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# Bond Yields and Return Calculations

After completing this reading, you should be able to:

• Distinguish between gross, and net realized returns and calculate the realized return for a bond over a holding period, including reinvestments.
• Define and interpret the spread of a bond and explain how to derive a spread from a bond price and a term structure of rates.
• Define, interpret, and apply a bond’s yield-to-maturity (YTM) to bond pricing.
• Compute a bond’s YTM, given a bond structure and price.
• Calculate the price of an annuity and perpetuity.
• Explain the relationship between spot rates and YTM.
• Define the coupon effect and explain the relationship between the coupon rate, YTM, and bond prices.
• Explain the decomposition of P&L for a bond into separate factors, including carry roll-down, rate change, and spread change effects.
• Explain the following four common assumptions in carry roll-down scenarios: realized forwards, unchanged term structure, consistent yields, and realized expectations of short-term rates; and calculate carry roll down under these assumptions.

## Gross vs. Net Realized Returns

The gross realized return on investment has two components: Any increase in the price of the asset plus income received while holding the investment. When dealing with bonds,

$${ \text{Gross realized return} }_{ \text{t}-1,\text{t} }=\cfrac { \text{Ending value}+ \text{Coupon – Beginning value} }{ \text{Beginning value} }$$

What is the gross realized return for a bond that is currently selling for $1,060 if it was purchased exactly six-months ago for$1,000 and paid a 20 coupon today? Solution \begin{align*} { \text{Gross realized return} } & =\cfrac { \text{Ending value}+ \text{Coupon-Beginning value} }{ \text{Beginning value} } \\ & = \cfrac { 1,060+20-1000 }{ 1000 } =8\% \end{align*} When calculating the gross realized return for multiple periods, it’s essential to consider whether coupons received are reinvested. If the coupons are reinvested, they will earn some interest at a given rate. #### Example: Gross Realized Return over One Year With Reinvested Coupons A bond purchased exactly six months ago for1,000 paid a $20 coupon today. Suppose the coupon is reinvested at an annual rate of 4.4% for the next six months and that the bond is worth$1,080 after one year. What is the realized return on the bond over the one-year period?

Solution

\begin{align*} { \text{Gross realized return} } & =\cfrac { \text{Ending value}+ \text{Coupon}+\text{Coupon Investment}-\text{Beginning value} }{ \text{Beginning value} } \\ & = \cfrac { 1,080+20+20\times1.022-1,000 }{ 1000 } =12.04\% \end{align*}

In addition to reinvestment income, we can also consider borrowing costs. If the investor buys the bond using borrowed funds, they will be expected to pay some interest at the end of the investment period. In these circumstances, the net realized return is calculated as follows:

$$\text{Net realised return}=\cfrac{\text{Ending value} + \text{Coupon}-\text{Beginning value} -\text{Financing costs}}{ \text{Beginning value} }$$

#### Example: Example: Gross Realized Return over One Year With Reivested Coupons and Financing Costs

An investor purchased a bond exactly six months ago at $980 (per$1,000 nominal value). The purchase was entirely financed at an annual rate of 2%. Today, the bond is worth $995. Given that the bond paid a coupon of$20 today, determine the net realized return

Solution

## Question 2

On Jan 1 2017, Commercial Bank of India issued a six-year bond paying an annual coupon of 6% at a price reflecting a yield to maturity of 4%. As of Dec 31, 2017, interest rates remain unchanged. Holding all other factors constant, and assuming a flat term structure of interest rates, how was the bond’s price affected? The price:

1. Remained constant
2. Decreased
3. Increased
4. Increased, but only in the second half of the year

From the data given, it’s clear that the bond’s coupon is higher than the yield. As such, the bond must have traded at a premium – implying the price must have been higher than the face value. Provided the yield doesn’t change; a bond’s price will always converge to its face value. Since the price starts higher, it must decrease. This phenomenon is called ‘pulling to par.”

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