Bond Pricing and Yield to Maturity

A bond is simply a loan that has been turned into a tradable security. The Government of Pakistan raises money through Pakistan Investment Bonds (PIBs) and Sukuk, companies issue Term Finance Certificates, and governments worldwide sell treasury bonds. To trade or analyse any of them you need two skills: pricing a bond from its cash flows, and working backwards from a price to its yield to maturity.

The Anatomy of a Bond

A standard fixed-coupon bond is described by a few numbers:

• Face value (par value), \( F \): the amount repaid at maturity, often PKR 100 or PKR 1,000 for a single unit.
• Coupon rate, \( c \): the annual interest rate paid on the face value. A 10% coupon on a PKR 1,000 bond pays PKR 100 a year.
• Maturity, \( n \): the number of years until the face value is repaid.
• Coupon frequency: annual or semi-annual. Many government bonds, including PIBs, pay semi-annually.
• Yield to maturity, \( y \): the market's required return on the bond, which changes daily.

The coupon rate is fixed when the bond is issued. The yield moves with market conditions. The gap between them is what makes bond prices rise and fall.

The Bond Pricing Formula

A bond's price is the present value of everything it will pay: a stream of coupons (an ordinary annuity) plus the face value at maturity (a single sum). With annual coupon \( C = cF \):

$$ P = C \cdot \frac{1-(1+y)^{-n}}{y} + \frac{F}{(1+y)^n} $$

For semi-annual coupons, halve the coupon, halve the yield and double the number of periods:

$$ P = \frac{C}{2} \cdot \frac{1-(1+y/2)^{-2n}}{y/2} + \frac{F}{(1+y/2)^{2n}} $$

Worked Example 1: Pricing a Bond

A bond has face value PKR 1,000, a 10% annual coupon and 5 years to maturity. The market yield on similar bonds is 12%. Find its price.

Step 1: Annual coupon. \( C = 0.10 \times 1{,}000 = \text{PKR } 100 \).

Step 2: Annuity factor at 12% for 5 years. Since \( (1.12)^{-5} = 0.567427 \),

$$ \frac{1 - 0.567427}{0.12} = 3.604776 $$

Step 3: Present value of coupons.

$$ 100 \times 3.604776 = 360.478 $$

Step 4: Present value of face value.

$$ 1{,}000 \times 0.567427 = 567.427 $$

Step 5: Add.

$$ P = 360.478 + 567.427 = \text{PKR } 927.90 $$

The bond trades at a discount (below par) because its coupon of 10% is less attractive than the 12% investors can get elsewhere. The lower price compensates them.

The Same Bond at Other Yields

Repeating the calculation:

Three rules follow directly from the formula:

1. When the yield equals the coupon rate, the bond sells at par.
2. When the yield is above the coupon rate, the bond sells at a discount; below it, at a premium.
3. Price and yield move in opposite directions. A rise in yield from 12% to 13% cuts the price by about 3.6%.

Semi-Annual Version

If the same bond paid PKR 50 every six months, then with 10 half-year periods at 6%:

$$ P = 50 \times 7.360087 + 1{,}000 \times 0.558395 = 368.004 + 558.395 = \text{PKR } 926.40 $$

The price is slightly lower than with annual coupons, because a 12% yield compounded semi-annually is effectively a little more than 12% per year.

Yield to Maturity

Yield to maturity (YTM) is the single discount rate that makes the present value of the bond's cash flows equal to its current market price. It is the bond's internal rate of return, assuming you hold it to maturity and every payment is made on time.

As with IRR, there is no general algebraic solution. By hand we use an approximation formula and then refine with trial and interpolation.

Quick Approximation

$$ YTM \approx \frac{C + \dfrac{F - P}{n}}{\dfrac{F + P}{2}} $$

The numerator is the annual coupon plus the yearly share of the gain (or loss) to maturity; the denominator is the average amount invested.

Worked Example 2: Finding YTM

A 5-year bond with face value PKR 1,000 and a 10% annual coupon is trading at PKR 950. Find its YTM.

Step 1: Approximate.

$$ YTM \approx \frac{100 + \dfrac{1{,}000 - 950}{5}}{\dfrac{1{,}000 + 950}{2}} = \frac{100 + 10}{975} = 0.1128 = 11.28\% $$

Step 2: Bracket the answer. Price the bond at 11% and 12%.

At 11%: annuity factor \( = 3.695897 \), discount factor \( = 0.593451 \).

$$ P_{11\%} = 100 \times 3.695897 + 1{,}000 \times 0.593451 = 369.59 + 593.45 = 963.04 $$

At 12% we already have \( P_{12\%} = 927.90 \). The market price of 950 lies between these, so the YTM is between 11% and 12%.

Step 3: Interpolate.

$$ YTM \approx 11\% + \frac{963.04 - 950}{963.04 - 927.90} \times 1\% = 11\% + \frac{13.04}{35.14} \times 1\% = 11.37\% $$

A precise numerical solution gives 11.37%, matching our interpolation. The approximation formula (11.28%) was close but slightly low, which is typical for discount bonds.

Sanity check. The bond is priced below par, so its yield must exceed its 10% coupon. It does.

Current Yield Versus YTM

The current yield is just the annual coupon divided by the price:

$$ \text{Current yield} = \frac{100}{950} = 10.53\% $$

It ignores the PKR 50 capital gain the holder receives at maturity, so for a discount bond it understates the true return. YTM includes both the coupons and the gain, and accounts for timing.

Interest Rate Risk

Because prices fall when yields rise, bondholders face interest rate risk. Two features increase it:

• Longer maturity: distant cash flows are discounted over more periods, so they react more strongly to rate changes.
• Lower coupon: more of the value sits in the final repayment, far in the future.

The standard measure of this sensitivity is duration, a weighted average of the times at which cash flows arrive. A bond with a duration of about 4 years will lose roughly 4% of its value if yields rise by one percentage point (more precisely, modified duration is used for this estimate).

Sukuk and Islamic Bonds

Sukuk are certificates representing ownership in an underlying asset or venture, and their returns come from rent or profit rather than interest. In practice, investors still value Sukuk cash flows by discounting them, and the market reports a yield for them, so the same pricing mathematics applies.

Did You Know?

• A zero-coupon bond pays no coupons at all; its price is simply \( F/(1+y)^n \), so a 10-year PKR 1,000 zero at 12% costs about PKR 321.97.
• Some bonds have been issued with maturities of 100 years; very long bonds can lose a large share of their value when rates rise sharply.
• In 2020 and 2021, some government bonds in Europe and Japan traded at negative yields, meaning investors paid more than they would get back.

Key Takeaways

• Bond price = present value of coupons (annuity) + present value of face value.
• Price and yield move in opposite directions; yield above coupon means a discount.
• YTM is the bond's internal rate of return; estimate it, bracket it, then interpolate.
• Current yield ignores capital gains and timing, so YTM is the better measure.
• Longer maturity and lower coupons mean greater interest rate risk.

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