Calculating Bond and Stock Prices - Online Tutor, Practice Problems & Exam Prep

The present value of an investment, such as a bond or stock, is determined by future cash flows. For bonds, cash flows include periodic interest payments and the principal at maturity, calculated using the formula: PV1=FV+CF/(1+r)n. Stocks generate dividends, which are expected to grow, calculated as P=D1/(r-g). Understanding these concepts is crucial for valuing investments effectively.

Calculating Stock and Bond Price

Now, let's see how we can use the time value of money to calculate the price of a bond today and the price of stocks today. So when we go to value an investment to say what it is worth today, we want to think of all the cash it's going to bring in the future. We want to think of it as the present value of future cash flows. We could think that in this first case, a bond is going to pay us interest payments over the life of the bond and then finally pay off the principal. A timeline for a bond might look something like this. Let's say it's a 3-year bond, just for simplicity. Most bonds are longer than 3 years, but this will give us the idea. Okay. What's going to happen is we're going to pay some amount, and this will be the price today. Let me do it in a different color. We're going to pay the price of the bond today which is equal to what is going to be the cash flows in the future. After 1 year, it's going to make an interest payment. After another year, it'll make another interest payment. And in the final year, it'll make another interest payment, right? But that's not all. In the final year, they're also going to pay back the principal, the money that we lent to them, right? So we're also going to get the principal back.

If you think about our time value of money equation, when we're looking for the present value, remember that present value=future value(1+r)n. Now, we've got a few cash flows. We're looking for the present value of all these cash flows. We're not only going to get one payment here. We're going to get a payment here, a payment here, a payment here, right? There are multiple cash flows in this case. To get the value of this bond today, we need to know the value of all of these payments together. That's exactly what's going on here in the bond price. When we say coupon, that's the coupon interest, the interest that's being paid by the bond.

This first interest payment, we need to bring it back 1 year. So we're going to take 1 plus the interest rate, right? Whatever the interest rate is on this bond. Well, we're going to bring it back 1 year, right? And then the second coupon payment will bring it back 2 years because we have to wait 2 years to get it. What's the present value of a 2-year payment? Notice, all we're doing is using our present value equation from our time value of money and just having multiple payments all at once. And then finally, we would do that every year, right? The third year, fourth year, depending on how long, and then in the final year, we would get one more coupon payment and we bring it back the number of years that it's been outstanding as well as first place, and once we find the value of all of those today, that is the price of the bond, the present value of those future payments. Cool?

So the same thing happens here with a stock, except stocks don't pay interest and they don't have a principal amount, right? When you buy a piece of stock, they're not going to say, hey, alright. In 10 years, I'm going to pay you back the value of the stock. No. You're buying shares in the company, so what you're going to get is dividends. You're going to earn some dividends in the company. Okay? The idea here is that the company that instead of cash flows being from interest and principal payments, the cash flows come from the dividends that you get from the stock. So you're going to have dividends every year. Right? You buy the stock now, so this would be the price of the stock today would come from the dividends that they're going to pay you in the future. Right?

So you're going to get a dividend every year. Let's say, in a simple situation, you would get a dividend every year and that's what it would be worth today. However, when we think about a company, we expect the company to grow over time. So we would expect the dividends to grow with the company. So when we do our calculation here, we actually do a little bit of a different calculation. It looks similar to what we had above with our time value of money, but we're accounting for the growth here. Without going into too much detail of how this is derived, it's good to just understand that what we're doing is basically finding the value of these dividends as they're growing. So, this is dividend 1 and this is dividend 2, dividend 3, and we can expect that all of those are a little bit bigger. There's going to be some growth in the dividends over time. So the way we calculate that is we're going to take the value of the dividend and divide it. So this is the first dividend. Dividend 1 and then we're going to divide it by the interest rate minus the growth rate, right? So these numbers would have to be given to you, right? So it actually makes the calculation actually pretty easy, they'll tell you what the dividend is, the interest, and the growth rate; you just plug it into this formula and you get the stock price, okay? It takes into account those time value of money calculations, but it's at a little bit of a higher level. So for this class, it's good enough to just know the formula. Cool? Alright. With that being said, let's go ahead and do a practice problem related to this.

Stock Price = Divident (i - g) Where i = discount rate and g = dividend growth rate.

A stock currently pays a dividend of $1 per share. Dividends are expected to increase at a rate of 5% per year, while the discount rate is 8%. What is the current price of the stock?

• A. $12.50
• B. $20.00
• C. $33.33
• D. Cannot be determined

Here’s what students ask on this topic:

How do you calculate the present value of a bond?

To calculate the present value of a bond, you need to discount all future cash flows (interest payments and the principal repayment) to their present value. The formula used is:

${\mathrm{PV}}_1=\frac{\mathrm{FV}+\mathrm{CF}}{{(1+r)}^{-n}}$

Here, $\mathrm{PV}$ is the present value, $\mathrm{FV}$ is the future value (principal), $\mathrm{CF}$ is the cash flow (interest payments), $r$ is the interest rate, and $n$ is the number of periods. Sum the present values of all cash flows to get the bond's price today.

What is the formula for calculating the price of a stock with growing dividends?

The formula for calculating the price of a stock with growing dividends is:

$P=\frac{D_1}{r-g}$

Here, $P$ is the price of the stock today, $D_1$ is the dividend expected next year, $r$ is the required rate of return, and $g$ is the growth rate of the dividends. This formula accounts for the time value of money and the expected growth in dividends.

What is the time value of money and how does it relate to bond and stock pricing?

The time value of money (TVM) is the concept that money available today is worth more than the same amount in the future due to its potential earning capacity. This principle is crucial in bond and stock pricing. For bonds, TVM is used to discount future interest payments and the principal to their present value. For stocks, TVM is applied to discount future dividends, considering their expected growth. Understanding TVM helps investors determine the present value of future cash flows, enabling them to make informed investment decisions.

How do interest rates affect bond prices?

Interest rates have an inverse relationship with bond prices. When interest rates rise, the present value of a bond's future cash flows decreases, leading to a drop in the bond's price. Conversely, when interest rates fall, the present value of future cash flows increases, causing the bond's price to rise. This relationship is due to the fixed nature of bond payments; as market interest rates change, the attractiveness of the bond's fixed payments changes relative to new bonds issued at current rates.

What are the key differences between valuing bonds and stocks?

The key differences between valuing bonds and stocks lie in their cash flows and the methods used. Bonds have fixed interest payments and a principal repayment at maturity, which are discounted to present value using the time value of money. Stocks, on the other hand, generate dividends that are expected to grow over time. The stock price is calculated using the formula:

$P=\frac{D_1}{r-g}$

where dividends are expected to grow at a constant rate. Bonds are generally less risky and have a defined maturity, while stocks offer potential for growth but come with higher risk.