Zero-coupon securities - as it says on the tin - do not have a coupon attached Interest is not 'added on' Instead, at maturity, the investor receives the face value Therefore, in order to receive a return, she must pay less than the face value How much less can be calculated in one of two ways: Yield or Pure Discount Rate

1. Zero-Coupon
Yields and Pure Discount Rates
References:
Lex van der Wielen: A Guide to Treasury in Banking
ICMA-European-Repo-Council-Guide-to-Best-Practice-27-July-2015-FINAL.pdf

2. Contents
 Zero-Coupon? Yield? Pure Discount?
 OK I get it. Big deal - Why do I care?
 The principles of Finance
 The ‘Rocket Science' - convert between Yield and Pure Discount Rates
 Example: U.S T-Bill
 Who uses these ‘Pure Discount Rates’?
 Extra Credit: Repos - description
 Extra Credit: Repos – calculation
 Extra Credit: Repos - question & solution

3. Zero-Coupon? Yield? Pure Discount?
 Zero-coupon securities - as it says on the tin - do not have a coupon attached
 Interest is not 'added on'
 Instead, at maturity, the investor receives the face value
 Therefore, in order to receive a return, she must pay less than the face value
 How much less can be calculated in one of two ways:
 1) Yield
A rate which is applied to the PV (present value) i.e. the purchase price, in
order to arrive at the face value
 2) Pure Discount rate
A rate which is applied to the FV (Future Value) i.e. the face value in order to
determine the purchase price

4. OK I get it. Big deal - Why do I care?
 1) You want to invest in short-term securities priced by different
methods.
E.g. You are considering U.S and U.K commercial paper of equivalent
credit rating and risk. You need to compare ‘apples with oranges’
 2) The same idea applies to other valuations as well.
For example, Repo initial margin/haircut calculation

5. The principles of Finance
 1) The Time Value of Money
A dollar is worth more in my hand today than tomorrow because I can
earn interest - even overnight (assuming interest rates are not negative)
 2) No Arbitrage
A fair price for a security must not allow for arbitrage
For example, the return on a zero-coupon security calculated by Yield or
by Pure Discount rate, must be the same
If it were otherwise, I would buy the security priced by the cheaper
method and sell by the dearer method

6. The ‘Rocket Science'
Convert between Yield and Pure Discount Rates
Given that we assert that the yield and pure discount rates represent the return on
a security and - although the rates are different – the return must be the same (no
arbitrage) then there must be a conversion formula
yield discount
----- equals -------- (note the difference in sign)
1 + yield 1 - discount
0.2 (20%) 0.1667 (16.67%)
---------- = 0.1667 (16.67%) --------------- = 0.2 (20%)
1 + 0.2 1 - 0.1667
£ 100 X 1 + 0.2 = £ 120 £ 120 X 0.8333 = £ 100

7. Example: U.S T-Bill
Let’s say we buy USD 10 million of on-the-run U.S T-bills at issue
If the discount rate is 2.38% then the price we pay is
USD 10,000,000 X 1 – 0.0238 = USD 9,762,000
The equivalent price by the yield calculation
Yield = Discount rate / 1 - Discount rate = 0.0238 / 1 – 0.0238 = 0.02438 (2.438%)
Price = 10,000,000 / 1 + 0.02438 = USD 9,762,000

8. Who uses these ‘Pure Discount Rates’?
 Pure discount rates are only used for pricing securities in U.S Money Markets
(generally, securities with a maturity of less than one year)
 These securities include Treasury bills, U.S commercial paper, Bills of exchange,
Bank Bills
They do not include CDs (certificates of deposit) as these have interest added
on to the face value
 Everything else uses yield
Including U.K Gilts, German Bunds, Japanese commercial paper etc.. etc..

9. Extra Credit: Repos - description
 When two parties enter into a Repo (Repurchase agreement) one - the seller -
offers securities as collateral in order to borrow cash. At a later date the
securities will be returned to the seller and the cash repaid to the buyer
 In most cases the purchase price of the securities will not be valued at market
value (dirty price) but at a lower value because of the liquidity risk.
They may not be able to be sold at face value in a credit event
 The difference can be calculated in two ways:
'Haircut' or 'Initial Margin‘ (aka Margin Ratio)
 Haircut is the discount applied to the market value of the securities to
determine the purchase price of the repo
 Initial Margin is the premium applied to the purchase price to arrive at the
market value
So, as in the previous example we can think of the purchase price as the PV and the
market value as the FV
Also, we can think of the Initial Margin as the Yield and the Haircut as the Pure
Discount Rate

10. Extra Credit: Repos - calculation
Example: MV = 20,000,000. Initial Margin = 2%
Purchase Price = 20,000,000 / 1 + 0.02 = 19,607,843
What rate of Haircut would be equivalent?
MV = 20,000,000. Haircut = 2% (remember we multiply by the discount)
Purchase Price = 20,000,000 X 1 - 0.02 = 19,600,000 - Oops!
Remember it's how much do we discount the FV (market value) by to get the PV
(purchase price)
Convert the Initial Margin (Yield) into a Haircut (Pure Discount Rate)
Yield 0.2
--------- = ------- = 0.01960784
1 + Yield 1.02
MV = 20,000,000. Haircut = 1.96%
Purchase Price = 20,000,000 X 1 - 0.0196 = 19,607,843

11. Extra credit: Repos - question & solution
 Question
A repo in French government bonds is quoted 0.67-71% for 10 days with a required initial margin of 0.2%
The dirty price of the bonds is 103
The face value of the bonds is EUR 20 million
We wish to buy the Repo. What is the purchase price? What collateral is demanded? What is the repurchase price?
 Solution
If the face value is 20 million then the dirty price is 20 million X 103/100 = 20,600,000, This will be the collateral
The initial margin is 0.2% so the purchase price = 20,600,000 / 1 + 0.002 = 20,558,882.24
This is the amount of cash that will be lent in exchange for the securities
 Check: what if we had been quoted a haircut?
The equivalent haircut = margin / 1 + margin = 0.002 / 1 + 0.002 = 0.001996 (0.1996%)
Purchase price = 20,600,000 X 1 - 0.001996 = 20,558,882
 Now we have to be careful
We have received a quote therefore we are dealing with a market maker so we always get the worst price of the spread,
whichever way round they are quoted ask-bid (U.K) or bid-ask (rest of the world)
 But who is lending the securities and who is lending the cash?
In repos the party offering the securities is known as the seller and the party offering the cash is known as the buyer
Therefore the market maker is lending the securities and we are lending the cash so we get the lowest rate = 0.67%
Another way to remember this is that generally the buyer of a security is the one who receives the fixed rate so in this case we
are receiving the fixed rate of interest on the loan of cash
 So we just have to calculate the interest due on the purchase price (cash lent) for 10 days, right?
Hmm, one more little problem. What day count convention should we use?
The bonds are denominated in Euros so it will be ACT/360 (If they were denominated in Sterling it would be ACT/365)
 So the repurchase price (cash lent + interest) = 20,558,882 X 1 + (10/360) X 0.0067 = EUR 20,562,708