Understanding the math behind a 20 year treasury

[atl2005]

I'm looking at a 20 year Treasury in my Fidelity brokerage account and I'm trying to understand the math behind the investment. I'm looking to invest 250k in a 20 year bond with these numbers. Ask 95.016, Yield 5.03, Coupon 4.625.

1. The coupon will give me 11,562.50 a year divided in 2 payments every 6 months. (Total from coupon 231,250)
2. The final yield of 5.03 should give me 251,500 in total interest based on a 250k investment.
3. What confuses me is where does the additional 20,970 come from? Is it based on the discount where I buy the 250 bonds? I'm buying them at a 50 dollar discount but even at 250 bonds that would only give me 12,500 at the maturity of the bond.

Where am I going wrong with my math?

[typical.investor]

I'm looking at a 20 year Treasury in my Fidelity brokerage account and I'm trying to understand the math behind the investment. I'm looking to invest 250k in a 20 year bond with these numbers. Ask 95.016, Yield 5.03, Coupon 4.625.

1. The coupon will give me 11,562.50 a year divided in 2 payments every 6 months. (Total from coupon 231,250)
2. The final yield of 5.03 should give me 251,500 in total interest based on a 250k investment.
3. What confuses me is where does the additional 20,970 come from? Is it based on the discount where I buy the 250 bonds? I'm buying them at a 50 dollar discount but even at 250 bonds that would only give me 12,500 at the maturity of the bond.

Where am I going wrong with my math?

$250k will buy about 263 bonds [250000 / 95.016 /10] ***, so at maturity you will receive $263,000. Also, you will receive $12,025 [$263,000 * 4.625%] per year (half every six months) in interest.

I say about 263 bonds because there may be accrued interest that you have to pay for, but then get back. So maybe you can only afford to buy 262 bonds for $250k.

*** the price is quoted at 95.016, but you need $950.16 for 1 bond. Thus taking the available amount $250000 and dividing by the price 95.016 is 10 times as many bonds as you could actually buy. That why it is divided by 10.

[Valuethinker]

I'm looking at a 20 year Treasury in my Fidelity brokerage account and I'm trying to understand the math behind the investment. I'm looking to invest 250k in a 20 year bond with these numbers. Ask 95.016, Yield 5.03, Coupon 4.625.

1. The coupon will give me 11,562.50 a year divided in 2 payments every 6 months. (Total from coupon 231,250)
2. The final yield of 5.03 should give me 251,500 in total interest based on a 250k investment.
3. What confuses me is where does the additional 20,970 come from? Is it based on the discount where I buy the 250 bonds? I'm buying them at a 50 dollar discount but even at 250 bonds that would only give me 12,500 at the maturity of the bond.

Where am I going wrong with my math?

So 250k of face value will cost you 250k*0.95016

Coupon payments = (4.625/2) per 100 of face amount (so (4.625/2)*(250k/100))

Yield To Maturity is the Internal Rate of Return of the cash flows so IRR = (purchase price) + coupon payments + 250k

With the XIRR function in Excel you give it the dates of each cash flow =XIRR(values, dates). You need to do this because the semi-annual feature makes things slightly more complicated.

That should give you the Yield to Maturity -- a complication is "bonds are priced clean and bought dirty". There's accrued interest between coupon dates and you have to pay for that. So I think you need to use the actual bond yield formula in Excel to get exactly the same answer as a Bloomberg terminal (or your broker) would give you.

[typical.investor]

I'm looking at a 20 year Treasury in my Fidelity brokerage account and I'm trying to understand the math behind the investment. I'm looking to invest 250k in a 20 year bond with these numbers. Ask 95.016, Yield 5.03, Coupon 4.625.

1. The coupon will give me 11,562.50 a year divided in 2 payments every 6 months. (Total from coupon 231,250)
2. The final yield of 5.03 should give me 251,500 in total interest based on a 250k investment.
3. What confuses me is where does the additional 20,970 come from? Is it based on the discount where I buy the 250 bonds? I'm buying them at a 50 dollar discount but even at 250 bonds that would only give me 12,500 at the maturity of the bond.

Where am I going wrong with my math?

So 250k of face value will cost you 250k*0.95016

Coupon payments = (4.625/2) per 100 of face amount (so (4.625/2)*(250k/100))

Yield To Maturity is the Internal Rate of Return of the cash flows so IRR = (purchase price) + coupon payments + 250k

With the XIRR function in Excel you give it the dates of each cash flow =XIRR(values, dates). You need to do this because the semi-annual feature makes things slightly more complicated.

That should give you the Yield to Maturity -- a complication is "bonds are priced clean and bought dirty". There's accrued interest between coupon dates and you have to pay for that. So I think you need to use the actual bond yield formula in Excel to get exactly the same answer as a Bloomberg terminal (or your broker) would give you.

Yes, but that is $250k face (or par) value.

I'm looking to invest 250k

At a price of 95.016, you will end up with a higher face value than $250k when you invest $250k. If you were to only purchase $250k in face value, you are going to have a lot of cash and uninvested money left over.

[123]

...At a price of 95.016, you will end up with a higher face value than $250k when you invest $250k. If you were to only purchase $250k in face value, you are going to have a lot of cash and uninvested money left over.

But if you're looking for $250K of par value at maturity the current cost is $237,540 (plus some added amount for accrued interest since the last interest payment).

[atl2005]

I'm looking at a 20 year Treasury in my Fidelity brokerage account and I'm trying to understand the math behind the investment. I'm looking to invest 250k in a 20 year bond with these numbers. Ask 95.016, Yield 5.03, Coupon 4.625.

1. The coupon will give me 11,562.50 a year divided in 2 payments every 6 months. (Total from coupon 231,250)
2. The final yield of 5.03 should give me 251,500 in total interest based on a 250k investment.
3. What confuses me is where does the additional 20,970 come from? Is it based on the discount where I buy the 250 bonds? I'm buying them at a 50 dollar discount but even at 250 bonds that would only give me 12,500 at the maturity of the bond.

Where am I going wrong with my math?

$250k will buy about 263 bonds [250000 / 95.016 /10] ***, so at maturity you will receive $263,000. Also, you will receive $12,025 [$263,000 * 4.625%] per year (half every six months) in interest.

I say about 263 bonds because there may be accrued interest that you have to pay for, but then get back. So maybe you can only afford to buy 262 bonds for $250k.

*** the price is quoted at 95.016, but you need $950.16 for 1 bond. Thus taking the available amount $250000 and dividing by the price 95.016 is 10 times as many bonds as you could actually buy. That why it is divided by 10.

Ok I wasn't taking into account that I can actually buy more than 250 bonds with 250k. I'm still confused how 250k at 5.03 yield can return 263k that seems like a 5.2 % yield based on 250k at 20 years. Is it just that I will be returned 1k per bond at 263 bonds? The annual coupon is based on the number of bonds or the amount invested?

[typical.investor]

$250k will buy about 263 bonds [250000 / 95.016 /10] ***, so at maturity you will receive $263,000. Also, you will receive $12,025 [$263,000 * 4.625%] per year (half every six months) in interest.

I say about 263 bonds because there may be accrued interest that you have to pay for, but then get back. So maybe you can only afford to buy 262 bonds for $250k.

*** the price is quoted at 95.016, but you need $950.16 for 1 bond. Thus taking the available amount $250000 and dividing by the price 95.016 is 10 times as many bonds as you could actually buy. That why it is divided by 10.

Ok I wasn't taking into account that I can actually buy more than 250 bonds with 250k. I'm still confused how 250k at 5.03 yield can return 263k that seems like a 5.2 % yield based on 250k at 20 years. Is it just that I will be returned 1k per bond at 263 bonds? The annual coupon is based on the number of bonds or the amount invested?

4.625% is the actual coupon that each $1k bond yields. 5.03% is the effect yield of buying a 4.625% coupon bond for a price of $950.16 for 1 bond.

$250,000 invested at a rate of 5.03% will mean in 20 years you have $501,500. (of course, what you do with the interest payments along the way and how much you earn on them is another story. Treasury interest does not compound)

Total Interest = $250000 × 5.03% × 20
= $251,500.00

End Balance = $250000 + $251,500.00
= $501,500.00

-----------------------------------

Since the price is 95.016, you can actually buy 263 bonds that yield 4.625%

Total Interest = $263000 × 4.265% × 20
= $224,339.00

End Balance = $263000 + $224,339.00
= $487,339.00

However, since you only spend $250,000 to buy that $263,000 par value, there is an additional $13,000 you receive at maturity which is in addition to the coupon payments.

$487,339 + 13,000 = $500,339

The reason $500,339 is less than $501,500 is that you aren't actually investing all your money when you buy 263 bonds. There will be some amount left over (as you can only buy in $1k increments) and that amount and any interest it could earn over 20 years isn't accounted for in the $500,339

[Tom_T]

$250k will buy about 263 bonds [250000 / 95.016 /10] ***, so at maturity you will receive $263,000. Also, you will receive $12,025 [$263,000 * 4.625%] per year (half every six months) in interest.

I say about 263 bonds because there may be accrued interest that you have to pay for, but then get back. So maybe you can only afford to buy 262 bonds for $250k.

*** the price is quoted at 95.016, but you need $950.16 for 1 bond. Thus taking the available amount $250000 and dividing by the price 95.016 is 10 times as many bonds as you could actually buy. That why it is divided by 10.

Ok I wasn't taking into account that I can actually buy more than 250 bonds with 250k. I'm still confused how 250k at 5.03 yield can return 263k that seems like a 5.2 % yield based on 250k at 20 years. Is it just that I will be returned 1k per bond at 263 bonds? The annual coupon is based on the number of bonds or the amount invested?

Besides the nice explanation that was just given, remember that the coupon is a fixed amount. You get 5.03% of every $1000 of face value, or $50.03, every year. This has no relationship to what you paid.

Now, if you paid $950 for the bond, then that $50.03 is obviously more than 5.03% of $950, so you're effectively getting 5.2% on your investment.

[typical.investor]

Ok I wasn't taking into account that I can actually buy more than 250 bonds with 250k. I'm still confused how 250k at 5.03 yield can return 263k that seems like a 5.2 % yield based on 250k at 20 years. Is it just that I will be returned 1k per bond at 263 bonds? The annual coupon is based on the number of bonds or the amount invested?

Besides the nice explanation that was just given, remember that the coupon is a fixed amount. You get 5.03% of every $1000 of face value, or $50.03, every year. This has no relationship to what you paid.

Now, if you paid $950 for the bond, then that $50.03 is obviously more than 5.03% of $950, so you're effectively getting 5.2% on your investment.

Hmmm, I don't think that is right. The coupon here is fixed at 4.265%, so you get 4.265% of every $1,000 of face value.

Now if you paid $950 for the bond, you will effectively get 5.03% which is 4.265%/year + the $50 at maturity ($1,000 returned for your $950 spent).

Nobody is getting 5.2% on this 20 year bond. Don't be mad at me for delivering the bad news

[atl2005]

Thanks everyone I understand it now. Was confused about the cheaper price of the bond and how it added built in value upon maturity. The 20 year time horizon fits in perfectly for me with my retirement age and I plan to use the interest each year to buy either a 1 year brokered cd or 12 month security of some type to keep earning the most interest possible.

[LotsaGray]

Thanks everyone I understand it now. Was confused about the cheaper price of the bond and how it added built in value upon maturity. The 20 year time horizon fits in perfectly for me with my retirement age and I plan to use the interest each year to buy either a 1 year brokered cd or 12 month security of some type to keep earning the most interest possible.

If you wanted (which I would at least consider each year) you could look at using the interest to purchase more of the same bond (or others) in secondary market (which you appear to already be buying in). What I would not do is limit my self to 1 yr securities/CDs. In year 15 you would be purchasing some (close to) $250K of 1 yr instruments. Each year the previous year's 12 month instrument will mature plus the next yrs interest.

[atl2005]

Thanks everyone I understand it now. Was confused about the cheaper price of the bond and how it added built in value upon maturity. The 20 year time horizon fits in perfectly for me with my retirement age and I plan to use the interest each year to buy either a 1 year brokered cd or 12 month security of some type to keep earning the most interest possible.

If you wanted (which I would at least consider each year) you could look at using the interest to purchase more of the same bond (or others) in secondary market (which you appear to already be buying in). What I would not do is limit my self to 1 yr securities/CDs. In year 15 you would be purchasing some (close to) $250K of 1 yr instruments. Each year the previous year's 12 month instrument will mature plus the next yrs interest.

I guess I was just going to do 1 year for simplicity which would allow me to add the previous years interest to my maturing security. Multiple years seems like it might be a lot keep up with since I don't know if I get notifications when these investments mature inside a 401.

[#Cruncher]

... if you're looking for $250K of par value at maturity the current cost is $237,540 (plus some added amount for accrued interest since the last interest payment).

The original poster seems to be referring to either the 4-5/8% bond maturing 5/15/2044 auctioned May 2024 or the 4-5/8% bond maturing 11/15/2044 auctioned November 2024. As shown on row 11 below, the accrued interest for both of these bonds is $1,884.50 making the total cost $239,424.50. Rows 14-22 show how the yield to maturity [*] is the discount rate that makes the present value of the coupons and principal equal this amount.

Code: Select all

Row                     Col A        Col B       Col C  Formulas in Col B Copied to Col C
  2                Face value      250,000     250,000
  3                Settlement    1/13/2025   1/13/2025
  4                   Matures    5/15/2044  11/15/2044
  5                    Coupon       4.625%      4.625%
  6                     Price       95.016      95.016
  7    Previous interest date   11/15/2024  11/15/2024  =COUPPCD(B3,B4,2,1)
  8  Next interest date (NID)    5/15/2025   5/15/2025  =COUPNCD(B3,B4,2,1)
  9            Days in period          181         181  =B8-B7
 10    Days before settlement           59          59  =B3-B7
 11          Accrued interest     1,884.50    1,884.50  =B2*(B5/2)*(B10/B9)
 12         Cost of principal   237,540.00  237,540.00  =B2*(B6/100)
 13   Cost incl accr interest   239,424.50  239,424.50  =B12+B11

Code: Select all

 14         Yield to maturity      5.0306%     5.0243%  =YIELD(B3,B4,B5,B6,100,2,1)
 15     Days after settlement          122         122  =B8-B3
 16  Number full 6 mo periods           38          39  =COUPNUM(B3,B4,2,1)-1
 17   Present value $1 on NID     0.389072    0.379980  =1/(1+B14/2)^B16
 18   PV annuity of $1 on NID    24.288493   24.680803  =(1-B17)/(B14/2)
 19       PV principal on NID    97,268.12   94,995.05  =B2*B17
 20         PV coupons on NID   146,199.10  148,467.14  =B2*(B5/2)*(B18+1)
 21           Total PV on NID   243,467.22  243,462.19  =B19+B20
 22    Total PV at settlement   239,424.50  239,424.50  =B21/(1+B14/2)^(B15/B9))

* As calculated on row 14 with the Excel YIELD function based on the 95.016 price in the original post.

[Thesaints]

What you want is a 20-year ZC. Maybe that bond can be found under strips, with all the coupons taken off.
Alternatively, you could invest the dividends from yr1 in a 19-year bond. The cumulative dividends from yr2 in a 18-year bond, and so on.

[typical.investor]

... if you're looking for $250K of par value at maturity the current cost is $237,540 (plus some added amount for accrued interest since the last interest payment).

The original poster seems to be referring to either the 4-5/8% bond maturing 5/15/2044 auctioned May 2024 or the 4-5/8% bond maturing 11/15/2044 auctioned November 2024. As shown on row 11 below, the accrued interest for both of these bonds is $1,884.50 making the total cost $239,424.50. Rows 14-22 show how the yield to maturity [*] is the discount rate that makes the present value of the coupons and principal equal this amount.

Code: Select all

Row                     Col A        Col B       Col C  Formulas in Col B Copied to Col C
  2                Face value      250,000     250,000
 
[/quote]

As always #Cruncher, you do great work.

The OP was looking to invest $250k though (as a total cost), and not looking for $250k par/face value.

[atl2005]

What you want is a 20-year ZC. Maybe that bond can be found under strips, with all the coupons taken off.
Alternatively, you could invest the dividends from yr1 in a 19-year bond. The cumulative dividends from yr2 in a 18-year bond, and so on.

Advantage of a zero coupon over a 20 year Treasury ? It wouldn't provide interest along the way for me to reinvest ?

[dbr]

What you want is a 20-year ZC. Maybe that bond can be found under strips, with all the coupons taken off.
Alternatively, you could invest the dividends from yr1 in a 19-year bond. The cumulative dividends from yr2 in a 18-year bond, and so on.

Advantage of a zero coupon over a 20 year Treasury ? It wouldn't provide interest along the way for me to reinvest ?

It wouldn't give you the problem of finding as good a deal to reinvest as you had in the original bond. With zero coupon the interest effectively accumulates in the value of the bond in the sense that the time to maturity is constantly decreasing and the price of the bond tends (up) to the redemption value. If you look at the formula for yield to maturity you would find a way to see that.

https://speckandcompany.com/yield-to-maturity/