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 Short-term (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 inflation-protected 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 zero-coupon 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 over-the-counter. Callable bonds: a feature of bonds which allows the bond issuer to purchase back the bond at a specific price within a period. Mortgages Adjustable-rate mortgage: the interest rate is adjusted periodically. Mortgage-backed 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 (zero-coupon). 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 price-yield 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 time-to-maturity, and the columns are the yields. Clearly, as time-to-maturity 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%, 30-year bond with 6-month 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 30-year, 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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