative summary of the amount of interest rate risk involved.

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