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Part 3: Trading

Lesson 09

Simple Rate and Compound Rate

Suppose that you place JPY 1,000,000 one-year deposit at the bank and the agreement stipulates that the interest
rate on the deposit is 10% simple interest per annum.
This means that you will receive the following amount on the due date.
The principal:

JPY 1,000,000

Interest:

JPY 1,000,000 * 10%

Total amount:

JPY 1,000,000

JPY100,000

JPY 100,000

JPY 1,100,000

Compound interest, also called interest on interest, is applied both to the principal and to the accumulated
interest made during previous periods.
With the same example above, let’s see how the compounding frequency will affect the total amount of payment.
(Principal: JPY 1,000,000, interest rate: 10% per annum, investment period: one year)
When the interest rate is measured with semiannual compounding, it means that 5% (=10%/2) is earned every
6 months, with the interest being reinvested. Then the invested money will grow to the following amount at theend of one year.
JPY 1,000,000 * 1.05 * 1.05

JPY 1,102,500

When the interest rate is measured with quarterly compounding, then 2.5% (=10%/4) is earned every 3 months,
with the interest being reinvested. JPY 1,000,000 then grows toJPY 1,000,000 * 1

0.025 4

JPY 1,103,812

As you have seen here, the frequencies of the compounding increases, the total amount of the money at the endof the investment period also increases.
A day count convention is presented in the form of “number of days in the accrual period/number of daysin the year”. If a deposit has a 30/360 basis, it means that the number of accrued days is counted on the basis of360 days per year and 30 days per month.
For most of onshore JPY transactions, Actual/365 (A/365) methods are being applied. The Actual/365 basis
divides the actual number of accrual days by 365.

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Chapter 2 Interest Rate and Bond Market

Suppose that you borrow JPY 100,000,000 from the bank at 5% per annum (A/365) and pay it back in 124 days.
What would be the total amount that you need to pay to the bank?
The total amount to pay

100,000,000 * 1

5% * 124/365

JPY 101,698,630(Please note that the any interest amount below JPY 1 is usually rounded down to zero as you can see in the aboveexample.)

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Part 3: Trading

Lesson 10

Discount Factor and Present Value

Receiving JPY 1,000,000 now is worth more than JPY 1,000,000 after one year from now. Why? An investor
can invest the JPY 1,000,000 in one year deposit today and presumably earn an interest on it (unless the interest
rate is negative).
Future value (FV) is the amount of money which can be achieved at a given date in the future by investing themoney at a given interest rate, assuming compound re-investment of any interest payments received before theend.
Conversely, if we know the cashflow in the future (FV), we can calculate Present value (PV) of this cashflow
by “discounting” the FV with a given interest rate. This applied interest rate is called as “Discount rate”.
Q.

One year interest rate: 10% and PV is JPY 1,000,000, then what the FV in one year time will be?

A. FV

JPY 1,000,000 * 10.01

JPY 1,100,000.

The ratio of the PV to the FV is called as Discount factor (DF), so the present value can be calculated bymultiplying the future value by the relevant discount factor. For the above example,
DF

PV/FV

JPY 1,000,000/JPY 1,100,000

PVFV * DF

JPY 1,100,000 * 1.0/1.1

1.0/1.1

JPY 1,000,000

The interest rate to be applied to the present value calculation is called as the discount rate. As we shall discusslater, there are many choices for the interest rate depending on the associated credit or liquidity risks. The selectionof the interest rate is essential as the present value would be higher (lower) with the lower (higher) discount rate.
In risk-neutral valuation of derivatives, a risk-free rate (RFR) is used as the discount rate.
For USD, the rates on Treasury bills, Treasury notes, Treasury bonds or LIBOR (London Inter-Bank OfferedRate) have been thought to be the risk-free rates as many consider it extremely unlikely that the US government
or major banks will ever default and the market for these transactions are most liquid among others.
(About LIBOR and its successor, see also Lesson 13.)

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Chapter 2 Interest Rate and Bond MarketLesson 11Bond Yield（１） Characteristic of bonds
The below are the main characteristic of bonds and all these items will be taken into consideration whenevaluating the bond prices.
１．Face value (Par value)
Face value (par value) is the money amount the bond will be worth at maturity. When the bond matures,investors will receive the same amount as the face value of the bond, regardless of the price they bought it.
２．Coupon rate
Coupon rate is the rate of interest the bond issuer will pay on the face value of the bond, expressed as apercentage.
A bond can be purchased for its face value (at par), more than its face value (at a premium), or less than its
face value (at a discount), which will change the yield an investor earns on the bond.
The coupon rate is one of the methods (and the simplest method) to measure a bond yield, which is calculatedby dividing its annual total of the coupon payments by the face value of the bond.
Coupon Rate

Annual Coupon Payment / Bond Face Value

Suppose that the one-year bond has a face value of $1,000 and will pay $100 (annually), then the coupon rate
will be:Coupon Rate

$100 / $1,000

10% per annum

If the market interest rate for similar investments rises to 12%, how would the price of this bond react to thismarket change?
Buyer of the bond will still earn a coupon payment of $100 but this would be not attractive to investors who canbuy one-year bonds that pay $120 as their coupon. The price of the original bond would need go down so that theyield equals to the market level, which is 12% in this case.
The bond price$100 / 0.12

$833.33

Thus, if market interest rates rise, the bond's price would fall and the further rates rise, the higher the bond's
price will fall.

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Part 3: Trading

On the contrary if market interest rates fall, the bond's price would rise because its coupon payment is moreattractive. The further rates fall, the higher the bond's price will rise.
３．Coupon dates
Coupon dates are the dates on which the bond issuer will make interest payments. Payments can be made in
any interval, but the standard is semiannual payments (i.e., twice a year).
４．Maturity date (Redemption date)
Maturity date or redemption date is the date on which the bond will mature and the bond issuer will pay thebondholder the face value of the bond (with the final interest if any).
５．Issue price
Issue price is the price at which the bond issuer originally sells the bonds. In most cases, bonds are issued atpar but can be issued at over-par or under-par.
６．Credit Quality of the issuer
If the issuer has a poor credit rating, the risk of default is greater, and hence the investors will require the higherinterest for these bonds.
Credit ratings for a company and its bonds are generated by credit rating agencies like Standard and Poor’s,Moody’s, and Fitch Ratings.
The very highest quality bonds are called “investment grade (IG)” and include debt issued by the U.S.
government or companies with high credit quality.
Bonds that are not considered investment grade but are not in default are called “speculative grade” or “highyield (HY)”. High yield bonds are also called “junk” bonds. These bonds have a higher risk of default in the futureand investors demand a higher coupon payment to compensate them for that risk.
７．Time to maturity
Time to maturity is another main determinant of the coupon rate of the bond. Bonds with longer maturity date
usually pay a higher interest rate. This higher compensation is because the bondholder is more exposed to default
risk, interest rate, inflation risks and liquidity risk for an extended period.

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Chapter 2 Interest Rate and Bond Market（２） Current Yield
The current yield is calculated by dividing the annual coupon payment by the current market bond price. Thisis simple measure is easy to calculate but the difference between the maturity face value and the current market
price of the bond is not taken into consideration.
Current Yield

Annual Coupon Payment / Current Bond Price（３） Yield to Maturity (simple interest)
This measure is conventionally used particularly in Japanese bond market. This simple interest method is easy
to calculate but the compounding effect on the investment is not considered in its calculation. Assuming that theremaining time to maturity of the bond is n (years), the formula is as follows:
YTD simple interest

Annual total coupon

Face value – Current price /n / Current price（４） Duration and Convexity
Duration is a measure of the sensitivity of the price of a bond or other debt instrument to a change in interestrates.
Generally speaking, the higher the duration, the more a bond’s price will drop as interest rates rise (and thegreater the interest rate risk). For example, if rates were to rise 1%, a bond or bond fund with a five-year average
duration would likely lose approximately 5% of its value. Traders and investors pay close attention to duration, as
it is the most basic measure of a bond’s riskiness. The longer the duration, the more the price of the bond is likely
to fluctuate before maturity.
The rate of change of a bond’s or bond portfolio’s sensitivity to interest rates (duration) is called as convexity.

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Part 3: Trading（５） Zero Rate/Spot Rate
A zero-coupon bond is a debt security that pays no intermediate payments but instead trades at a discount,rendering a profit at maturity, when the bond is redeemed for its full-face value.
N-year spot rate or n-year zero rate is the rate of interest earned on an investment to n-year zero coupon bond(assuming that the bond is held until maturity).

（６） Bond Price
The theoretical price of a bond can be calculated as the present value of all the cash flows that will be received
by the owner of the bond.
Sometimes bond traders use the same discount rate for all the cash flows underlying a bond, but a more accurate
approach is to use a different zero rate for each cash flow.

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Chapter 2 Interest Rate and Bond MarketLesson 12

Yield Curve

Interest rates of the bonds can be different for the different maturities even if they have equal quality of credit.
Expectations theory suggests that the current long-term interest rates are determined based on the market
expectation for the future short-term interest rates. However, there are many other forces which drive the longterm interest rates. Investors might ask for “risk premium” on the bonds with longer term as there would be higher
market risks.
A “yield curve” is a line that plots interest rates of bonds having equal credit quality but different maturities.
There are three main shapes of yield curve shapes:
normal (forward yield; upward sloping curve)

inverted (reverse yield; downward sloping curve)

flatThe below is the current shape of the yield curve of Japan Government Bonds (JGB), and you can see the rates
are higher for the longer maturities (normal curve).

Figure 2 The yield curve of JGB (as of 6 May 2023)

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Part 3: Trading

As shown in the above, the yield curve is drawn against two axes, the vertical showing yield and the horizontal
giving the maturity.
The shape of the yield curve varies from day to day. If long-term interest rates rise relative to short-term interest
rates, the curve is said to steepen. Especially when the long-term interest rates rise greater than the short-term rates,
it is called as “bear steepening” and when the short-term rates fall greater than the long-term rates, it is called as“bull steepening”.
And if short-term rates rise relative to long-term rates the curve flattens (it is also classified as “bull flattening”
and “bear flattening”).

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