So far we have seen the bond with coupons. But there are bonds without coupons at all. This kind of bond is called zero coupon bond. The only cash payment the investor receives is the face value of the bond on the maturity date. An example of zero-coupon bond is U.S. government bonds with a maturity of up to one year. It is also called Treasury bills. Zero-coupon bond has only one lump-sum cash flow at the maturity date. So the price of zero-coupon bond is less than its face value. That is, zero-coupon bonds trade at a discount, so they are also called pure discount bonds. Suppose that a one-year, risk-free, zero-coupon bond, with a $100,000 face value has an initial price of $99,000. If you had purchased this bond and held it to maturity, you'd have the following cash flows. Present value is equal to minus 99,000, and future value is 100,000. Although the bond pays no interest directly, investors are compensated for the time they lock their money by purchasing the bond at a discount to its face value. In the previous example, what is the return investors will earn from holding the bond to maturity and receiving the promised face value payment. This return is called Yield to Maturity. You can find this return using rate function in Excel. Let's start Excel and type in =RATE. For number periods type in 1. For payment per period, type in 0 since there is no cash flow other than 100,000 at the maturity. For present value, type in -99000. For future value, type in 100,000. The result is 1.01%. That is the yield to maturity of this bond is 1.01% because the bond is risk free. Investing in this bond and holding it to maturity is like earning 1.01% interest on your initial investment. Suppose the following zero coupon bonds are trading at the prices shown below on $100,000 of face value. Let's start Excel template, using rate function, you can get the yield to maturity for each maturity bond. We already calculated rate for a bond with one year maturity. Let's re-skip one year maturity. Let's calculate rate for a bond with two year maturity. Type in =RATE. For number of periods, type in 2, or select A4 cell. For payment for period, type in 0. For present value, type in -90,000 or select B4 cell. For future value, type in, 100,000. The result is, 1.015%, that is the maturity for this bond is 1.015%. If you copy and paste this cell to the cell below, then you can complete a table easily. The yeild to maturity for a bond, with three year maturity is 1.02% and four year maturity is 1.026%. You can see the treasury bill rates in the following website. As you can find in this website, a treasury bill rate is higher for long term maturity than short term maturity. There are some abnormal cases, but we are not going to talk about it in this class. You can also find yield to maturity of a coupon bond. Let's consider a U.S. Treasury Bond with 10 year maturity with a 1% coupon rate and semiannual coupons. It's face value is $1,000 and it's current price is $985. In order to find the yield to maturity of the U.S. Treasury Bond, you need to find the cash flows from the bond first. Since the bond pays $1,000 of face value at the maturity, you will receive $1,000 in year 10. Next you will receive a coupon payment every six months of $5. That is $1,000 x (1%/2) is equal to $5. So there are 20 periods and 20 periodic cash flows of $5. You can find the yield to maturity of the US Treasury bond using rate function in Excel. Let's start Excel and type in =RATE. For number of periods type in 20 or 10 times 2. The period is 20 because there are two payments per year, and there are 10 years of maturity. For payment for period, type in 5 because coupon rate is 1%, and coupon amount is 1% times 1,000, is equal to 10. And this is paid in two payments so payment per period is five. For present value, type in -985 because its price is $985. For future value, type in 1000 because it's face value is $1000. The result 0.58%. That is the six month yield to maturity for this bond is 0.58% because the bond pays coupon semi-annually. We can convert it to an annual percentage rate. It is also APR, for short. By multiplying by the number of coupon payments per year, thus, the bond has yield to maturity equal to 1.16% APR, with a semi-annual compounding. Now, let's learn how to find a price of a bond with semi-annual coupon payment. Let's consider again, the ten year, $1,000 bond, with a 1% coupon rate, and semi annual coupons. Suppose you're told, that it's yield to maturity is 1.5%. Yield to maturity, is expressed as an APR, with semi annual compounding. You can find the price of the bond doing PV function in Excel. Let's start Excel and type in =PV. For rate, type in 1.5%/2. The yield to maturity of the bond is 1.5%, so the discount rate for six-month period is the half of the YTM. For number of periods type in 10*2. It is 10 years times two periods per year. For payment amount, type in 10/2. It is $1,000 times 1% coupon rate divided by two periods per year. For future value, type in $1,000 The result is -953.73. That is the bond price is $953.73. We can confirm whether this price is right by finding YTM using this price. Again type in =RATE. For number periods, type in 20, or 10 times 2. For payment per period, type in 10/2, or 1,000 times 1%, divided by 2. For present value, type in -953.73. For future value type in 1000. The result is 0.75%, that is the 6 month yield to maturity for this bond is 0.75%. We can convert it to an annual percentage rate by multiplying by 2. Thus the bond has a yield to maturity equal to a 1.5% APR i with semi-annual compounding.