# Zero coupon bond excel formula

We calculated the rate an investor would earn reinvesting every coupon payment at the current rate, then determining the present value of those cash flows. The summation looks like this:. For most purposes, such as quickly estimating a yield to maturity, the approximation formula should suffice. The calculator internally uses the secant method to converge upon a solution, and uses an adaptation of a method from Github user ndongo. This makes calculating the yield to maturity of a zero coupon bond straight-forward:.

In addition, they will want to be compensated for the risks of the money having less purchasing power when the loan is repaid. These risks are systematic risks, regulatory risks and inflationary risks.

The first includes the possibility that the borrower will default or be unable to pay on the originally agreed upon terms, or that collateral backing the loan will prove to be less valuable than estimated. The second includes taxation and changes in the law which would prevent the lender from collecting on a loan or having to pay more in taxes on the amount repaid than originally estimated. The third takes into account that the money repaid may not have as much buying power from the perspective of the lender as the money originally lent, that is inflation, and may include fluctuations in the value of the currencies involved.

The inflation premium will compensate for the third risk, so investors seek this premium to compensate for the erosion in the value of their capital, due to inflation. Actual interest rates without factoring in inflation are viewed by economists and investors as being the nominal stated interest rate minus the inflation premium.

The Fisher equation in financial mathematics and economics estimates the relationship between nominal and real interest rates under inflation.

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## YIELD function

In economics, this equation is used to predict nominal and real interest rate behavior. This is not a single number, as different investors have different expectations of future inflation.

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• Coupon Bond Formula.

Since the inflation rate over the course of a loan is not known initially, volatility in inflation represents a risk to both the lender and the borrower. In finance and economics, nominal rate refers to the rate before adjustment for inflation in contrast with the real rate. The real rate is the nominal rate minus inflation. In the case of a loan, it is this real interest that the lender receives as income. Real and nominal : The relationship between real and nominal interest rates is captured by the formula.

In this analysis, the nominal rate is the stated rate, and the real rate is the rate after the expected losses due to inflation. Since the future inflation rate can only be estimated, the ex ante and ex post before and after the fact real rates may be different; the premium paid to actual inflation may be higher or lower. This time may be as short as a few months, or longer than 50 years. Once this time has been reached, the bondholder should receive the par value for their particular bond. The issuer of a bond has to repay the nominal amount for that bond on the maturity date.

After this date, as long as all due payments have been made, the issuer will have no further obligations to the bondholders. These dates can technically be any length of time, but debt securities with a term of less than one year are generally not designated as bonds. Instead, they are designated as money market instruments. Money market interest rates : Interest rates of one-month maturity of German banks from to Most bonds have a term of up to 30 years.

That being said, bonds have been issued with terms of 50 years or more, and historically, issues have arisen where bonds completely lack maturity dates irredeemables. In the market for United States Treasury securities, there are three categories of bond maturities:. Because bonds with long maturities necessarily have long durations, the bond prices in these situations are more sensitive to interest rate changes. In other words, the price risk of such bonds is higher. Although this present value relationship reflects the theoretical approach to determining the value of a bond, in practice, the price is usually determined with reference to other, more liquid instruments.

In general, coupon and par value being equal, a bond with a short time to maturity will trade at a higher value than one with a longer time to maturity. This is because the par value is discounted at a higher rate further into the future. Finally, it is important to recognize that future interest rates are uncertain, and that the discount rate is not adequately represented by a single fixed number this would be the case if an option was written on the bond in question stochastic calculus may be employed. Where the market price of a bond is less than its face value par value , the bond is selling at a discount.

Conversely, if the market price of bond is greater than its face value, the bond is selling at a premium.

The yield to maturity is the discount rate which returns the market price of the bond. YTM is the internal rate of return of an investment in the bond made at the observed price. To achieve a return equal to YTM i. What happens in the meantime? Over the remaining 20 years of the bond, the annual rate earned is not Payment frequency can be annual, semi annual, quarterly, or monthly; the more frequently a bond makes coupon payments, the higher the bond price.

The payment schedule of financial instruments defines the dates at which payments are made by one party to another on, for example, a bond or a derivative. It can be either customised or parameterized. Payment frequency can be annual, semi annual, quarterly, monthly, weekly, daily, or continuous. Bond prices is the present value of all coupon payments and the face value paid at maturity.

I have also included a cell B6 that provides a place to specify the number of payments per year. This way, we can set up the formula without making assumptions regarding the payment frequency, which adds some flexibility since not all bonds pay semiannually. To calculate the value of the bond, in B8, we use the PV function:. Take notice of the "-" in front of the function. If I didn't put that there, then the function would have returned a negative value.

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Technically, that would be correct because you would have to pay a cash outflow that amount. However, we tend to think in terms of positive dollars, not negative. Also note that the required return and annual payment are converted in the function to semiannual values by dividing by the payment frequency. Similarly, the number of years to maturity is converted to the number of semiannual periods by multiplying by the payment frequency.

## Value and Yield of a Zero-Coupon Bond | Formula & Example

Notice that the bond is currently selling at a discount i. This discount must eventually disappear as the bond approaches its maturity date. A bond selling at a premium to its face value will slowly decline as maturity approaches. In the chart below, the blue line shows the price of our example bond as time passes. The red line shows how a bond that is trading at a premium will change in price over time.

Both lines assume that market interest rates stay constant. In either case, at maturity a bond will be worth exactly its face value.

## Excel ODDFYIELD Function

Keep this in mind as it will be a key fact in the next section. In the previous section we saw that it is very easy to find the value of a bond on a coupon payment date. However, calculating the value of a bond between coupon payment dates is more complex. As we'll see, the reason is that interest does not compound between payment dates. That means that you cannot get the correct answer by entering fractional periods e. We are going to go through the whole process here, but you can jump directly to the section that uses the Price function if you don't care about the details.

Let's start by using the same bond, but we will now assume that 6 months have passed. That is, today is now the end of period 1. What is the value of the bond at this point? To figure this out, note that there are now 5 periods remaining until maturity, but nothing else has changed.

### Bond Yield to Maturity Formula

Therefore, simply scroll up to B5 and change the value to 2. Notice that the value of the bond has increased a little bit since period 0. As noted previously, this is because the discount must eventually vanish as the maturity date approaches. Now, is there another way that we might arrive at that period 1 value? Of course. First reset B5 to 3. Remember that your required return is 4. Therefore, the value of the bond must increase by that amount each period. Put this formula in a blank cell to prove it:. Wait a minute! That's not the same answer. However, remember that this is the total value of your holdings at the end of period 1.

If we subtract that, you can see that we do get the same result:. This is one of the key points that you must understand to value a bond between coupon payment dates. Let me recap what we just did: We wanted to know the value of the bond at the end of period 1.