# fixed-income securities

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Fixed-income securities. A variety of fixed-income securities, I. Interest-bearing bank deposit: (1) saving account, (2) certificate of deposit (CD, a time deposit), (3) money market account. Commercial paper. Eurodollar deposits and CDs. A variety of fixed-income securities, II. T-bills. - PowerPoint PPT Presentation

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• Fixed-income securities

• A variety of fixed-income securities, IInterest-bearing bank deposit: (1) saving account, (2) certificate of deposit (CD, a time deposit), (3) money market account.Commercial paper.Eurodollar deposits and CDs.

• A variety of fixed-income securities, IIT-bills.T-notes.T-bonds.T-strips.

• A variety of fixed-income securities, IIIMunicipal bonds.Corporate bonds.Callable bonds.Put bonds.Zeros (zero-coupon bonds).Mortgage-backed securities (MBS): publicly traded bond-like securities that are based on underlying pools of mortgages.

• Bond pricing formulaP = C { [ 1 1 / (1 + i)N ] / i } + FV / (1 + i)N .FV is the face (par) value of the bond.C is coupon payment.i is the period discount rate.If coupons are paid out annually, i = YTM. If coupons are paid out semiannually, i = YTM/2.N is the number of periods remaining.The first term, C { [ 1 1 / (1 + i)N ] / i }, is the present value of coupon payments, i.e., an annuity.The second term, FV / (1 + i)N, is the present value of the par.

• Yield (yield to maturity, YTM)The (quoted, stated) discount rate over a year.Determined by the market.Time-varying.

• Bond pricing example, ISuppose that you purchase on May 8 this year a T-bond matures on August 15 in 2 years. The coupon rate is 9%. Coupon payments are made every February 15 and August 15. That is, there are still 5 coupon payments to be collected: August this year, 2 payments next year, and 2 payments the year after next year. The par is \$1,000. The YTM is 10%. What is the fair price of the bond?

• Bond pricing example, II

Sheet1

Discount rate

Cash flow0.05PV

08/15/Y04542.8571428571

02/15/Y14540.8163265306

08/15/Y14538.8726919339

02/15/Y24537.0216113656

08/15/Y21045818.7848439595

Bond Quotation978.3526166468

Accrued Interestcoupon payment * (days since last payment /

20.521978022days in this period)

Actual Payment% Par

998.874594668899.8874594669

• Bond pricing example, IIICalculator: 45 PMT; 1000 FV; 5 N; 5 I/Y; CPT PV. The answer is: PV = -978.3526.Bond quotations ignore accrued interest.Bond buyer will pay quoted price (\$978.3526) and accrued interest (\$20.5220), a total of \$998.8746, to the seller.

• YTM example Northern Inc. issued 12-year bonds 2 years ago at a coupon rate of 8.4%. The semiannual-payment bonds have just make its coupon payments. If these bonds currently sell for 110% of par value, what is the YTM?Calculator: 42 PMT; 1000 FV; 20 N; -1100 PV; CPT I/Y. The answer is: I/Y = 3.4966.YTM = 2 3.4966 = 6.9932 (%).

• A few observationsBond price is a function of (1) YTM, (2) coupon (rate), and (3) maturity.The YTMs of various bonds move more or less in harmony because the general interest rate environment (e.g., Fed policies) exerts a market-wide force on every bonds.As YTMs move (in harmony), bond prices move by different amounts.The reason for this is that every bond has its unique coupon (rate) and maturity specification.It is therefore useful to study price-yield curves for different coupon rates or different maturities.

• Price-yield curves and coupon ratesNegative slopes: price and YTM have an inverse relation.When people say the bond market went down, they mean prices are down, but interest rates (yields, YTMs) are up.When coupon rate = YTM, the bond has a price of 100%.

• Price-yield curves and maturityEverything else being equal, bonds with longer maturities have steeper price-yield curves. That is, the prices of long bonds are more sensitive to interest rate changes, i.e., higher interest rate risk.

• AssignmentUse Excel to duplicate both Figure 3.3 and 3.4. Due in a week.

• Current yield (CY) vs. YTMCY = annual coupon payment / bond price.CY is a measure of the annual return of the bond (if it is held to maturity).CY and YTM move in the same direction. When the bond price falls, CY and YTM rise.YTM is a more sensible measure of return because it is the current return rate implies by the entire cash flow stream.

• Managing a portfolio of bonds: horse racingSuppose that you are a bond manager and your (your companys) goal is to have good relative performance with respect to a 20-year bond index. After studying the interest rate environment, you believe that interest rates will fall in the near future (and your belief is not widely shared by investors yet). Should you have a bond portfolio that has an average maturity longer or shorter than 20 years? What if you believe interest rates will rise in the near future?

• [extra] Managing a portfolio of bonds: immunization, IFor many institutional bond portfolios, their goals are not to out-perform the market. The usual purpose is, in fact, to use a bond portfolio to meet a series of future cash obligations.For example, UVM may want to invest and hold a bond portfolio to meet a \$100 million expansion in 10 years.

• [extra] Managing a portfolio of bonds: immunization, IIA simple strategy is to invest in 10-year zero-coupon bonds that will pay exactly \$100 million at maturity.Say the current YTM for 10-year zero-coupon T-bonds is 8%. That means you need to invest \$45.6387 million in 10-year zero-coupon T-bonds today. 45.6387 * (1.04)20 = 100.No re-investment risk.

• [extra] Managing a portfolio of bonds: immunization, IIIBut what if you would like to earn more than 8%, or equivalently, invest less than \$45.6387 million?You may need to invest in more risky bonds, such as corporate bonds or emerging market bonds, which usually do not have zeros.This is where we would need immunization.

• [extra] Managing a portfolio of bonds: immunization, IVSuppose that UVM is interested in bonds that have 9% YTMs.UVM may acquire a portfolio having a value equal to the present value of the \$100 million obligation @9%, i.e., \$41.4642 million. 41.4642 * (1.045)20 = 100.If the YTM does not change over the next 10 years, the total value of the portfolio, including the re-investment of coupons, will be \$100 million at the end of the 10 year horizon. Therefore, UVM will meet the obligation exactly.

• [extra] Managing a portfolio of bonds: immunization, VA problem with this present-value-matching technique arises if YTM changes.The value of the bond portfolio and the present value of the obligation will both change in response, but probably by amounts that differ from one another, i.e., no longer present-value matching.Recall that bonds with longer maturities are more sensitive to interest rate risk.

• Managing a portfolio of bonds: immunization, VI[extra] Immunization: the procedure that immunizes the bond portfolio value against interest rate (YTM) changes.For immunization, we need a measure to capture the average maturity of all cash flows (coupons and principles) for all bonds in the portfolio. A popular measure is called (Macaulay) duration.[extra] Immunization is to make the duration of the bond portfolio equal to that of the obligation so that when there is an interest rate change, the change in the value of the bond portfolio is roughly equal to that in the present value of the obligation.Duration is a weighted average of the times that cash flows are made. The weighting coefficients are the present values of the individual cash flows.

• Duration (3-year bond, coupon rate 7%, YTM 8%)D = [PV(t0)* t0 + PV(t1)* t1 + + PV(tn)* tn ] / total PV.

Sheet1

Discount factor

YearCF, % par0.08PV, % parWeightA*E

0.53.50.96153846153.36538461540.0345596790.0172798395

13.50.9245562133.23594674560.03323046060.0332304606

1.53.50.88899635873.11148725530.03195236590.0479285489

23.50.8548041912.99181466860.03072342880.0614468576

2.53.50.82192710682.87674487370.02954175840.0738543961

3103.50.790314525781.79755341310.83999230732.5199769219

97.378931571612.7537170245

Duration

• Duration of a portfolioSuppose that all the bonds have the same yield (a reasonable assumption for an institutional portfolio), the duration of a portfolio is a weighted sum of the durations of the individual bondswith weighting coefficients proportional to the market values of individual bonds.For n bonds, V = V1 + V2 + + Vn.D = (V1 / V) * D1 + (V2 / V) * D2 + + (Vn / V) * Dn.

• [extra] Immunization example, IBack to UVMs 10-year, 100-million obligation.Suppose that UVM is planning to hold a 2-bond portfolio and both bonds have a 9% YTM.Bond 1: 6% coupon rate; mature in 30 years. Thus, bond price is 69.04% of par, and D1 = 11.44 (year).Bond 2: 11% coupon rate; mature in 10 years, Thus, bond price is 113.01% of par, and D2 = 6.54 (year). Please verify these numbers (Excel).

• [extra] Immunization example, IIThe present value, PV, of the obligation @ 9% is \$41.4642 million.The immunized portfolio is found by solving the following two equations:V1 + V2 = 41.4642.(V1 / 41.4642) * D1 + (V2 / 41.4642) * D2 = 10. The second equation states that the duration of the bond portfolio equals the duration of the obligation.Solution: V1 = 29.2789 and V2 = 12.1854.

• [extra] Immunization example, III

Sheet1

YTM (%)

8910RateMaturity

Bond 16%30

Value32814297.49980122927890026353308.9491432

Price773.7651002552690.4621.4142094986

Shares42408.603707995442408.603707995442408.6037079954

Bond 211%10

Value12980668.47151541218540011454459.861552

Price1203.85489517451130.11062.3110517127

Shares10782.585611892810782.585611892810782.5856118928

Obligation100000000100000000100000000

PV45638694.620129241464285.968458537688948.2873001

Difference156271.35118737114.0315415487118820.523395203

• [extra] To immunize or not

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