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# For What is the Yield to Maturity (YTM) Used?

## YTM Gives Us the Bond’s Price

Given the YTM and a bond’s cash flows, we can calculate the bond’s price. Say a 10-year bond pays an annual $50 coupon and has a 3% YTM. Then the bond’s value is (using the present value of an annuity formula): V_0 = \$50(\frac{1 - \frac{1}{1.03^{10}}}{0.03}) + \frac{\$1000}{1.03^{10}} = \$1170.60

## How Do We Get the YTM?

To get the YTM, we solve for the rate which makes the present value of the bond’s payments equal to the bond’s price. So say a bond’s price is $900 and it pays an$80 annual coupon for the next 5 years. Then the YTM is:

\$900 = \$80(\frac{1 - \frac{1}{(1 + YTM)^{5}}}{YTM}) + \frac{\$1000}{(1+YTM)^{5}} We can solve this equation for YTMYTM using methods such as Newton-Raphson. In this case the YTM=10.68%YTM=10.68%. ## Circular? So we get the bond’s price be using the YTM, but we get the YTM by observing the bond’s price. Isn’t this circular? Seemingly, yes. What the YTM actually does is quote the bond’s return to you. We quote a bond’s return, instead of its price, because it is much more informative. • We do not use the YTM to determine how much to pay for a bond (as in bond valuation). We can only use it to know what the present price is. ## Why Quote a Bond’s Return? Different 5-year U.S. Treasury Notes, as one example, may each have different prices. But all will have the same YTM. For instance, say we have the following three 5-year notes: • Note A: 30-year bond issued 25 years ago with a 15% coupon rate. • Note B: 10-year bond issued 5 years ago with a 9% coupon rate. • Note C: 5-year bond issued today with a 1% coupon rate. By arbitrage, each bond with have the same 1% YTM. So this means the bond prices are: • Note A:$1679.48
• Note B: $1388.27 • Note C:$1000.00

## Interactive App

The following app allows you to calculate the bond’s price given the YTM, or alternatively calculate the YTM given the bond’s price.

## So How do You Value a Bond?

There are a couple of approaches to determine how much we would pay for a bond.

• For a coupon bond with no default risk, we can discount each cash flow by the market-determined zero coupon rate. If the bond is not default-risk free, we can adjust the rate for risk.

• Often to value a corporate bond, you can use the YTM of a bond with similar risk.

## Credits and Collaboration

Click the following links to see the codeline-by-line contributions to this presentation, and all the collaborators who have contributed to 5-Minute Finance via GitHub.

#### Also in Fixed Income and Other Debt

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Yield to Maturity