Fixedrate 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 yieldstomaturity on zerocoupon 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 “noarbitrage value.”
Calculating the Price of a Bond using Spot Rates
Suppose that:
 The 1year spot rate is 3%;
 The 2year spot rate is 4%; and
 The 3year spot rate is 5%.
The price of a 100par value 3year bond paying 6% annual coupon payment is 102.95.
Time period 
1  2  3  
Calculation  \(\frac{$6}{(1+3\%)^1}\)  \(\frac{$6}{(1+4\%)^2}\)  \(\frac{$106}{(1+5\%)^3}\)  
Cash flow  $5.83  + $5.55  + $91.57 
= $102.95 
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 Yieldtomaturity of a Bond using Spot Rates
Continuing on the same example, this 3year bond is priced at a premium above par value, so its yieldtomaturity must be less than 6%. We can now use the financial calculator to find the yieldtomaturity 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 yieldtomaturity is found to be 4.92%, which we can confirm with the following calculation:
Time period 
1  2 
3 

Calculation  \(\frac{$6}{(1+4.92\%)^1}\)  \(\frac{$6}{(1+4.92\%)^2}\)  \(\frac{$106}{(1+4.92\%)^3}\)  
Cash flow 
$5.719  + $5.450  + $91.770 
= $102.95 
Question
An investor wants to buy a 3year 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 yieldtomaturity is closest to:
A. 3.47%
B. 2.55%
C. 4.45%
Solution
The 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 3year 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 52 LOS 52c:
Define spot rates and calculate the price of a bond using spot rates