Fixed-rate bonds are discounted by the market discount rate but the same rate is used for each cash flow. Alternatively, different market discount rates called spot rates could be used. Spot rates are yields-to-maturity on zero-coupon bonds maturing at the date of each cash flow. Sometimes, these are also called “zero rates” and bond price or value is referred to as the “no-arbitrage value.”

## Calculating the Price of a Bond using Spot Rates

Suppose that:

- The 1-year spot rate is 3%;
- The 2-year spot rate is 4%; and
- The 3-year spot rate is 5%.

The price of a 100-par value 3-year bond paying 6% annual coupon payment is 102.95.

$$

\begin{array}{l|cccccc}

\text{Time Period} & 1 & 2 & 3 \\

\hline

\text{Calculation} & \frac {\$6}{{\left(1+3\%\right)}^{ 1 } } & \frac { \$6 }{ { \left( 1+4\% \right) }^{ 2 } } & \frac { \$106 }{ { \left( 1+5\% \right) }^{ 3 } } \\

\hline

\text{Cash Flow} & \$5.83 & +\$5.55 & +\$91.57 & =\$102.95 \\

\end{array}

$$

The general formula for calculating a bond’s price given a sequence of spot rates is given below:

\({ PV }_{ bond }=\frac { PMT }{ { (1+{ Z }_{ 1 }) }^{ 1 } } +\frac { PMT }{ { (1+{ Z }_{ 1 }) }^{ 2 } } +…+\frac { PMT+Principal }{ { (1+{ Z }_{ n }) }^{ n } } \)

## Calculating the Yield-to-maturity of a Bond using Spot Rates

Continuing on the same example, this 3-year bond is priced at a premium above par value, so its yield-to-maturity must be less than 6%. We can now use the financial calculator to find the yield-to-maturity using the following inputs:

- N = 3;
- PV = -102.95; (Since this is a cash outflow)
- PMT = 6; (Since this is a cash inflow for the investor)
- FV = 100; (Since this is a cash inflow for the investor)
- CPT => I/Y = 4.92 (Which signifies 4.92%)

The yield-to-maturity is found to be 4.92%, which we can confirm with the following calculation:

$$

\begin{array}{l|cccccc}

\text{Time Period} & 1 & 2 & 3 \\

\hline

\text{Calculation} & \frac {\$6}{{\left(1+4.92\%\right)}^{ 1 } } & \frac { \$6 }{ { \left( 1+4.92\% \right) }^{ 2 } } & \frac { \$106 }{ { \left( 1+4.92\% \right) }^{ 3 } } \\

\hline

\text{Cash Flow} & \$5.719 & +\$5.450 & +\$91.770 & =\$102.95 \\

\end{array}

$$

QuestionAn investor wants to buy a 3-year 4% annual coupon paying bond. The expected spot rates are 2.5%, 3%, and 3.5% for the 1

^{st}, 2^{nd}, and 3^{rd}year, respectively. The bond’s yield-to-maturity isclosestto:A. 3.47%

B. 2.55%

C. 4.45%

SolutionThe correct answer is A.

\(\frac{$4}{(1.025)^1}+\frac{$4}{(1.03)^2}+\frac{$104}{(1.035)^3}=$101.475\)

Given the forecast spot rates, the 3-year 4% bond is priced at 101.475.

Here again, we can find the yield to maturity using the financial calculator:

- N = 3;
- PV = -101.475;
- PMT = 4;
- FV = 100;
- CPT => I/Y = 3.47%
And finally, we can confirm this is correct using the following formula:

\(\frac{$4}{(1.0347)^1}+\frac{$4}{(1.0347)^2}+\frac{$104}{(1.0347)^3}=101.475$$\)

*Reading 44 LOS 44c: *

*Define spot rates and calculate the price of a bond using spot rates*