Short Dated Valuation

## Valuation of Short Dated Instruments

### Discount Rate Based

Treasury bills, Bank Negara bills, commercial papers and banker’s acceptances are considered discount instruments. In general, these are traded in the market using discount rate that is used for the calculation of proceed. The discount rate is not the same as the rate used in in any of our previous equation. $$r$$ was referred to as effective or true yields whereas $$r_d$$ is a discount rate.

For comparison purposes discount rate need to be converted to an effective of true yield and vice versa.

CALCULATOR: T-BILLS PROCEED

#### Example 1-3

A bank sold a 30-day Treasury bill at a discount rate of 3.00% with a face value of MYR10,000,000 for value spot. The proceed can be calculated using Equation 1-11;

\begin{align} P &= 10,000,000\left(1-\dfrac{0.03 \textrm{x} 30}{365}\right)\\ &=9,975,342.47 \end{align}

The calculator on the right can help reader to simulate the proceed of discount instrument with different tenors, rate, maturity or day count convention. 'Days To Maturity' should be less than the number of days in a year. Effective or true yield is also calculated for convenience.

The bank will receive MYR9,975,342.47 two business days from the transaction date. The effective or true yield of the T-bills can be calculated using $$P = \dfrac{FV}{\left(1+\frac{rt}{365}\right)}$$;

\begin{align} 9,975,342.47 &= \dfrac{10,000,000} {\left(1+ \dfrac{r \textrm{ x }30} {365}\right)}\\ r &= \left(\dfrac{10,000,000}{9,975,342.72}-1\right)\dfrac{365}{30} \\ r&=3.0074\textrm{%} \end{align}

The above calculation can be generalised into the following:

\begin{align} r &= \left(\dfrac{FV}{\textrm{T-bill proceed}}-1\right)\dfrac{365}{t}\\ r &= \left(\dfrac{FV}{FV\left(1-\frac{r_dt}{365}\right)}-1\right)\dfrac{365}{t} \end{align}

and finally;

\begin{align} r=\left(\dfrac{r_dt}{365-r_dt}\right)\dfrac{r_dt}{365} \end{align}

Equation 1-12

### Yield Based

Rate is a very general term that is only applicable to a simple deposit or investment such as money market borrowing and/or lending and fixed deposit. In these instances, it refers to the rate of return of the investment for the purpose of calculating interest payment on maturity of the placement.

Yield on the other hand refers to the current market rate of return and is often used in the secondary market for trading of coupon bearing instrument such as bond, a topic that will be discussed in detail in Chapter 4. A bond-like short-term deposit instrument is NIDs. When a bank issues NID to investors, the certificate will have among other things the following information;

1. Face value – the original principal amount invested;
2. Coupon rate – the periodic interest rate that will be paid. However, only one coupon payment will be made for NID with original tenor of less than a year;
3. Maturity date – the maturity of the deposit and also the payment the principal and the last coupon interest; and
4. Coupon frequency – number of coupon payments made in a year. This is not applicable to NID with original maturity of less than a year.

#### Example 1-4

A bank issued an NID maturing in 90 days bearing a coupon rate of 3.30% with a face value of MYR1,000,000 to an investor. On NID issue date, the investor paid the bank MYR1,000,000 and on maturity of the NID he is expected to receive the face value (MYR1,000,000) and coupon interest of MYR8,136.99 (0.033 x 1 million x 90/365).

After holding the NID for 20 days, the investor decided to sell the NID to another bank.

CALCULATOR: Example 1-4
##### Scenario 1

Interest rate has risen since the investor purchased the NID. The bank is quoting a yield of 4.50% to purchase the NID. Using modified Equation 1-3 for 'Actual/365 Fixed' convention;

\begin{align} \textrm{Proceed} &= \dfrac{1,000,000+8,136.99}{\left(1+\frac{0.045\textrm{ x }70}{365}\right)}\\ &= 999,511.07 \end{align}

The investor will be receiving MYR999,511.07, an amount lower than the face value of the NID.

##### Scenario 2

Interest rate has dropped since the investor purchased the NID and the bank is quoting a yield of 2.50% to purchase the NID. Using modified Equation 1-3 for 'Actual/365 Fixed' convention;

\begin{align} \textrm{Proceed} &= \dfrac{1,000,000+8,136.99} {\left(1+\frac{0.025 \textrm{ x }70}{365}\right)} \\ &= 1,003,326.52 \end{align}

The investor will be receiving MYR1,003,326.52, a higher amount than the face value of the NID.

The formula for the secondary NID proceeds is basically the same as Equation 1-3 as shown in the scenarios above. However, BNM showed a different formula in its “Guidelines on Negotiable Instrument of Deposit (2006)” which will be shown as the same formula presented in a different manner.

In both scenarios, the future value or cash flows are being discounted to the present, to obtain the current value. The future value of the NID comprises of the face value and coupon of the NID.

\begin{align} \textrm{Proceed} &= \dfrac{FV + C}{\left(1+ \frac{yt}{365}\right)}\\ \textrm{where } FV &= \textrm{face value of the NID}\\ C &= \textrm{coupon interest}\\ y &= \textrm{yield of the NID}\\ t &= \textrm{remaining tenor of the NID} \end{align}

Adjusting Equation 1-1 for the convention, the above formula can be expanded into:

\begin{align} \textrm{Proceed} &= FV\dfrac{\left(1 + \frac{rt_o}{365}\right)}{\left(1+ \frac{yt}{365}\right)}\\ \textrm{where } r &= \textrm{coupon rate of the NID}\\ t_o &= \textrm{original tenor in number of days} \end{align}

A step by step simplication of the above is shown below and reflect calculation of proceeds for short term NID as per BNM Guidelines on Negotiable Instrument of Deposit (2006).

\begin{align} \textrm{Proceed} &= FV\dfrac{\left(1 + \frac{rt_o}{365}\right)}{\left(1+ \frac{yt}{365}\right)}\\ &= FV\dfrac{\left( \frac{365 + rt_o}{365}\right)}{\left( \frac{365 + yt}{365}\right)} \end{align}

which will finally reveal the BNM equation;

$$\textrm{Proceed} = FV\left( \dfrac{365 + rt_o}{ 365 + yt}\right)$$

Equation 1-12

### Rate Conversion

Conversion rate from discount rate to effective yield or from Actual/360 to Actual/365 are done for comparability in a portfolio. Yield or effective yield mentioned earlier is also referred to as simple interest rate. Equation 1-10 introduces compounding, despite being used for termination of a deposit. Compounding is an important concept in bonds and continuous compounding are used in option pricing models.

In all cases, there is one common factor that can be used to convert the required rate type i.e. discount factor. Converting the yield or rate to compounding rate or rate with another interest convention should preserve the discount factor of the original rate. The basic formula for discount factor was provided as Equation 1-9. With 'Actual/365 Fixed' convention the discount factor in Example 1-1.

\begin{align} DF &= \dfrac{1}{\left(1+\frac{0.03\textrm{ x }30}{365}\right)}\\ &= 0.997540 \end{align}

The current value of 1 should be the same regardless of convention used. Otherwise, there will be window for arbitrage to take place. Hence an arbitrage free iterates that:

\begin{align} DF_{365F} &= DF_{360} \\ \textrm{where }DF_{365F} &= \textrm{discount factor using Actual/365 Fixed} \\ DF_{360} &= \textrm{discount factor using Actual/360} \end{align}

Expanding $$DF_{360}$$;

$$DF_{365F} = \dfrac{1}{\left(1+\frac{r_{360}t}{360}\right)}$$

will solve the rate for Actual/360 convention;

$$r_{360} = \left(\dfrac{1}{DF_{365}}-1\right)\dfrac{360}{t}$$

Equation 1-13

CALCULATOR: RATE CONVERSION

We can now feed the numbers into the equation

\begin{align} r_{360} &= \left(\dfrac{1}{0.997540}-1\right) \dfrac{360}{30}\\ &= 2.958904\textrm{%} \end{align}

The equivalent rate for 'Actual/360' convention is 2.958904% The equivalent compounding rate can obtained using the following formula:

\begin{align} DF&=\dfrac{1}{(1+r_c)^{t/365}}\\ r_c&=\sqrt[\frac{365}{t}]{\dfrac{1}{DF}} - 1 \end{align}

The compounded rate based on 30-day compounding for one year andis equivalent to 3.04168%. The figure can be checked using the calculator on the right.In practice, however, only annual, semi-annual and quarterly compounding are commonly used. Continuous compounding rate on the other hand is commonly used in option pricing models and can be calculated from the earlier discount factor using following equation.

\begin{align} DF&=e^{-r_et}\\ r_e&=-\dfrac{\textrm{ln }DF}{t} \end{align}

$$t$$, is fraction of a year (30/365) and $$r_e$$ is calculated to be 2.996307%.