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A central concept of bond pricing is the present value. So what is the present value?
The present value is the value calculated today of a series of expected cash flows discounted at a given interest rate.
And the important thing to notice is that the present value is always less or equal than the future value.
It’s always less or equal than the future value, because money has an interest-earning potential.
And this fundamental principle of finance is called the time value of money. And simply stated, the time value of money says that a dollar received today is always worth more than a dollar received in the future.
Because a dollar received today can start earning interest immediately, and therefore, it will grow to a larger amount by the time when we will get another dollar in the future. So that was the present value.
And how do you use the present value to derive the bond pricing formula? So for the sake of the argument, consider a bond which pays coupons periodically with a frequency n.
So a little n is basically the number of intra-year coupon payments offered by the bond.
So the cash flows consist in each period of a given coupon. So we have in the bottom part of this figure the periods. And in each period the bond is paying a given cash flow. So in the first period, the bond is paying a coupon. In the second period, the bond pays a second coupon, and so on. And all the way up until maturity, which happens in year T. And at time T, or in period n times t, the investor receives both the last coupon and the principal amount invested originally.
So this is basically the stream of cash flows that you receive by holding a bond maturing in T years where the frequency of payments is n. And the relevant question at this stage is the following.
So how much is this stream of cash flows worth today? So to answer this question, we need to calculate the present value of this stream of cash flows. And the present value is, basically, the price that we are willing to pay today to purchase this bond, the price that we are willing to pay today to receive these streams of expected payments in the future. So the present value of these streams of payments is calculated by discounting all the cash flows that we are getting in every period. And how do we discount them? Well, we will discount each of these cash flows by a given interest rate, which for the moment we will call little y. So in the first period, the discounted coupon or cash flow would be equal to C divided by 1 plus y. In the second period, the second discounted coupon would be equal to C over 1 plus y squared.
And so on up until maturity, when we’ll have to discount the cash flow C plus M by 1 plus y elevated to the n times T. And, of course, we should divide the y by n, which is the frequency of payments.
So if we discount the cash flows in this way, and then we sum them over, then we are able to derive the bond pricing formula. So in the bond pricing formula, we are going to have two components.
The first component is the sum of the discounted coupons. And the second component is the principal amount discounted back to the present. What do we notice here? P is the bond price, C is the coupon payment, n is the number of coupon payments that we receive each year when we hold the bond, and big T is the number of years to maturity. Then we have little y, which is the interest rate used to discount the cash flows. It is also called yield to maturity, or market-required yield
And finally, M is the par value or face value or maturity value of the bond. So if we write down again the bond pricing formula, what do we have? We have that the bond price is equal to what? to the present value of all the cash flows that you receive if you hold the bond until maturity. So the price of the bond is equal to the first discounted coupon plus the second discounted coupon in year two, and all the way up to maturity. So we can rewrite again the formula as follows: it’s the sum of C over 1 plus y over n to the s
plus the principal amount discounted back to present. And this first component of the discounted cash flows can also be rewritten as follows. And pay attention here, because we are going to use this formula time and again during the exercises. So this is the first component, which are the sum of the discounted coupons. And then the second component is the principal amount discounted back to present. Now that we have introduced the bond pricing formula, we can distinguish three special cases.
The first case is when we have a zero-coupon bond. In this case, C is equal to 0, and therefore, the bond pricing formula reduces to this, so that the price of the zero-coupon bond is simply equal to the discounted principal back to present. If the principal is equal to 0, so if the bond is irredeemable, then the bond pricing formula is reduced to an annuity. And an annuity consists of security when you receive in each period a given coupon for a finite number of periods. And if in the annuity you impose that T is going to infinity, then this security is called a perpetuity. And perpetuity is simply equal to the coupon discounted by the yield to maturity. So just to sum up, the bond pricing formula is telling us two important things. The higher the coupon rate, the higher the coupon payments and, therefore, the higher the bond price. And secondly, the higher the yield to maturity, the lower the bond price.
So the higher the yield to maturity, the more you are discounting the cash flows that you’re receiving, and this ultimately lowers the bond price.
