Calculate Zero-coupon Bond Purchase Price

Enter the face value of a zero-coupon bond, the stated annual percentage rate (APR) on the bond and its term in years (or months) and we will return both the upfront purchase price of the bond, its nominal return over its duration & its yield to maturity.

• Entering Years: For longer duration bonds enter the number of years to maturity.
• Entering Months: For shorter duration bonds enter the number of months to maturity.
• Entering Both: If you enter a set number of years AND months then those figures are additive. For example:
  • 5 years + 60 months = 10 years
  • 3.5 years + 3 months = 3 years 9 months

Example Zero-coupon Bond Formula

P = M / (1+r)ⁿ

variable definitions:

• P = price
• M = maturity value
• r = annual yield divided by 2
• n = years until maturity times 2

The above formula is the one we use in our calculator to calculate the discount to face value every half-year throughout the duration of the bond's term.

Here is an example calculation for the purchase price of a $1,000,000 face value bond with a 10 year duration and a 6% annual interest rate.

$1,000,000 / (1+0.03)²⁰ = $553,675.75

Calculating Yield to Maturity on a Zero-coupon Bond

YTM = (M/P)^1/n - 1

variable definitions:

• YTM = yield to maturity, as a decimal (multiply it by 100 to convert it to percent)
• M = maturity value
• P = price
• n = years until maturity

Let's say a zero coupon bond is issued for $500 and will pay $1,000 at maturity in 30 years. Divide the $1,000 by $500 gives us 2. Raise 2 to the 1/30th power and you get 1.02329. Subtract 1, and you have 0.02329, which is 2.3239%.

Advantages of Zero-coupon Bonds

Most bonds typically pay out a coupon every six months. This makes typical bonds a great source of income, though it limits their capital appreciation if & when bond yields fall (as they often do during recessions, deflation & strong disinflation).

Bond yields & price move inversely. Thus if interest rates fall, any outstanding bond which pays an interest rate above the current prevailing rate enjoys capital appreciation, since it is paying a higher rate than an investor could obtain by buying another similar bond at current rates.

Since zero coupon bonds do not pay a coupon, any capital appreciation remains in the bond. Since they sell at a discount to their stated maturation value they are known as discount bonds. In a falling rate envirnoment zero-coupon bonds appreciate much faster than other bonds which have periodic coupon payments.

Bonds with a longer duration are more sensitive to the impact of interest rate shifts.

Economist Gary Shilling mentioned holders of 30-year zero-coupon bonds purchased in the early 1980s outperformed the S&P 500 with dividends reinvested by 500% over the subsequent 30-years as interest rates fell from around 14.6% to around 3%.

I started investing in 30 Year zero coupon treasuries. Now, zero coupon bonds don't pay any interest, but they are issued at a discount. And the interest in effect is in effect built in the difference between the issue price which is below 100 and they're expiring at 100. It's built-in. Now, the fact that it's built-in, it has big advantages when interest rates come down. You don't have a reinvestment risk. In other words, if you invest it, let's just take an example. Let's say you invest in a 10% yielding security and the rates dropped to 5%. Well, you've got to reinvest at 5%, you no longer can invest at 10%, that's gone. But the zero coupons build that in, so you get actually about twice as much appreciation for given declining interest rates with a zero coupon, as with a coupon bond, and the longer the maturity, the more bang for the buck.

Now, it works both ways. You'll lose more money if rates go up. But actually, I started in with the zero coupon bonds from my own account in 1981. And by the mid-80s, the Shilling family, on that one investment, had achieved financial independence. And it's been a tremendous asset, as a matter of fact, since the early '80s, and we have documented that these zero coupon bonds have outperformed the S&P 500 by five times- that's including dividends in the S&P, but a lot of people, they think that Treasury bonds are for little old ladies and orphans. Well, I've never, never, never bought Treasury bonds for yield.

I couldn't care less what the yield is as long as it's going down. Because when it goes down, they increase in price, and I bought it for the same reason most people buy stocks. Most people don't buy stocks for dividends, you have some for utilities and real estate investments, but most people are looking for appreciation. And that's what my interest is in Treasury bonds." - economist Gary Shilling

If 30-year interest rates are 14% a person would only need to spend $17,257.32 to buy a $1,000,000 face-value zero coupon bond. With interest rates at 3% that math changes drastically, requiring a $409,295.97 payment to buy the same instrument. That difference in price is capital appreciation. The following table shows how interest rates & term impact the price & nominal return of various bond investments.

Face Value	Interest Rate	Term	Purchase Price	Nominal Return
$1,000,000	10%	30 years	$53,535.52	$946,464.48
$1,000,000	5%	30 years	$227,283.59	$772,716.41
$1,000,000	3%	30 years	$409,295.97	$590,704.03
$1,000,000	2%	30 years	$550,449.62	$449,550.38
$1,000,000	1%	30 years	$741,372.20	$258,627.80
$1,000,000	10%	10 years	$376,889.48	$623,110.52
$1,000,000	5%	10 years	$610,270.94	$389,729.06
$1,000,000	3%	10 years	$742,470.42	$257,529.58
$1,000,000	2%	10 years	$819,544.47	$180,455.53
$1,000,000	1%	10 years	$905,062.90	$94,937.10
$1,000,000	10%	7 years	$505,067.95	$494,932.05
$1,000,000	5%	7 years	$707,727.20	$292,272.80
$1,000,000	3%	7 years	$811,849.28	$188,150.72
$1,000,000	2%	7 years	$869,962.97	$130,037.03
$1,000,000	1%	7 years	$932,556.46	$67,443.54

Disadvantages of Zero-coupon Bonds

There are two major disadvantages of zero-coupon bonds.

The first disadvantage is they do not throw off any income as the capital is stored in the bond. In some countries the imputed interest may be taxed as income even though the bond has not yet been redeemed or reached maturity. The IRS requires zero-coupon bond holders to pay tax on the "phantom" imputed interest income just as they would if they had received coupon payments, even though there wasn't any interest paid to the bond holder.

To calculate imputed interest, begin with the starting value of your instrument & then mutiply it by the yield to maturity to obtain the imputed interest for that year. For example, if you paid $5,000 for a 5-year bond & it has an imputed interest of 2.337% then for the first year you would calculate imputed interest as 2.337% of $5,000, or $116.85. For the subsequent years you would start with the base from prior years to calculate the new imputed interest value.

year	initial value	imputed rate	imputed interest	final value
1	$5,000	2.337%	$116.85	$5,116.85
2	$5,116.85	2.337%	$119.58	$5,236.43
3	$5,236.43	2.337%	$122.38	$5,358.81
4	$5,358.81	2.337%	$125.23	$5,484.04
5	$5,484.04	2.337%	$128.16	$5,612.20

The above table presumes the insturment was held for throughout the duration of each calendar year. If the instrument was bought in the middle of the year then imputed interest would need to be calculated for 6 different years with the first & final years being partial year calculations based on the percentage of the year where the instrument was held.

The second major disadvantage is when interest rates rise significantly they can see a drastic decline in capital value, as they have a significant duration risk because no capital is paid out until the bond reaches maturity (risk remains embedded in the instrument until it is redeemed). Bonds can be traded on the secondary market, with valuations reflecting the current interest rate envirnoment. If you want to see what can happen in a rising rate environoment, look to the table above and see how a 30 year bond issued at 3% for $409,295.97 would only be worth around $227,283.59 if rates suddenly rose to 5%. If rates remained relatively flat for 10 years & then went to 5% the 30-year bond (with 20 years remaining) would only be worth $372,430.62 - meaning the bond owner would have paid income taxes on imputed interest for a decade only to see the bond be worth less than they paid for it after holding it for a decade.

Treasury Instruments

Bills, Notes, Bonds

Many people refer to any fixed-income treasury instrument as a bond, however the duration determines the specific name.

• Bills: These mature in 1 year or less. Due to their short duration they do not throw of a steady stream of interest payments, rather they sell at a discount to their face value & mature at par.
• Notes: These mature between 2 to 10 years & pay interest semi-annually.
• Bonds: These have maturities beyond 10 years & pay interest semi-annually.

Longer duration bonds are more sensitive to shifts in interest rates. And zero-coupon long duration bonds are more sensitive to rate shifts than bonds which regularly pay interest.

Typically the yield curve is upward sloping with longer duration bonds offering a higher return to compensate for the added risk. When shorter duration bonds offer a higher yield than longer duration bonds that is called yield curve inversion. If investors are willing to lock their money up for a longer period of time at low rates it usually indicates they think there might be significant economic risks ahead. Inversion frequently happens anywhere from about 6 to 18 months before a recession happens. The most widely watched segment is the 2-year versus the 10-year.

10-year Treasury plays an important role in the economy, as 30-year fixed-rate mortgage rates tend to closely follow shifts in the 10 year Treasury note, trading at a slightly higher rate than the 10-year. Banks pay short-term deposit rates & lend across longer duration loans, so when the yield curve significantly inverts it can hurt their profit margins & make them less willing to lend.

Investors are allowed to invest up to $5 million in each marketable security type when they bid in a noncompetitive auction. This limitatio does not apply to rollover reinvestment.

Current Yields

Here are stated rates from June 20, 2019:

Instrument Type	Yield (APR %)
Treasury bills, notes, bonds
1 month Treasury bills	2.17%
2 month Treasury bills	2.15%
3 month Treasury bills	2.14%
6 month Treasury bills	2.04%
1 year Treasury bills	1.91%
2 Year Treasury notes	1.72%
3 year Treasury notes	1.69%
5 year Treasury notes	1.74%
7 year Treasury notes	1.87%
10 year Treasury notes	2.01%
20 year Treasury bonds	2.31%
30 year Treasury bonds	2.53%
Corporate bonds
Moody's Seasond Aaa Corporate Bond Yield	3.33%
Moody's Seasond Baa Corporate Bond Yield	4.39%
ICE Bank of Americ Merrill Lynch High Yield B Effective Yield	6.29%
ICE Bank of Americ Merrill Lynch High Yield CCC or Below Option-Adjusted Spread	9.39%
Mortgages
5-1 adjustable rate mortgage	3.48%
15 year fixed rate mortgage	3.25%
30 year fixed rate mortgage	3.84%

Rate data sources: Treasury.gov, FRED & Freddie Mac's PMMS.

Some municipal bonds are exempt from income taxes, which boost their equivalent yield when compared against other bonds.

Other Treasury Marketable Securities

Investors can also purchase inflation-protected Treasury securites (TIPS), US Treasury floating rate notes (FRN), as well as Treasury Strips.

• TIPS: TIPS pay a regular semi-annual coupons while the principal can be redeemed at the greater of the original principal amount or their inflation-adjusted equivalents.
• FRN: Floating rate notes pay quarterly interest based on discount rates for 13-week treasury bills, with the principal paid at maturity.
• STRIPS: Strip bonds are explained in more detail in the following section.

Strip Bonds

Investment bankers & bond dealers have the ability to separate the components of a traditional coupon-paying bond into the coupon & the principal (or residue). The coupon payments & residue can be sold separately to investors, creating additional zero-coupon bonds.

This process is called stripping & STRIPS stands for "Separate Trading of Registered Interest and Principal Securities." It makes a lot of sense to hold these types of instruments in tax-deferred retirement accounts so that they are not subject to annual income taxes based on imputed interest.

Savings Bonds

• Series EE: accrual-type security with a fixed interest rate
• Series I: accrual-type security which combines both a fixed interest rate & the rate of inflation

Each person may invest in up to $10,000 of Series EE or Series I bonds. Gift purchases are attributed to the recipient.

The Treasury also offers zero-percent certificate of indebtedness (C of I) which can be used to fund TreasuryDirect purchases.

ETFs

There are mutliple popular long-duration bond ETFs for investors seeking to track the market in a liquid form without purchasing bonds directly.

Ticker	Sponsor	Fund Name
TLT	iShares	iShares 20+ Year Treasury Bond
BLV	Vanguard	Vanguard Long-Term Bond

Investors who believe longer duration rates are likely to fall can also buy exposure to long duration zero-coupon Treasuries using ETFs.

Ticker	Sponsor	Fund Name
ZROZ	PIMCO	PIMCO 25+ Year Zero Coupon US Treasury Index
EDV	Vanguard	Vanguard Extended Duration Treasury

If rates fall longer duration zero-coupon bonds will increase in value significantly more than shorter duration federal government bonds & federal bonds which pay a regular coupon. If rates rise the converse is true - zero-coupon bonds will be hit much harder than other bonds.

Negative Yields

After the financial crisis of 2008-2009 central banks became far more aggressive participants in financial markets. Their goal was to provide liquidity & push investors out on the risk curve, hoping that asset price inflation would drive wealth effect spending that stimulates the economy. In some cases not only was the short end of the curve driven to zero, but some countries like Germany & Japan have negative rates going out 10 years.

Negative yields mean the lender pays the borrower to borrow. This has never happened at a large scale in the history of humanity & in June of 2019 over $13 trillion in bonds have negative yields globally.

A negative yielding zero coupon bond would have an investor buying it at above par, paying more than face value. As crazy as it sounds, negative yielding bonds can still appreciate if rates go more negative than they already have, because that would mean bonds currently in circulation have higher yields than newly issued bonds.

Resources & References

• SEC fast answers - compares how ZCBs performn against other bonds
• Finra one minute guide - quick overview of ZCBs & a reminder on the implications of imputed interest
• IRS Publication 1212 - Guide to Original Issue Discount (OID) Instruments
• TreasuryDirect - price Series EE, Series E, and Series I savings bonds, and Savings Notes