UK gilt calculations

nonolerigolo

I am calculating the redemption rate annually for the gilt TG31 - 31/07/31 coupon 0.25%. 

Yield Gimp website gives a redemption of 4.20%. Current price is £76.96.

My calculation is: 100 (par value) -76.96 ‎ = 23.04

23.04\6.75 (year to maturity) =3.41 

3.41 + 0.35 coupons= 3.76% 

So approximately 0.36% difference. 

Where does the difference come from?

Thank you 

Hoenir

Gross redemption yield assumes that the Gilt is held to maturity and all coupons paid are reinvested back into the Gilt. 

SnowMan

The 3.41% bit is wrong because the £23.04 excess of the redemption amount over the price comes from an investment of £76.96 not from an investment of £100 as you are assuming. So doing it your way the first bit should be 3.41% x 100/76.96. But that then doesn't allow for compound interest so overstates that bit of the calculation

The rough return ignoring the coupons and allowing for compounding (i.e your 3.41% bit corrected for these things) is

(100/76.96)^(1/6.75) - 1 = 3.96%pa ignoring coupons.

But you can't just add in the running yield from the coupons (i.e 100/76.96 x 0.25% = 0.32%) to get the redemption yield because you are assuming that the coupons you are valuing are increasing at the overall gross redemption yield you are trying to calculate, in reality they aren't increasing.

However if you accept it's an over-estimate it gives you a redemption yield of 3.96% + 0.32% = 4.28% which compares with the accurate 4.2%pa you quote.

Identifying all the cashflows and dates and then using the XIRR function in excel would probably be the easiest way to reproduce the accurate figure.

nonolerigolo

Thank you all. What I do not understand is for me a gilt does not compounds, especially when the coupon is low. I am misunderstanding it? 

SnowMan

nonolerigolo said:

Thank you all. What I do not understand is for me a gilt does not compounds, especially when the coupon is low. I am misunderstanding it? 

The coupon remains the same for the next 6.75 years. That's not a problem, it's just a design feature of the gilt. The price and redemption yield of the gilt reflect that design feature.

Let's suppose you were going to use £1,000 to buy that gilt. Think of the redemption yield of 4.2%pa as being the interest you would need to earn on a fixed rate savings account where you saved £1,000 for 6.75 years to reach an amount in that savings account with interest at 2031, equal to the redemption amount of the gilt (adding into the redemption amount the coupons from the gilt, which you assume also go into a different hypothetical savings account earning interest at 4.2%pa from when they are paid up to 2031). 

Under that hypothetical comparator fixed rate savings account paying 4.2%pa it's assumed the interest would compound each year like a normal savings account, that is you would get more interest in later years because you would have interest on interest. That's what the talk of compounding is about, the compounding on that hypothetical comparator savings account, not the coupons from your actual investment, the gilt, which stay the same each year up to 2031.

masonic

nonolerigolo said:

Thank you all. What I do not understand is for me a gilt does not compounds, especially when the coupon is low. I am misunderstanding it? 

That doesn't mean that you are forced to earn nothing on the coupons until the gilt matures. Given that the duration may be decades, it would not be satisfactory to measure the return without considering the time value of money.

Linton

nonolerigolo said:

Thank you all. What I do not understand is for me a gilt does not compounds, especially when the coupon is low. I am misunderstanding it? 

The gilt does not compound. However compounding is a way of looking at the time value of money. A coupon payment today is more valuable than the same coupon paid in 10 years time as you could invest today’s coupon in the same gilt and get a higher value 10 years later. If you are going to fairly and accurately compare 2 different gilts  you need to take such effects into consideration.

OldScientist

An approximate formula can be found at https://corporatefinanceinstitute.com/resources/fixed-income/yield-to-maturity-ytm/

Plugging in the numbers gives about 4.14% (fairly close to the yieldgimp value).

A more accurate function, yield, can be found in excel (or other spreadsheets)

Note that the maths is non-trivial, but easy enough to implement with programming skills.

Finally, yield to maturity does not imply reinvestment of coupons. However, total return does.

A simple example will show why. Imagine a 4% coupon bond bought at par.

Trivially, in this case the yield to maturity is 4%.

The annualised total return (assuming no price or yield changes and two coupons per year) is 4.04% (i.e., 1+0.02)^2

SnowMan

This should be the accurate answer for the gross redemption yield 4.26% pa. But I must have misunderstood the precise nature of the cashflows as it's different to yieldgimp which was 4.2%. The first line is what you shell out, the purchase price, and the following lines are what you get back, the coupons and redemption amount. I've left the accrued interest out of the next coupon and price but that error will be trivial.

OldScientist

SnowMan said:

This should be the accurate answer for the gross redemption yield 4.26% pa. But I must have misunderstood the precise nature of the cashflows as it's different to yieldgimp which was 4.2%. The first line is what you shell out, the purchase price, and the following lines are what you get back, the coupons and redemption amount. I've left the accrued interest out of the next coupon and price but that error will be trivial.

Return is not the same as yield (except for zero coupon bonds).

Using the excel (actually libreoffice) yield function (see https://support.microsoft.com/en-gb/office/yield-function-f5f5ca43-c4bd-434f-8bd2-ed3c9727a4fe for details and an example), we have



where annualised return=100*((1+yield/200)^2)-1)

In other words, XIRR calculates the annualised return whereas yield calculates the yield to maturity!

EthicsGradient

Odd thing about the YIELD function on LibreOffice: I'd think that, if the coupon is zero, then the frequency of payment shouldn't make a difference.
But
=YIELD("04/11/2024","31/07/2031",0,76.96,100,2,1) gives me 3.924%, while
=YIELD("04/11/2024","31/07/2031",0,76.96,100,1,1) gives me 3.962%.
Why?

To sort of answer my own question: half of 3.924 is 1.962; 1.01962 squared is 1.3962. So it's as if the function is giving the answer to "what gross rate, paid twice yearly, gives the same amount?", rather than "what AER...".

And if you do the YIELD calculation 
=YIELD("04/11/2024","31/07/2031",0.0025,76.96,100,1,1)
you get 4.255% - or 4.26%, to the 2 decimal places used until now.