Subject 4. Different Yield Measures of a U.S. Treasury Bill

Money market instruments are low-risk, highly liquid debt instruments with a maturity of one year or less. There are two types of money market instruments: interest-bearing instruments (e.g., bank certificates of deposit), and pure discount instruments (e.g., U.S. Treasury bills).

Pure discount instruments such as T-bills are quoted differently than U.S. government bonds. They are quoted on a bank discount basis rather than on a price basis:

  • rBD = the annualized yield on a bank discount basis
  • D = the dollar discount, which is equal to the difference between the face value of the bill, F, and its purchase price, P
  • t = the number of days remaining to maturity
  • 360 = the bank convention of the number of days in a year.

Bank discount yield is not a meaningful measure of the return on investment because:

  • It is based on the face value, not on the purchase price. Instead, return on investment should be measured based on cost of investment.
  • It is annualized using a 360-day year, not a 365-day year.
  • It annualizes with simple interest and ignores the effect of interest on interest (compound interest).

Holding period yield (HPY) is the return earned by an investor if the money market instrument is held until maturity:

  • P0 = the initial price of the instrument
  • P1 = the price received for the instrument at its maturity
  • D1 = the cash distribution paid by the instrument at its maturity (that is, interest).

Since a pure discount instrument (e.g., a T-bill) makes no interest payment, its HPY is (P1 - P0)/P0.

Note that HPY is computed on the basis of purchase price, not face value. It is not an annualized yield.

The effective annual yield is the annualized HPY on the basis of a 365-day year. It incorporates the effect of compounding interest.

Money market yield (also known as CD equivalent yield) is the annualized HPY on the basis of a 360-day year using simple interest.

Example

An investor buys a $1,000 face-value T-bill due in 60 days at a price of $990.

  • Bank discount yield: (1000 - 990)/1000 x 360/60 = 6%
  • Holding period yield: (1000 - 990)/990 = 1.0101%
  • Effective annual yield: (1 + 1.0101%)365/60 - 1 = 6.3047%
  • Money market yield: (360 x 6%)/(360 - 60 x 6%) = 6.0606%

If we know HPY, then:

  • EAY = (1 + HPY)365/t - 1
  • rMM = HPY x 360/t

If we know EAY, then:

  • HPY = ( 1 + EAY)t/365 - 1
  • rMM = [(1 + EAY)t/365 - 1] x (360/t)

If we know rMM, then:

  • HPY = rMM x (t/360)
  • EAY = (1 + rMM x t/360)365/t - 1

User Contributed Comments 27

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Aimy: what is "CD"?
tony1973: certificate of deposit
hrai1: how do you calculate money market yield from HPY?
chandos: MMY = HPY * 360/N
hagi10: how do you do all these on the calculator?
Shyam17: Hey, How to remember so many formulas???
nsmwaura: Hummm buffles me too
yly13: MM = "simple" annualized HPY,
EAY = "compounded" annualized HPY,
HPY = simple ratio
0is4eva: The rMM formula:

Recall that rBD = D/F * 360/t ==> D = F * rBD * t / 360

Start with F / (F - D), end value through start value, and deduct 1 (to get the "interest rate"):

F/(F - D) - 1 = F/(F-F*(rBD*t/360)) - 1 = abbreviate F = 1/(1-(rBD*t/360)) - 1
= 360/(360 - rBD*t) - 1 =
= 360 - (360 - rBD*t) / (360 - rBD*t) =
= rBD * t / (360 - rBD * t)

Simple! (But not immeiately apparent, at least not to me).
raychow: Money market yield is annualized HPY
could it be 1.0101%*6=6.0606% ?
it's easier, isn't it?
laozi: 0is4eva, your rMM formula was great except you need to add one more step, that is, you need to annualize the interest rate on a 360 day basis. Basically, you just need to multiply your formula by 360/t to arrive at (360 * rBD)/(360 - rBD * t).
surob: Laozi, raychow is right. He is using this formula:
Rmm = HPY*360/t = 1.0101*360/60 = 6.0606%
epizi: sometimes that you may not have to remember many formulae you may need to know just the main fomular and do the conversions when necessary
jjack: i took the exam before,as i knew, this part is pretty important...
hillrat: why doesn't their MM yield work right I get 1.2
hillrat: never mind Order of Operations. They should right it like (360 * BDY)/ (360-(60 * 6%))
ASADRANA: OH! I JUST CANT REMEMBER A LOT OF FORMULAS!!!!
Tommytang: I am definitely gonna bomb this part :(
Mosobalaje: This is one of the few things you need to cram. The way I learnt this is:

BDY - I call this the BaD Yield - Because it has 3 things wrong with it. 1. It uses a 360 day year, 2. Simple interest and 3. Face value in the denominator.

MMY - Is pretty bad as well but not as bad as the "BaD Yield". The only thing it does better is that it uses the FV - Discount in the denominator. Other than that, they are pretty much the same yield.

EAY - Further builds on the MMY and corrects 2 more of the issues. 1. It annualizes the HPY using compounding (1+HPY)^365/t instead of just multiplying by 360/t. And 2. it uses 365 days instead of 360.

So BDY - Bad Yield, MMY - Better than BDY but only cos of denominator and EAY is flawless.
samuelong: Try this:

Since BDY = (F - P)/F * 360/t
Where F = Face Value, P = Price and t = time,

and

Since Money market yield is simply = (F - P)/P * 360/t

then

BDY = Rmm * F/P
johntan1979: You guys GOT to be joking right? This is only Study Session 2, Reading 6.

I'm not a fan of memorizing formulas myself, but please DO NOT complain about too much to remember even before reaching the halfway point, which is Reading 34: Accounting Shenanigans
Rachelle3: for EAY I kept getting this wrong and could not understand why until I undid % to calculate and worked out (the power to) 1st I like to choose up to 3 decimal points.
1st work out 365 divided by 60 = 6.083
(1 + 0.010101) ^ 6.083 then - 1 = 0.063043618
after this 0.063043618 * 100 = 6.3043618 or 6.3047% rounded up I hope this helps
Rachelle3: tommytang read mine
Rachelle3: I am close because I followed the equation but I made a mistake at last part I am unsure why my calculation is not exact but I understand this formula can someone correct my method so I am exact?
This is the yield I can never get 100% correct in calculation but I find it easier to remember than the rest.
darryl: this part of quants is abit confusing but if you are able to understand what you are doing then you will love it,

money market instruments are two of kinds ( interest bearing and non-interest)
1- for non-interest bearing instruments such as T-bills
a- we use bank discount yield.
2- for interest bearing instruments such as CD's
a- we use Holding period yield

there are two ways to anualize the holding period return:
i- Effective Annual Yield; 365 days + compounding interest
ii- Money Market Yield; 360 days + simple interest
Joshua3964: In the text, money market yield is the annualized HPY (not the annualized BDY). The formula given is MMY = (HPY)(360/t). (See textbook Reading 7, Question 15) . Is this equal to the formula given in the notes above (i.e., am I missing something obvious), or is this a mistake in the notes?
931129: Read & Understand when brain is fresh.