### Calculate the Price of a Bond using Spot Rates

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:

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

## 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:

$$\begin{array}{l|cccccc} \text{Time Period} & 1 & 2 & 3 \\ \hline \text{Calculation} & \frac {\6}{{\left(1+4.92\%\right)}^{ 1 } } & \frac { \6 }{ { \left( 1+4.92\% \right) }^{ 2 } } & \frac { \106 }{ { \left( 1+4.92\% \right) }^{ 3 } } \\ \hline \text{Cash Flow} & \5.719 & +\5.450 & +\91.770 & =\102.95 \\ \end{array}$$

## Question

An 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 1st, 2nd, and 3rd year, respectively. The bond’s yield-to-maturity is closest to:

A. 3.47%

B. 2.55%

C. 4.45%

Solution

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

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:

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

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

Isha Shahid
2020-11-21
Literally the best youtube teacher out there. I prefer taking his lectures than my own course lecturer cause he explains with such clarity and simplicity.
Artur Stypułkowski
2020-11-06
Excellent quality, free materials. Great work!
2020-11-03
One of the best FRM material provider. Very helpful chapters explanations on youtube by professor James Forjan.
Rodolfo Blasser
2020-10-15
The content is masters degree-level, very well explained and for sure a valuable resource for every finance professional that aims to have a deep understanding of quantitative methods.
Mirah R
2020-10-15
Priyanka
2020-09-29
Analyst Prep has actually been my soul guide towards this journey of FRM.I really appreciate the videos ad they are ALIGNED , good speed, and the Professor just keeps everything Super CASUAL. If I Clear my exams Ultimately credit goes to you guys. Keep sharing. God bless.
Sar Dino
2020-09-29
Had a test on actuarial science coming up and was dead on all the concepts (had to start from ground zero). came across the channel as it had small bits of FM chapters consolidated by the professor Stephen paris. this made it easy for me to look at the chapters i was having trouble with (basically everything lol). I love the way he explains the questions, and the visualization! its so helpful for me to see the diagrams and how the formulas move around. he really did a great job explaining, and i understand so much better. 7 weeks worth of lessons condensed into 3 days of binge watching their videos.... Amazing and i am truly baffled as to why the videos have not gained traction as they should have!

Share:

#### Related Posts

##### A Bond’s Price given a Market Discount Rate

Bond pricing is an application of discounted cash flow analysis. The general approach...

##### Risks in Relying on Ratings from Credit Rating Agencies

Investors overwhelmingly believe that credit rating agencies do a good job assessing credit...