###### Structures of Securitizations

Internal credit enhancements are common for securitizations called subordination (or credit tranching). Note... **Read More**

Investors in fixed-rate bonds achieve returns through the following:

- Receipt of anticipated coupon and principal payments.
- Reinvestment of coupon payments.
- Capital gains or losses are realized when the bond is sold prior to its maturity.

Discount bonds feature a coupon rate below the current market rate, while premium bonds have a coupon rate above the market rate. Over time, the book value of a bond is amortized to match its face value upon reaching maturity. The carrying value represents the bond’s purchase price, adjusted for any amortized discount or premium. Rising interest rates decrease bond prices (and vice versa). This affects the total return, specifically if the bond is sold before maturity.

This metric is crucial for bond investors. If an investor holds a bond until maturity, avoids any bond defaults, and consistently reinvests coupons at the prevailing interest rate, the YTM accurately reflects the investor’s actual rate of return.

The investment horizon is critical in assessing interest rate risks and returns. The interest rate risk comprises two offsetting risks:

- Coupon reinvestment risk.
- Market price risk.

Reinvestment risk pertains to the possibility that an investor may not be able to reinvest the cash flows from an investment at a rate matching the investment’s existing rate of return (yield to maturity). Two factors affect the degree of reinvestment risk:

**Maturity**: The longer the bond’s maturity, the higher the reinvestment risk. This is because of the high possibility that interest rates will be lower than they were at the time the bond was purchased;**The coupon rate of the bond**: The higher the coupon rate, the higher the payments that have to be reinvested and, consequently, the higher the reinvestment risk. In fact, a bond selling at a premium is more dependent on reinvestment income than another bond selling at par. The only fixed-income instruments that do not have reinvestment risk are zero-coupon bonds since they have no interim coupon payments.

Bond market prices will decrease in value when the prevailing interest rates rise. In other words, if an investor wishes to sell the bond prior to maturity, the sale price will be lower if rates are higher.

As noted earlier, these two risks offset each other to an extent. The dominant risk depends partially on the investment horizon. The lower the investment horizon, the lower the reinvestment risk, but the higher the market risk.

This metric delves deeper, offering insight into an investor’s internal rate of return (IRR). It considers the total return, which is composed of reinvested coupons and the sale/redemption amount divided by the purchase price of the bond.

\[r = \left( \frac{\text{FV} + \text{Sale}/\text{Redemption Amount}}{PV} \right)^{\frac{1}{T}} – 1\]

Where:

- \(FV =\) Future value of reinvested coupons.
- \(PV =\) Purchase price of the bond.
- \(T =\) Holding period.

An investor initially buys a 5-year, \(8\%\) annual coupon payment bond at the price of 85.00 per 100 of par value.

**Case 1: Holding the Bond Until Maturity**

The yield to maturity of the bond is calculated as follows.

\[85 = \frac{8}{(1 + r)^{1}} + \frac{8}{(1 + r)^{2}} + \frac{8}{(1 + r)^{3}} + \frac{8}{(1 + r)^{4}} + \frac{108}{(1 + r)^{5}};\ r = 12.18\]

The bond’s yield-to-maturity is \(12.18\%\). The easiest way to determine the value of \(r\) is to use the financial calculator:

\[n = 5;PV = – 85;PMT = 8;FV = 100;CPT = > I/Y = 12.18\]

So the investor receives the series of 5 coupon payments of 8 (per 100 of par value), a total of 40, plus the redemption of principal (100) at maturity. Besides collecting the coupon interest and the principal, there is an opportunity to reinvest the cash flows. If the coupon payments are reinvested at \(12.18\%\) immediately after they are received, the coupon’s future value on maturity date will amount to 51 per 100 par value, calculated as per the following table.

$$\begin{array}{|c|c|c|c|c|} \hline \textbf{End of Year 1} & \textbf{End of Year 2} & \textbf{End of Year 3} & \textbf{End of Year 4} & \textbf{End of Year 5} \\ \hline \$ 8 \times (1.1218)^{4} & \$ 8 \times (1.1218)^{3} & \$ 8 \times (1.1218)^{2} & \$ 8 \times (1.1218)^{1} & \$ 8 \times (1.1218)^{0} \\ \hline \end{array}$$

The \(1^{\text{st~}}\) coupon payment of \(\$ 8\) is reinvested at \(12.18\%\) for 4 years until the end of the \(5^{\text{th~}}\) year, the \(2^{\text{nd~}}\) is invested for 3 years, and so forth. The amount in excess of the coupons, \(11\ ( = \ 51\ –\ (5 \times \ 8)),\) is called “interest-on-interest” gain from compounding.

The investor’s total return is 151, the sum of reinvested coupons (51), and the redemption of principal at maturity (100). Therefore, the realized rate of return is \(12.18\%\).

\[85 = \frac{151}{(1 + r)^{5}};\ r = 12.18\%\]

As case 1 demonstrates, the yield-to-maturity at the time of purchase equals the investor’s rate of return under the following three assumptions:

- The investor holds the bond to maturity.
- There is no default by the issuer.
- The coupon interest payments are reinvested at that same rate of interest.

**Case 2: Selling the Bond Before Maturity**

If another investor buys the same bond but chooses to sell it after four years and reinvests all coupon payments at 12.18%, the future value of these reinvested coupons will be 38.3356% of the bond’s face value at the end of the fourth year. This is calculated as follows:

$$\begin{array}{|c|c|c|c|} \hline \textbf{End of Year 1} & \textbf{End of Year 2} & \textbf{End of Year 3} & \textbf{End of Year 4} \\ \hline \$ 8 \times (1.1218)^{4} & \$ 8 \times (1.1218)^{3} & \$ 8 \times (1.1218)^{2} & \$ 8 \times (1.1218)^{1} \\ \hline \end{array}$$

Total \(= \$ 38.3356\)

The interest-on-interest gain from compounding is \(6.3356( = 38.3356 – 32)\).

At the time the bond is sold, it has one year remaining until maturity. If the yield-to-maturity remains \(12.18\%\), the sale price of the bond (calculated as the PV of anticipated cash flows) is:

\[\text{Price}_{t = 4} = \frac{108}{1.1218} = 96.2738\]

Therefore, the total return is \(134.6094( = 38.3356 + 96.2738)\), and the realized rate of return is \(12.18\%\).

\[85 = \frac{134.6094}{(1 + r)^{4}};\ r = 12.18\%\]

Case 2 demonstrates that the realized horizon yield matches the original yield-to-maturity provided two conditions are met:

- Coupon payments are reinvested at the same interest rate as the original yield-to-maturity.
- The bond is sold at a price on the constant-yield price trajectory, i.e., the investor does not have any capital gains or losses when the bond is sold. The price trajectory is the time series of a bond’s prices from some date (usually the date on which the bond is purchased) until its maturity.

QuestionFor a fixed-rate bond, what will most likely happen to its market price if interest rates rise?

- The market price will rise.
- The market price will remain unchanged.
- The market price will fall.

SolutionThe correct answer is

C.Bond prices and interest rates have an inverse relationship. Thus, when interest rates increase, bond prices tend to fall.

A is incorrect: As mentioned, bond prices and interest rates have an inverse relationship. So, the market price won’t rise with rising interest rates.

B is incorrect: Bond prices are sensitive to changes in interest rates, so they will not remain unchanged.