# Yields Used to Value Money Market Instruments

Money market instruments are Financial instruments that mature in less than a year. Examples are Treasury bills, commercial papers, or municipal notes. Most T-bills have a maturity of either 91 days or 180 days.

Money market instruments yield measures differ  from longer-term fixed-rate bonds in the following ways:

• bond yields-to-maturity are annualized and compounded. However, yield measures in money markets are annualized but not compounded. The rate of return is stated on a simple interest basis;
• bond yields-to-maturity are calculated using the standard time value of money. In contrast, money market instruments are often quoted using nonstandard interest rates such as discount rates or add-on rates;
• money market instruments with different times-to-maturities have different periodicities. However, bond yields-to-maturity are given for a common periodicity for all times-to-maturity.

## Money Market Rates

Money market rates are quoted as discount rates or add-on rates.

### Discount Rates

Note that we conventionally define discount rate as the interest rate used to compute the present value of payment(s). However, in the money market, the discount rate is a different type of rate used to value short-term money market instruments.

### Pricing Money Market Instruments Using Discount Rates

The pricing of the money market instrument formula is given by:
$$PV=FV×(1-\frac { Days }{ Year } ×DR)$$

Where:

$$PV$$ = the price of the money market instrument (present value)
$$FV$$ = face value of the money market instrument (future value paid at maturity)
$$Days$$ = count of days between settlement and maturity
$$Year$$ = number of days in a year
$$DR$$ = annualized discount rate

#### Example: Calculating the Price of the Money Market Instrument Using Discount Rates

A 91-UK T-bill (Treasury bill)  with a face value of 20 million euros at a discount rate of 2.5%. Assuming that a year has 360 days, Calculate the price of the T-bill.

Solution

The information given in the question is as follows:

$$PV$$ = ?
$$FV$$ = 20,000,000
$$Days$$ = 91
$$Year$$ = 360
$$DR$$ = 2.5%

Now using the formula provide:

$$PV=20,000,00×(1-\frac { 91 }{ 360 } ×0.25)=19,873,611.11$$

We can also transform the  formula above to make the discount rate the subject:

$$DR=(\frac{Year}{Days})×(\frac{FV-PV}{FV})$$
The variables are similarly defined as above. However, you note that $$\frac{Year}{Days}$$ is the periodicity of the annual rate, and FV-PV  is the interest earned on the money-market instrument.

#### Example: Calculating the Money Market Discount Rate

A 91-US T-bill (Treasury bill)  with a face value of USD 5 million at a discount rate of 2.5% and a price of USD 4.9 million. Assuming that a year has 360 days, Calculate the discount rate assumed by the T-bill.

Solution

The information given in the question is as follows:

PV = 1,500,000
FV = 5,000,000
Days = 91
Year = 360
DR =?

Now using the formula provide:

$$DR=(\frac{Year}{Days})×(\frac{FV-PV}{FV})=(\frac{360}{91})×(\frac{5,000,000-4,900,000}{5,000,000})=0.07912=7.912 \%$$

The money market instrument is computed on an add-on rate basis; the interest rate is added to the principal to calculate the future value of the money market instrument.

### Pricing Money Market Instruments Using Add-On Rates

The pricing formula for the money market instruments quoted on an add-on rate basis is given by:

$$PV=\frac{FV}{(1+\frac{Days}{Year}×AOR)}$$

Where:

$$PV$$ = the price of the money market instrument (principal amount or the present value)
$$FV$$ = the redemption amount paid at maturity, including interest (future value)
$$Days$$ = number of days between settlement and maturity
$$Year$$ = number of days in the year
$$AOR$$ =  annualized add-on rate

#### Example: Calculating the Price of a Money Market Instrument Quoted on an Add-On Rate Basis

A US-based insurance company purchases a 90-day banker’s acceptance (BA) with a quoted add-on rate of 5% and a redemption value of USD 10 million. Assuming that the year has 365 days, calculate the price of the BA.

Solution

Using the information given in the question, we have:

$$PV$$ =?
$$FV$$ = 5,000,000
$$Days$$ = 90
$$Year$$ = 365

$$AOR$$ =  5%

The price of the BA is given by:

$$PV= \frac{5,000,000}{(1+\frac{90}{365}×0.05)}=4,939,106.90$$

You can also be asked to calculate the redemption value of the money market instrument when you are provided with a price (initial principal) and the variables. If we make FV the subject in the above formula, we have:

$$FV=PV+PV(\frac{Days}{Year}×AOR)$$

Looking at the resulting formula  (for FV), it is easy to see that the redemption value is an initial principal (PV) plus the interest $$PV(\frac{Days}{Year}×AOR)$$.

The interest is the principal times the product of the fraction of the year and the add-on rate.

Assume that we do not know the redemption value in our example above. Therefore:

$$FV=4,939,106.90+4,939,106.90(\frac{90}{365}×0.05)≈5,000,000$$

Needless to say, the interest earned on the BA is:

$$4,939,106.90(\frac{90}{365}×0.05)=60,893.09877$$

Additionally, we can also make AOR the subject of the formula such that:

$$AOR=(\frac{Year}{Days})×(\frac{FV-PV)}{PV}$$

If you look at the formula, the add-on rate is a more reasonable yield measure for the money market instrument as compared to the discount rate since the $$\frac{Year}{Days}$$ is the periodicity  and the  $$\frac{FV-PV)}{PV}$$ is the interest rate earned divided by the initial principal (PV). The interest earned in the discount rate is divided by the reaction value (FV), which understates the rate of return to the investor and the cost of borrowed funds to the investor since PV < FV provided that DR ≥ 0.

#### Example: Calculating the Add-On Rate

A US-based insurance company purchases a 135-day banker’s acceptance (BA) with a redemption value of USD 20 million and a price of 19,951,106.90. Assuming that the year has 365 days, calculate the rate of the BA.

Solution

Using the information given in the question, we have:

PV = 19,951,106.90
FV = 20,000,000
Days = 90
Year = 365

AOR =?

The rate of return (AOR) from the BA is given by:

$$AOR=\left(\frac{Year}{Days}\right)\times \left(\frac{FV-PV)}{PV}\right)=\left(\frac{365}{135}\right)\times \left(\frac{20,000,000-19,951,106.90)}{19,951,106.90}\right)=0.00662=0.662\%$$

### Comparison of Money Market Instruments Using Bond Equivalent Yield

Analyzing money market investment is a bit difficult because:

• some money market instruments are quoted using discount rates, and others by add-on rates;
• some assume 360 days in a year, while others assume 365 days in a year;
• the amount quoted for the discount rate face value is paid at maturity, while for the add-on rate, it is the principal (price at the issue date).

So, it is essential to compare the money market instrument discount and add-on rate on the same basis by converting one rate to another to get the bond equivalent yield, usually quoted on a 365-day add-on rate basis.

#### Example: Calculating the Bond Equivalent Yield

A 91-day commercial paper is quoted at a discount rate of 5.5% for a year assumed to have 360 days. Calculate the bond equivalent yield rate given the price of the instrument is paid 100 per face value.

Solution

We need to calculate the price of commercial paper using the discount rate. That is:

$$PV=FV×(1-\frac{Days}{Year}×DR)=100×(1-\frac{91}{360}×0.055)=98.610$$

We now need to calculate the AOR  using the formula:

$$AOR=\frac { Year }{ Days } ×\frac { (FV-PV) }{ PV }=\frac { 365 }{ 91 } ×\frac { (100-98.610) }{ 98.610 }=5.655 \%$$

So, the bond equivalent yield is 5.655%.

### Question 1

A 180-day US Treasury bill with a face value of 100 has a quoted discount rate of 4.5%. Its bond equivalent yield is closest to:

1. 3.33%
2. 4.25%
3. 4.67%

Solution

$$PV=100×(1-\frac { 180 }{ 360 } ×4.5\% ) =97.750$$

And the bond-equivalent yield for a 365-day year is 4.67%.

$$AOR=\frac { 365 }{ 180 } ×\frac { (100-97.75) }{ 97.75 } =4.67\%$$

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