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Actuary.FMExamAppl History

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June 09, 2007, at 09:47 PM by 216.80.126.250 -
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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:
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May 24, 2007, at 12:53 PM by 140.192.67.70 -
Added lines 78-82:

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:
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May 22, 2007, at 02:46 PM by 140.192.67.70 -
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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.

to:
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.
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May 22, 2007, at 02:44 PM by 140.192.67.70 -
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= sum of the differences between the coupon payments and interests.

to:

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

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

to:

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

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May 22, 2007, at 02:43 PM by 140.192.67.70 -
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amount of premium = sum of the differences between the coupon payments and interests.\\

to:

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

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where P_0 is the book price right after the last payment.

to:
where P_0 is the book price right after the last payment.
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May 22, 2007, at 02:41 PM by 140.192.67.70 -
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The initial value is P, obtained by the following

to:
  • The initial value is P, obtained by the following
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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.

to:
  • 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.
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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.

to:
  • 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.
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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

to:
  • 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.

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May 22, 2007, at 02:22 PM by 140.192.67.70 -
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Variables:

to:

Setup

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  • Value of a bond
    • The initial value is P, obtained by the following
to:

Value of a bond

The initial value is P, obtained by the following

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  • 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, which knocks down the bond price a little bit with each coupon payment.

The series of differences forms an amortization of the remium:

to:

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.

The series of differences forms an amortization of the remium:
Changed lines 66-68 from:
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:

to:

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

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

  • Price-plus-accrued of the bond is
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May 22, 2007, at 02:20 PM by 140.192.67.70 -
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Price= present value of coupons + present value of redemption value\\

to:

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

Changed lines 58-60 from:
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:
to:
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:

Changed lines 64-66 from:
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:
to:
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:

Changed lines 70-73 from:
to:
  • 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

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May 22, 2007, at 02:04 PM by 140.192.67.70 -
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Price= present value of coupons + present value of redemption value
to:

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

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

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

Changed line 65 from:
amount of discount = sum of the differences between the coupon payments and interests.
to:

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

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

to:
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May 22, 2007, at 02:03 PM by 140.192.67.70 -
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  • The initial value is P, obtained by
to:
  • 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.)

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May 22, 2007, at 01:38 PM by 140.192.67.70 -
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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
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May 21, 2007, at 02:47 PM by 140.192.196.53 -
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Bonds

to:

Bonds

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

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May 21, 2007, at 02:43 PM by 140.192.196.53 -
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Cash flow analysis

to:

Cash flow analysis

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Investment Funds

to:

Investment Funds

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Loans

to:

Loans

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Bonds

to:

Bonds

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May 21, 2007, at 02:41 PM by 140.192.196.53 -
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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

Setup: the fund is worth F_0 at time 0. For k=1, 2, \ldots, n, c_k\, amount of money is invested in the fund at time t_k\,. The fund is worth F_T\, at time T. We want to measure the rate of return on this fund. There are two methods.
  • 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\,.

When T is small, we can approximate a solution by using the binomial approximation
(1+i)^p \approx 1+pi.
The dollar-weighted rate depends on the amount of money c_k\, invested and the time t_k\, at which the investments are made .
  • 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}.
The time-weighted method minimizes the roles of the time periods between payments and the amount of payments.

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

Either method should give us the same answer, but it seems like the retrospective method is more often used.

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

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Page last modified on June 09, 2007, at 09:47 PM