Chapter 3. Fixed Income Securities


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1 IE Chapter 3. Fixed Income Securities
2 IE Financial instruments: bills, notes, bonds, annuities, futures contracts, mortgages, options,...; assortments that are not real goods but they carry values by the promises they represent. Securities: financial instruments that are traded in well developed markets. Fixed income securities: securities that promise definite cash flow streams.
3 IE The market for future cash The only uncertainty in holding a fixed income security is that the issuer may default. There are various forms of fixed income securities. Savings deposits Certificate of deposit (CD): issued in standard denominations such as $10,000. Large CD s can be traded in the market. Money market instruments Shortterm (1 year or less) loans by corporations and banks. Commercial papers: unsecured (without collateral) loans. A banker s acceptance: If A sells goods to B, and B promises to pay within a fixed time. Some bank may accept the promise by promising to pay the bill on behalf of B. A can then sell the banker s acceptance at a discount before expiration. Eurodollar deposits: deposits denominated in dollars but held in a bank outside US.
4 IE US government securities Treasury bills: issued in denominations of $10,000 or more with fixed terms to maturity of 13, 26, and 52 weeks. Treasury notes: maturities of 1 to 10 years, and sold in denominations as small as $1,000. The owner of notes also receives a coupon payment every 6 months until maturity. Treasury bonds: maturities more than 10 years. Treasury inflationprotected securities (TIPS): the principal value changes with the Consumer Price Index (CPI), but the coupon rate does not change in time. Treasury strips: each coupon payment is sold as separate security. Treasury strips are also known as zerocoupon bonds.
5 IE Example 3.3. Terms to learn: APR (Annual Percentage Rate); Points (the percentage of the loan amount charged for providing the mortgage, not including other possibly fees and expenses). A typical mortgage broker advertisement: Rate Pts Term Max amt APR yr $203, yr $203, yr $600, yr $203, yr $600,
6 IE What does the advertisement say about the mortgage expenses? For example, using the formula (A/P, 7.883%/12, 30 12) we can compute the monthly payment of a loan amount of $203,150 to be $1,474. Now, what is the implied expense? If we use the annual rate of 7.625% for the monthly payment of $1,474, then the principal would have been $1, 474 (P/A, 7.625%/12, 30 12) = $208, 253. The difference, $208,253$203,150=$5,103, is the total cost. The loan fee itself is 1% $203, 150 $2, 032. Therefore, the other expense is $5,103$2,032=$3,071.
7 IE Other bonds Municipal bonds: issued by agencies of state and local governments. Corporate bonds: issued by corporations. Some are traded on an exchange, many are traded overthecounter. Callable bonds: a feature of bonds which allows the bond issuer to purchase back the bond at a specific price within a period. Mortgages Adjustablerate mortgage: the interest rate is adjusted periodically. Mortgagebacked securities: individual mortgages are bundled into large packages and traded among institutions.
8 IE Details of a bond Face value (or par value). Coupon payments. The bid price: the price the bond is sold. The ask price: the price the bond is bought. Accrued interest: AI = # of days since last coupon # of days in current coupon coupon amount. Quality rating: Moody s (Aaa, Ba, etc.); Standard & Poors (AAA, BB, etc.).
9 IE Example of accrued interest: Suppose we purchased on May 8 a U.S. Treasury bond. The coupon rate is 9% per year, to be paid on February 15 and August 15 each year. Hence, AI = = This value will be added to the quoted price. If the face value is $1,000, then $20.50 would be added to the quoted price.
10 IE Quality Ratings Rating Classifications: Moody s Standard & Poor s High grade Aaa AAA Aa AA Medium grade A A Baa BBB Speculative grade Ba BB B B Default danger Caa CCC Ca CC C C D
11 IE Yield of a bond: Its internal rate of return (IRR). Consider a bond with face value F, coupon payment C per annum to be paid m times, mature in n/m years, and the purchase price P. Then, its yield is λ, satisfying P = = n F (1 + λ/m) n + k=1 F (1 + λ/m) n + C λ C/m (1 + λ/m) k { 1 1 [1 + (λ/m)] n }.
12 IE The yield is not explicitly computable in general. One exceptionally simple case is when C = 0 (zerocoupon). In that case, ( ) n F λ = m P 1. Another interesting case is when C/F = λ (i.e. the coupon rate is exactly the yield). Then, and the bond is said to be at par. P = F In general, the priceyield curve is convex. The steepness of the curve appears to be related to the length of the period.
13 IE Prices of 9% coupon bonds: 5% 8% 9% 10% 11% 1 yr yr yr yr yr where the rows are timetomaturity, and the columns are the yields. Clearly, as timetomaturity increases, the price of the bond tends to depend more sensitively to the change of yield.
14 IE Duration: For a cash flow {(F 1,..., F n ) F i occurs at t i, i = 1,..., n}, its duration is the weighted average of the payment dates D = n k=1 P V (F k)t k P V Macaulay Duration of a bond: D = n k=1 n k=1 F k (1+λ/m) k F k (1+λ/m) k k m Explicitly, for a bond with coupon c, paid m times, and yield rate y: D = 1 + y my 1 + y + n(c y) mc[(1 + y) n 1] + my.
15 IE Example: Consider a 7% bond with 3 years to maturity. Suppose that the yield is 8%. Then, the breakdown of the duration of its cash flows will be: Year Payment Discount Factor PV Weight Duration Total
16 IE Example 3.7. Consider a 10%, 30year bond with 6month coupons. Suppose it is at par (yield is 10%). We then compute from D = 1 + y my 1 + y + n(c y) mc[(1 + y) n 1] + my that D = 1 + y my = [ 1 [ 1 ] 1 (1 + y) n ] 1 (1.05) 60 =
17 IE If we denote then P (λ) = n F k /(1 + λ/m) k, k=1 D(λ) = P (λ) P (λ) (1 + λ/m). We call to be the modified duration. D M (λ) = P (λ) P (λ) The duration measures the sensitivity of the price relative to the change of interest rate. The most sensitive bond will be zero coupon bond with a long maturity period.
18 IE Example. Consider a 30year, 10% coupon bond, which is at par with price $100. The duration is D = Hence, D M = 9.94/1.05 = The slope of the curve at that point is dp/dλ = 947. The straight line approximation suggests if the yield changes to 11%, then the change in price is P = D M 100 λ = = Hence the estimated new price is $ On the other hand, if the bond does not carry any coupons. Then, we have D = 30, and D M 27. If the yield changes to 11%, then the estimated new price will be $73, which is a big change!
19 IE It is important to control the risk of a portfolio with respect to the interest rate risk. Let us consider what happens if we hold two bonds, A and B, in a portfolio. We have D A = n k=0 P V A k t k P A and D B = Observe that the total PV is n P = (P V A k=0 The duration of the portfolio is n k=0 D = (P V k A + P V k B)t k P A + P B = k n k=0 P V B k t k P B. + P V B k ) = P A + P B. P A P A + P B DA + In other words, it is a convex combination of the two! P B P A + P B DB.
20 IE In general, if we have m fixed income securities, each with price P i and duration D i, i = 1, 2,..., m, then the portfolio will have price P and duration D: P = P 1 + P P m D = w 1 D 1 + w 2 D w m D m where w i = P i /(P 1 + P P m ), i = 1, 2,..., m.
21 IE Immunization: managing the interest rate risk. Example. The X Corporate has an obligation to pay $1 Million in 10 years. It wishes to invest in some bonds in order to meet this obligation. The following three bonds are under consideration: Rate Maturity Price Yield Bond 1 6% 30 yrs % Bond 2 11% 10 yrs % Bond 3 9% 20 yrs % We calculate that D 1 = 11.44, D 2 = 6.54, D 3 = 9.61, and P V = 414, 643. We decide to combine Bond 1 and Bond 2, and set P V = V 1 + V 2 10P V = D 1 V 1 + D 2 V 2 leading to V 1 = $292, and V 2 = $121,
22 IE Immunization results: 9% 8% 10% Bond 1 price shares value 292, , ,535 Bond 2 price shares value 121, , ,515 obligation value 414, , ,889 difference 19 1,562 1,162
23 IE Convexity of a bond: it is possible to improve the immunization by using a second order approximation. Let C = P (λ) P (λ). In the case of a cash flow with payments c k, C = n k=1 c k k(k + 1) (1 + λ/m) k m 2 P (1 + λ/m) 2. We have P D M P λ + CP 2 ( λ)2.
24 IE It is possible to use a combination of bonds to fit the PV, the duration, and the convexity of the obligation. If we hold a bond portfolio (P 1,, P m ), then its convexity is D = w 1 D w m D m where w i = P i /(P P m ), i = 1, 2,..., m. Back to the problem of the X Corporate, if the convexity is to be matched as well, then we can consider the following equation P V = V 1 + V 2 + V 3 D P V = D 1 V 1 + D 2 V 2 + D 3 V 3 C P V = C 1 V 1 + C 2 V 2 + C 3 V 3. One will need three bonds to do the matching.
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