## What is an 'Average Price'

The average price of a bond is calculated by adding its face value to the price paid for it and dividing the sum by two. The average price is sometimes used in determining a bond's yield to maturity where the average price replaces the purchase price in the yield to maturity calculation.

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## BREAKING DOWN 'Average Price'

In basic mathematics, an average price is a representative measure of a range of prices that is calculated by taking the sum of the values and dividing it by the number of prices being examined. The average price reduces the range into a single value, which can then be compared to any point to determine if the value is higher or lower than what would be expected. In situations where there is a range of prices it can be useful to calculate the average price to simplify a range of numbers into a single value. For example, if over a four-month period you earned \$104, \$105, \$110, and \$115 from your investments, the average return on your portfolio will be (\$104 + \$105 + \$110 + \$115) / 4 = \$108.50.

In the finance sector, the average price is mostly attributed to bonds. Bondholders that want to know the total rate of return they will get from a bond that is held until maturity can calculate a metric known as the yield to maturity (YTM). An estimate of the yield to maturity can be calculated using the bondâ€™s average rate to maturity (ARTM). The ARTM determines the yield by measuring the proportion of the average return per year to the average price of the bond. For a coupon bond, the yield to maturity can be calculated as:

YTM = C + [(F â€“ P)/n] Ã· (F + P)/2

Where C = coupon rate

F = face value

P = purchase price

n = number of years

For example, consider an investor that purchased a corporate bond at a premium to par for \$1,100 and annual coupon rate of 5% with 6 years to mature. Annual coupon payments will, therefore, be 5% x \$1,000 face value of corporate bond = \$50. YTM is:

Â = \$50 + [(\$1,000 - \$1,100)/6] Ã· (\$1,000 + \$1,100)/2

= \$33.33 / \$1,050 = 3.17%

The logic behind the formula is that the premium amount over par, that is, F â€“ P = \$1,000 - \$1,100 = -\$100 is divided over the number of years to maturity. Hence, -\$100/6 = -\$16.67 is the amount that reduces the coupon payment per year. Hence, even though the investor receives \$50 coupon per year, his real or average return is \$50 - \$16.67 = \$33.33 per year since he bought the bond for a price above par. Dividing the average return by the median or average price is the bondholderâ€™s yield to maturity.

Note that if the bond was purchased at a discount to par, the investorâ€™s average return per year will be higher than the coupon payment. Furthermore, if an investor bought the bond at par, his average return per year will equal the coupon rate. In this case, the YTM will also equal the coupon rate after dividing the average return per year by the average price of the bond.

Although the average price of a bond is not the most accurate method to find its YTM, it does give investors a rough and simple gauge to find out what a bond is worth.

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