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

Bond pricing is an application of discounted cash flow analysis. The general approach to bond valuation is to utilize a series of spot rates to reflect the timing of future cash flows.

## Bond Pricing with a Market Discount Rate

For option-free or fixed rate bonds, future cash flows are a series of coupon interest payments and a repayment of principal at maturity. The price of the bond at issuance is the present value of future cash flows discounted at the market discount rate. The market discount rate, also called required yield or required rate of return, is the rate of return required by investors based on the risk of the investment. The present value of a single cash flow can be calculated as follows:

The present value of a single cash flow can be calculated as follows:

$${ \text{Present value} }_{ t }=\frac {\text{Expected cash flow at time t}}{ { (1+i) }^{ t } }$$

Where:

t = number of years until the cash flow is received;

i = discount rate

Thus, the present value of a bond is simply:

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

Where:

PMT = coupon payment per period

FV = face value of the bond at maturity

i = market discount rate

For example, suppose that the coupon rate is 5% and the payment is made once a year for 5 years. If the market discount rate is 6%, the price of the bond is 95.788 for 100 of par value.
$$\begin{array}{l|cccccc} \text{Time Period} & 1 & 2 & 3 & 4 & 5 \\ \hline \text{Calculation} & \frac { \5 }{ { \left( 1+6\% \right) }^{ 1 } } & \frac { \5 }{ { \left( 1+6\% \right) }^{ 2 } } & \frac { \5 }{ { \left( 1+6\% \right) }^{ 3 } } & \frac { \5 }{ { \left( 1+6\% \right) }^{ 4 } } & \frac { \105 }{ { \left( 1+6\% \right) }^{ 5 } } \\ \hline \text{Cash Flow} & \4.717 & +\4.450 & +\4.198 & +\3.960 & +\78.462 & =\95.788 \\ \end{array}$$
In this example, the bond is trading at a discount as the price is below par value.

Note that the easiest way to do this calculation is with the help of the financial calculator (BA II plus) with the following input:

• N = 5;
• I/Y = 6; (You have to enter “6” and not 0.06)
• PMT = 5;
• FV = 100;
• CPT => PV = -95.7876 (Note the negative sign since this is a cash outflow.)

Now what if the coupon rate changed to 6%, paid annually, and the market discount rate still remains at 6%. Then, the price of the bond would be 100, and the bond would be trading at a par.

$${ PV }_{ bond }=\frac { 6 }{ { 1.06 }^{ 1 } } +\frac { 6 }{ { 1.06 }^{ 2 } } +\frac { 6 }{ { 1.06 }^{ 3 } } +\frac { 6 }{ { 1.06 }^{ 4 } } +\frac { 106 }{ { 1.06 }^{ 5 } } =100$$

If the bond happens to have a coupon rate of 7% and the market discount would still be 6%, its price would be 104.21, and it would be trading at a premium.

$${ PV }_{ bond }=\frac { 7 }{ { 1.06 }^{ 1 } } +\frac { 7 }{ { 1.06 }^{ 2 } } +\frac { 7 }{ { 1.06 }^{ 3 } } +\frac { 7 }{ { 1.06 }^{ 4 } } +\frac { 107 }{ { 1.06 }^{ 5 } } =104.21$$

As these examples demonstrate, the price of a fixed-rate bond, relative to par value, depends on the relationship between the coupon rate (Cr) and market discount rate (Mdr). In summary:

• If Cr < Mdr, then the bond is priced at a discount below par value.
• If Cr = Mdr, then the bond is at par value.
• If Cr > Mdr, then the bond is at a premium.

European bonds make annual payments whereas Asian and North American bonds generally make semiannual payments. For semiannual payments, semiannual coupon payments are discounted by one half of the market discount rate (Mdr).

## Yield-to-Maturity

If the market price of a bond is known, the discounted cash flow equation can be used to calculate its yield-to-maturity, or in other words, the internal rate of return of the cash flows. The yield-to-maturity is also the implied market discount rate.

For example, if a 3-year 4% annual coupon payment bond is priced at 104, the yield-to-maturity is the solution for the rate that results in the sum of the discounted cash flows to 104, which is 2.60%.

$$\begin{array}{l|cccc} \text{Time Period} & 1 & 2 & 3 \\ \hline \text{Calculation} & \frac {\4}{{\left(1+r\%\right)}^{ 1 } } & \frac { \4 }{ { \left( 1+r\% \right) }^{ 2 } } & \frac { \104 }{ { \left( 1+r\% \right) }^{ 3 } } \\ \hline \text{Cash Flow} & \3.899 & +\3.800 & +\96.301 & =\104.000 \\ \end{array}$$We can note that the bond trades at a premium because its coupon rate is greater than the yield required by investors.

Again, the easiest way to do this calculation is with the help of the financial calculator (BA II plus) with the following input:

• N = 3;
• PV = -104; (Negative sign since it is a cash outflow.)
• PMT = 4;
• FV = 100;
• CPT => I/Y = 2.6 (Here, 2.6 signifies a 2.6% yield-to-maturity.)

## Question

A 2-year, 5% semiannual coupon payment bond with a price of \$102 trading at:

B. Discount

C. Par value

Solution

The yield-to-maturity is the rate that makes the sum of the discounted cash flows to 102, which is 1.98%, compounded semi-annually. The bond trades at a premium because its coupon rate of 5% / 2 = 2.5% is greater than the yield required by investors. (Cr > Mdr)

Calculate a bond’s price given a market discount rate

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