Derivatives Basics

This is when we trade financial instruments whose prices are based on prices of other things. The three kinds of derivates can be summarized in six positions:

• long positions: Someone in these positions benefits from the price going up.
• short positions:

= sum of the differences between the coupon payments and interests.\\

amount of discount
= sum of the differences between the coupon payments and interests.\\

amount of premium
= sum of the differences between the coupon payments and interests.

• The initial value is {$P$}, obtained by the following

• The book value of the bond (right after a coupon payment at any time) is obtained by doing the same thing with the remaining coupons.

Think of a bond as the bond issuer trying to pay back a loan to the bond holder. The loan interest rate is the yield rate, the loan payments are the coupons and the redemption payment at the end.

• When a bond is bought at a premium, the price is larger than the redemption amount, so that the coupon rate is higher than the yield rate. So each coupon payment is bigger than the interest earned, which knocks down the bond price a little bit with each coupon payment.

• When a bond is bought at a discount, the price is lower than the redemption amount, so the coupon rate is lower than the yield rate. Each coupon payment is lower than the interest payment. That leaves a little bit of interest to accumulate in the bond price.

• Price {$P_t$} at a fractional {$t$} periods after a payment is determined by two methods:
  • Price-plus-accrued of the bond is

{$$ P_0(1+j)^t, $$} where {$P_0$} is the book price right after the last payment.

• Price is

{$$ P_0(1+j)^t-t(Fr) $$} Note this works for {$t=1$} as well.

Setup

Value of a bond

The initial value is {$P$}, obtained by the following

The book value of the bond (right after a coupon payment at any time) is obtained by doing the same thing with the remaining coupons.

When a bond is bought at a premium, the price is larger than the redemption amount, so that the coupon rate is higher than the yield rate. So each coupon payment is bigger than the interest earned, which knocks down the bond price a little bit with each coupon payment.

When a bond is bought at a discount, the price is lower than the redemption amount, so the coupon rate is lower than the yield rate. Each coupon payment is lower than the interest payment. That leaves a little bit of interest to accumulate in the bond price.

Price {$P_t$} at a fractional {$t$} periods after a payment is determined by two methods:

• Price-plus-accrued of the bond is

Price= present value of coupons + present value of redemption value

When a bond is bought at a premium, the price is larger than the redemption amount, so that the coupon rate is higher than the yield rate. So each coupon payment is bigger than the interest earned, which knocks down the bond price a little bit with each coupon payment.

The series of differences forms an amortization of the remium:

When a bond is bought at a discount, the price is lower than the redemption amount, so the coupon rate is lower than the yield rate. Each coupon payment is lower than the interest payment. That leaves a little bit of interest to accumulate in the bond price.

The series of differences form an accumulation of discount:

• Price {$P_t$} at a fractional {$t$} periods after a payment is determined by two methods:
  • Price-plus-accrued of the bond is

{$$ P_t = P_0(1+j)^t, $$} where {$P_0$} is the book price right after the

Price= present value of coupons + present value of redemption value\\

amount of premium = sum of the differences between the coupon payments and interests.\\

amount of discount = sum of the differences between the coupon payments and interests.\\

• The initial value is {$P$}, obtained by the following

{$$Price= present value of coupons + present value of redemption value$$}

• The book value of the bond (right after a coupon payment) is obtained by doing the same thing with the remaining coupons.

When a bond is bought at a premium, the price is larger than the redemption amount, so that the coupon rate is higher than the yield rate. So each coupon payment is bigger than the interest earned. The series of differences forms an amortization of the premium:

{$$amount of premium = sum of the differences between the coupon payments and interests.$$} (Take difference to be positive, i.e., coupon - interest).

When a bond is bought at a discount, the price is lower than the redemption amount, so the coupon rate is lower than the yield rate. Each coupon payment is lower than the interest payment. The series of differences form an accumulation of discount:

{$$amount of discount = sum of the differences between the coupon payments and interests.$$} (Take interest - coupon.)

Variables:

• Face amount (par amount) {$F$} vs. redemption amount {$C$} vs. price of bond {$P$}. Usually {$C=F$}. The bond is bought (sold) at par if {$P=C$}; at a premium if {$P>C$}; at a discount if {$P<C$}.
• Coupon rate {$r$} vs. yield rate {$j$} (usually a semiannual effective rate).
• Value of a bond
  • The initial value is {$P$}, obtained by

Bonds

[FIGURE OUT HOW TO PRICE BOND BETWEEN COUPON PAYMMENTS!]

Cash flow analysis

Investment Funds

Loans

Bonds

FM Exam Application

Cash flow analysis

• The net present value is

NPV = present value of inflow - present value of outflow

• The internal rate of return is the interest rate that gives an NPV of 0.

Investment Funds

• The dollar-weighted rate of interest is the interest {$i$} computed via the following

accumulated value of the initial value of the fund
+ accumulated values of each payment
= the value of the fund at time {$T$};
{$$ F_0(1+i)^T +c_1(1+i)^{T-t_1} +c_2(1+i)^{T-t_2} +c_3(1+i)^{T-t_3} +\cdots+c_n(1+i)^{T-t_n} =F_T\,.$$}

{$$ (1+i)^p \approx 1+pi. $$}

• We can also find the time-weighted rate of interest. Here we no longer need to know the {$t_k$} values, but we do need to know the values of the fund {$F_{k}\,$} right before each investment {$c_{k}\,$} is added. The interest {$i$} is computed via the following formula

{$$(1+i)^T = \frac{F_1}{F_0} \cdot \frac{F_2}{F_1+c_1} \cdot \frac{F_3}{F_2+c_2} \cdot \cdots \cdot \frac{F_T}{F_n+c_n}.$$}

Loans

Two different ways to pay back a loan.

Amortization method

• Part of each payment is used for the interest and the rest goes to reducing the loan principal.
• The interest portion will decrease over time because the principal is reduced over time.
• The present value of the loan equals the present value of the loan payments, which form an annuity.
• To figure out how much of the loan is outstanding, use one of the following two methods:
  • The prospective method

remaining value of the loan
= future value of the total loan - future values of the payments so far

• The retrospective method

remaining value of the loan
= present value of payments not yet submitted

Sinking fund method

• A payment consists of the service portion and the sinking fund portion.
• The service portion is the interest of the loan; it stays constant throughout because the principal also stays constant and isn't paid off until at the very end.
• The other portion goes into a sinking fund that may or may not accrue interest at the same rate as the loan. The future value of the series of payments should equal the value of the loan.

Relationship

• When the two interest rates are the same in the sinking fund method, the payments are the same as in the amortization method.

{$$ \frac{1}{a_{\overline{n}|}} = i + \frac{1}{s_{\overline{n}|}}$$}

Bonds