Fixed-income securities

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.

YTM, like other discount rates, has 2 components: (1) risk-free component, and (2) risk premium.

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 (that is, the risk-free component).

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.

Relative performance vs. a benchmark

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?