How to use spot rate to calculate bond price?

Hi everyone, this is William Jiamin here. Let's talk about spot rates. Previously, we introduced the concept of spot rates. Today, I'm going to show you how to use spot rates.

Do you still remember the Excel sheet I showed you earlier? I've made some changes to it. I added a lot of dummy data and included a year column to draw a term structure of spot rates. Essentially, the term structure of spot rates is a graph that shows the spot rates over different maturities.

If you look at the x-axis, it represents the maturity time in years. You can think of it as year one, year two, year three, and so on. I used years just for simplicity, but in practice, you might come across spot rate data with different maturities. Don't worry, you can simply change it to match the maturity you're working with.

Of course, the term structure of spot rates doesn't necessarily look like the graph I showed you. That was just an example with random data. In reality, the term structure might have a curved structure, especially when spot rates vary. So, be aware that it can look different in real life.

Now, the reason I showed you the spot rate graph or the term structure of spot rates is because we need to use these different spot rates to calculate the price of a bond. You might ask, "Isn't the price of a bond just the face value?" Well, not exactly. Let me clarify this for you.

If you have a face value of 1,000 units, it doesn't mean much on its own. And if I tell you that the bond pays a 5% coupon rate, it only tells you the amount of cash flow you'll receive at the end of each year. The coupon rate doesn't provide any information about the relationship with spot rates or the bond's maturity. So, my suggestion is to not worry too much about the face value or the percentage the bond will give you. They are just soft information that helps us calculate the cash flows.

Once you have the bond's information, such as the face value and the maturity period (let's say 3 years), you can proceed with the calculation. This is where the term structure of spot rates becomes useful. For example, let's say the spot rate for the first year is 11.11%, the second year is 12.22%, and the third year is 14.51%.

Now, let's calculate the cash flows. The face value is 1,000 units, and the coupon rate is 5%. So, the annual cash flow will be 1,000 multiplied by 5%, which is 50 units.

For each year, you'll receive 50 units as coupons. Don't forget that at the end of the third year, you'll also receive the 1,000 units principal amount because the bond will mature. So, in total, you'll receive 50 units, 50 units, 50 units, and 1,000 units at the end of each year.

But we also need to consider the time value of money and the spot rates for discounting. Since spot rates change every year, we need to discount each cash flow accordingly. We divide each cash flow by one plus the respective spot rate raised to the power of the year.

For example, for the first year, we divide 50 units by 1 plus the spot rate of the first year raised to the power of one. We repeat this process for each year and add up all the discounted cash flows. This gives us the present value of the bond, which represents its fair value in the market.

In this case, because the spot rates are relatively high, the fair value of the bond is around 900 units, even though the face value is 1,000 units. This calculation allows us to monitor the bond price and its changes in the market.

So, feel free to try this out on your own. Use the term structure of spot rates to calculate the fair value of a bond and observe how it changes over time.