2.1a.1 Bonds

Bonds

Bonds are a type of fixed-income security that represents a loan made by a lender (generally a corporation or government) to a borrower. Bonds are used by entities to raise capital, and in return, the lender may receive regular coupon payments (interest). On the bond’s maturity date, the lender also receives the bond’s face value (principal payment), and stops earning interest. Special types of bonds include:

• Zero-coupon bonds, which pay a face value with no coupons.
• Treasury securities, which
• Mortgage bonds, which pay out coupon payments for a certain period without a face value.
• Callable bonds, which allow the issuer to buy back the bond before maturity. This usually occurs when the interest rates drop below the coupon rate.

The price of a bond is directed by the movement of the market interest rate. When a bond is issued, the yield-to-maturity and coupon rate of the bond are matched to the market rate (i.e., $y_0=\frac{C}{F}=r_0$). After issuance, market rates move all the time and coupon rates don’t change. Therefore, the only way for the bond to stay competitive is for its price to adjust, which changes the yield. The price of a bond anytime after issuance, known as the dirty price, is given as
$P=\underset{\text{Accrued Interest}}{\underbrace{\frac{C}{y}\left(1-\frac{1}{{\left(1+\frac{y}{n}\right)}^{nT}}\right)}}+\underset{\text{PV of Face Value}}{\underbrace{\frac{F}{{\left(1+\frac{y}{n}\right)}^{nT}}}},$
for some face value $F$ and coupon rate $C$ paid $n$ times per year over $T$ years.

If a bond is bought some time after issuance, the clean price is used as a measure of determining the present value of all future cash flows, subtracting the interest that has already been accrued.

When interest rates rise, existing bonds must lower their prices to boost their yields and stay competitive, ensuring that the opportunity cost of holding the bond aligns with current market returns. This means that:

• If $P=F⟺y=\frac{C}{F}$, then the bond is offered at par.

• If $P<F⟺y>\frac{C}{F}$, then the bond is offered at a discount.

• If $P>F⟺y<\frac{C}{F}$, then the bond is offered at a premium.

YTM Assumes Reinvestment $P$ for the yield-to-maturity of a coupon bond $y$ assumes that all coupons are reinvested, and, of course, that the bond is held to maturity.

Inverting

Measurements of Yield

(Current Yield) $Y_C=\frac{C}{P}$.

(Nominal Yield/Spot Rate) $Y_N=\frac{C}{F}$.

(Yield-to-Maturity/Yield) $y=n\left[{\left(\frac{F}{P}\right)}^{1/nT}-1\right]$.

(Yield-to-Call)

Measured as the bond’s return on investment if called at the earliest opportunity.

(Holding Period Return) $R=n\left[{\left(\frac{P_T+C_T}{P}\right)}^{1/nT}-1\right]$