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. T-notes. T-bonds. T-strips.

A variety of fixed-income securities, III

Municipal 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 formula

P = 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, I

Suppose 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

Discount rateCash flow 0.05 PV

08/15/Y0 45 42.8571402/15/Y1 45 40.8163308/15/Y1 45 38.8726902/15/Y2 45 37.0216108/15/Y2 1045 818.7848

Bond Quotation 978.3526

Accrued Interest20.52197802

Actual Payment % Par998.8745947 99.88745947

coupon payment * (days since last payment /days in this period)

Bond pricing example, III

Calculator: 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 observations

Bond 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 rates

Negative 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 maturity

Everything 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.

Assignment

Use Excel to duplicate both Figure 3.3 and 3.4.

Due in a week.

Current yield (CY) vs. YTM

CY = 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 racing

Suppose that you are a bond manager and your (your company’s) 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, I

For 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, II

A 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, III

But 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, IV

Suppose 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, V

A 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.

Discount factorYear CF, % par 0.08 PV, % par Weight A*E0.5 3.5 0.961538462 3.36538 0.0346 0.01731 3.5 0.924556213 3.23595 0.0332 0.0332

1.5 3.5 0.888996359 3.11149 0.032 0.04792 3.5 0.854804191 2.99181 0.0307 0.0614

2.5 3.5 0.821927107 2.87674 0.0295 0.07393 103.5 0.790314526 81.7976 0.84 2.52

97.3789 1 2.7537Duration

Duration of a portfolio

Suppose 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 bonds—with 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, I

Back to UVM’s 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, II

The 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

8 9 10 Rate MaturityBond 1 6% 30

Value 32814297.5 29278900 26353308.95Price 773.7651 690.4 621.4142095

Shares 42408.6037 42408.6037 42408.60371Bond 2 11% 10

Value 12980668.5 12185400 11454459.86Price 1203.8549 1130.1 1062.311052

Shares 10782.5856 10782.5856 10782.58561

Obligation 100000000 100000000 100000000PV 45638694.6 41464286 37688948.29

Difference 156271.351 14.0315415 118820.5234

YTM (%)

[extra] To immunize or not to immunize

If one decides not to immunize and put the entire $41.4642 million on the 6% coupon rate, 30 year bonds, the portfolio would be consist of 60,055.6436 bonds ($690.4297 * 60,055.6436 = $41.4642 million).

If the YTM subsequently raises to 10%, the bond price becomes $621.4142. The bond portfolio would have a market value of $37.3194 million.

This creates a relative big gap of about $370 k relative to the PV ($37.6889 million) of the obligation @ YTM=10%.

[extra] More about immunization

Immunization is a dynamic strategy. That is, this is a strategy that needs to be modify over time.

The duration of the bond portfolio (assets) and the duration of the obligation (liabilities) usually change by different amounts over time even without changes in interest rate.

If the difference (mismatch) becomes large enough, rebalancing of the bond portfolio is required.