In this article, we’re going to explore Zero Coupon Bonds, including what they are and how we can value them.
We’re going to assume that you have a basic understanding of what bonds are. If not, feel free to read our sister Article on What is a Bond.
What are Zero Coupon Bonds?
Zero Coupon Bonds, aka “Deep Discount Bonds”, or “ZCBs” are bonds (a type of ) that don’t pay any coupons (aka interest).
In other words, there is no . They pay a zero coupon.
Hence the name, zero coupon bond.
The only thing they do pay is the Par (aka “face value”) when the .
Put differently, a zero coupon bond is a bond that doesn’t pay any interest. Instead, it only pays a lump-sum payment at the end of the bond’s life. That is, at its maturity or expiration date; i.e., the date when the or expires.
Trading Status for Zero Coupon Bonds
The nature of the zero coupon bonds means that they will always trade at a discount.
Because, remember, the par is the only reward for investors, since there is no interest payment / coupon whatsoever.
So the price will always be less than the Par Value for all deep discount bonds.
If the price was greater than the par value, or even if it was equal to it, then there’d be no buyers.
Because if it’s equal to the par value, then you wouldn’t be getting compensated for the effects of the time value of money.
And if it’s greater than the par value, that would be even worse.
Interest rate of ZCBs
The interest rate (aka yield) of zero coupon bonds tends to be higher than the interest rate of say, straight / vanilla bonds. And that’s ultimately because for the most part, zero coupon bonds tend to be riskier securities.
The higher interest rate / higher yield is meant to compensate for, or pay for, the higher risk.
Investment Goals of holding ZCBs
The high risk of zero coupon bonds may well have you thinking, why even bother buying this debt instrument?
Buying zero coupon bonds can make sense if you’ve got a diversified portfolio already, and you’re looking to increase your expected return whilst accepting a possible increase in risk.
For instance, complimenting existing bonds with a relatively riskier one may well allow for greater returns with only marginal increases in risk,
Because again, the interest rate of ZCBs tends to be fairly high, given the relatively higher risk.
Alternatively, one might buy zero coupon bonds to increase the amount of diversification.
For instance, suppose an investor only holds equities / stocks in her portfolio with no . This investor may well benefit from buying a debt instrument like zero coupon bonds to diversify the types of securities she holds.
Naturally of course, investment goals are very much a function of a whole host of factors, including the investor’s individual situation, their income tax bracket / status, interest rate risk, and whether the economy’s in a rising interest rate environment, to name a few. So we’re really just about scratching the surface here, with the most broad / case.
But if an investor is looking to add zero coupon bonds to her / his portfolio, knowing how to value them is imperative.
Valuing Zero Coupon Bonds
Valuing a zero coupon bond is perhaps one of the easiest things you can do.
ZCB Cash flow Payoff Timeline
If you think of the payoff timeline for bond payoffs in the general case, this is what it looks like…
For deep discount bonds, coupons are equal to zero.
And therefore you’re essentially getting zero as your coupons, and a par value at maturity. The timeline thus changes to this…
And given that 0 is, as it implies, nothing, the timeline simplifies to this…
This Article features a concept that is covered extensively in our course on Bond Valuation Mastery.
If you’re interested in learning and mastering Bond Valuation, then you should definitely check out the course.
Equation to value Zero Coupon Bonds
We can use exactly the same thinking to get to the formula for the price of a zero coupon bond.
The general equation for the price of a bond is this:
Where the first part is discounting all the future coupons at each time back to the present (i.e., to today).
And the second part is discounting the Par value back to the present.
For a deep discount bond, the coupon is of course equal to 0. So the equation changes to this:
Which in turn simplifies to this, since 0 divided by anything is equal to 0.
In other words…
The value of a zero coupon bond is nothing but the Present Value of its Par Value.
Example Valuation (Swindon Plc)
Consider an example of Swindon PLC, which is issuing a zero coupon bond with a par value of £100 to be paid in one year’s time.
What is the price of this bond today, if the yield is 7%?
Here we have a par value of £100 pounds. The yield (aka Yield to Maturity, or the ) is 7%. And the time to maturity or is equal to 1.
The price of the bond is estimated as…
Plug in the numbers of Swindon Plc and you’ve got…
Solve for that, and you’ll find that the price of the bond is approximately equal to £93.46
This is how we go about calculating the price of a zero coupon bond manually.
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But we can also calculate it on Excel®.
Valuing Zero Coupon Bonds on Excel®
We’ll be using Excel’s “PRICE” function to value Swindon Plc’s bond.
The first thing you want to do is setup your spreadsheet with a pro-forma / template which consists of the all different variables you’ll need.
The “PRICE” function on Excel® requires:
- Settlement, which is the date today (or the date that you’re buying the bond)
- Maturity, which is when the bond expires
- Rate, which is the coupon rate.
- “YLD” refers to the yield (aka Yield to Maturity, or the )
- Redemption,which is the face value; and
- Frequency, which refers to the frequency of the payments (i.e., whether it’s annual, semi-annual, quarterly payments, for example)
So we’re just going to need to fill in the data for Swindon Plc’s bond like this…
We’ve got a par value for a hundred pounds for Swindon
Setting the Settlement & Maturities
Note that the question didn’t specify a date of purchase. However, Excel needs this information to value the bond.
The specific date we’ve used is completely irrelevant, and doesn’t matter.
The only thing that matters is the timeframe distance between the “settlement” and “maturity” parameters.
Remember that Swindon’s bond is a 1 year bond. And by setting the settlement date to the 1st of January 2020 and the maturity date to the 31st of December 2020, we’re telling Excel that this is a one year zero coupon bond.
The coupon rate, or “rate” is 0% because this is a zero coupon bond.
The yield (or “yld”) here is 7%, given that’s the yield for Swindon Plc. The frequency’s 1 because this is a one year annual zero coupon bond.
Then it’s just a simple case of applying Excel’s PRICE function and plugging in the variables.
And that’s it!
Discrepancies in price estimates
You’ve got a price of £93.47, which is nearly identical to what we had when we calculated it manually.
Now, if you’ve got a close eye, you’ll note that the estimate is nearly identical, but not identical!
There’s a minuscule difference in the estimate
We estimated it to be approximately £93.46 but Excel’s result is £93.47
Why’s there a £0.01 difference?
2020 was a leap year.
Meaning there’s 366 days, not 365.
And Excel’s incorporated that additional day into the estimate for the bond.
The full documentation for Excel’s PRICE function is viewable here.
Excel does make life a lot easier, but it’s always good to know what Excel is doing. Or what any other software is doing for that matter.
Which is why we would strongly encourage you to calculate the price of bonds manually, especially during the early stages on your journey to bond valuation mastery.
All right, hopefully this example makes sense.
Let’s take a look at another example.
Example Valuation (Warren B Inc)
Let’s now consider a hypothetical zero coupon bond that Warren B Inc is evaluating. Suppose bond promises to pay $1,000 in four years time.
Assume the bond has a yield equal to 8.2% and is currently priced at $814.39.
Your task is to advise Warren B Inc on whether it should buy this bond today.
Try solving this on your own before reading any further. Because the process is exactly the same as with Swindon Plc, except that we’re also asking you to analyse whether this is a good investment and provide investment advice fictitiously.
Alright, let’s solve this now then.
We know that the price of a zero coupon bond is estimated as…
Plug in the numbers of the bond Warren B Inc’s evaluating and you’ve got…
Solve for that, and you’ll find that the price of the bond is approximately equal to £93.46
Now, recall from the question – the bond was trading at $814.39. That means it’s overvalued by approximately $84.78
And therefore Warren B should not invest in this bond.
Because they’d be paying approximately $814 for something that’s only worth approximately $729.
Then we learned that zero coupon bonds are bonds that don’t pay any coupons. They only pay a Par at maturity.
And given this fact, zero coupon bonds will always trade at a discount.
In other words, the price will always be lesser than the Par for all zero coupon bonds.
Finally, we learned the value of a zero coupon bond is equal to the present value of its par value with the discount rate equal to its yield (aka Yield to Maturity).
Do you want to learn how to value bonds from scratch? And become a PRO at it while you’re at it?