• Chapter 1
  • 1.2 Interest
  • 1.3 Accumulation and amount functions
  • 1.4 Simple interest / Linear Accumulation Functions
  • 1.5 Compound interest (the usual case)
  • 1.6 interest in advance / the effective discount rate
  • 1.7 Discount function / The Time value of money
  • 1.8 Simple Discount
  • 1.9 Compound Discount
  • 1.10 nominal rates of interest and discount
  • 1.11 Friendly competition(constant force of interest)
  • 1.12 force of interest

Chapter 1

1.2 Interest

• Interest, an investment amount $K grow to an amount $S over a period, the interest is the difference $K - $S.
• Charging of interest is based on the economic productivity of capital .

1.3 Accumulation and amount functions

• Unit of money : dollar, euros, yen, gold pieces.

• Principal the investment amount $K of money, or the amount of money borrowed.

• Unit of time : year, month, day, quarter. Time t = 0, is the initial financial transaction, meaning $K is invested at t = 0.

NOTE: $A_K (t) = K a(t), A_K (0) = K, a(0) = 1$

• $A_K(t)$ and $a(t)$ are most often non - decreasing functions, if the interest is only paid at the end of each period, then amount functions are step functions.

An investment of $\$1000$ grows by a constant amount of $\$250$ each year for five year.
$A_1000(t) = \$1000 + \$250 t$.

interest paid continuously	interest paid at the end of every year

Effective Interest Rate for the internal $[t_1, t_2]$:

1.4 Simple interest / Linear Accumulation Functions

• $A_K(t) = K(1 + st)$, is the amount function for $\$K$ invested by simple interest at rate s. $a(t) = 1 + st$ , is the simple interest accumulation function at rate s.

• Decreasing sequence, converge to 0, rarely used for loans of long duration.
  
  

• Exact simple interest, actual/actual method, exact days/ days in a year.

• leap year, can divided by 4, if it can be divided by 100, it need to be divided by 400 to be a leap year.

• Ordinary simple interest, 30/360 rule.

• pretend each month has 30 days and each year has 360 days.

• Banker’s rule, actual/360, hybrid, exact days / 360

A borrows $5000 from B on October 14, 1998 at 8% exact simple interest and repay the loan on May 7, 1999. What is the amount of A’s repayment?

1.5 Compound interest (the usual case)

• want the effective interest rate for the n-th period to be independent of $n$, $i = i_1 = a(1) - 1$.

• No advantage nor disadvantages to close and instantly re-open the account.

• Compound interest accumulation function at interest rate $i : a(t) = (1 + i)^t, for ~ all ~ t \ge 0$.

• Greatest integer function， $[t]$, greatest integer $\le t$, $f[t]$ is a floor function.

deposit $12000 at bank, Money receive after 6.5 years is,

• Fiscal policy refers to the governor’s decisions about spending and taxation. Governments spend more will increase the money available to consumers, drives interest down.
• Monetary policy, regulation of the money supply and interest rates by a central bank. Federal Reserve (directly control of rates in the Unite States).
• Federal Funds Rate, charged for inter bank with U.S. Treasury Securities.
• Prime rate, a bank charges to its customer, Federal reserve has less direct influence.
  

1.6 interest in advance / the effective discount rate

• Discount rate, investor lends $K for one period at a discount rate D, then the borrower will have to pay $K * D in order to receive the used of $K. The quantity $KD is called the amount of discount.

• Effective discount rate,

• Equivalent rate,
  $i_n=\frac{d_n}{1-d_n}$, $d_n=\frac{i_n}{1+i_n}$, $(1+i_n)(1-d_n)=1$.

growth of money is governed by the accumulation function a(t) = (1.05)^(t/2) (1+ 0,025t). find d₄ and i₄ .

$a(4) = (1.05)^2 (1.1), a(3) = (1.05)^{\frac{3}{2}} (1.075)$
$d_4 = a(4) - \frac{a(3)}{a(4)}, i_4 = a(4)-\frac{a(3)}{a(3)}$.

1.7 Discount function / The Time value of money

1. discount function, $v(t)=\frac{1}{a(t)}$

2. discount factor , compound interest accumulation function $a(t) = (1+i)^t$, then we define discount factor $v=\frac{1}{1+i}$.
   

3. present value, present value of $L to be received in year t
   

4. Net Present Value, sequence of investment returns received at time 0, $t_1, t_2, t_3, \ldots, t_n$.
   use calculator, enter CASH FLOW value then press NPV
   
   

1.8 Simple Discount

1. Simple discount, when $v(t) = mt + b$ , $v(t)$ is linear.
2. increasing function with asymptote $t = \frac{1}{d}$, there fore the discount not he interval $[0, \frac{1}{d_1})$.

1.9 Compound Discount

effective discount rate,
$d=1-v, v=1-d$, $a(t)=(1+t)^t=(1-d)^{-t}$.

For borrower, choose low annual effective interest rate, low annual effective discount rate.

1.10 nominal rates of interest and discount

• bank credit interest more than once per year, m times per year.

• nominal (annual) interest rate of $i^{(m)}$, convertible, compounded, or payable M times per year.

• i is APY, annual percentage yield.

3. nominal (annual) discount rate of $d^{(m)}$, convertible, compounded, or payable M times per year.
   
   

$d^{(m)}= \frac{i^{(m)}}{1+ \frac{i^{(m)}}{m}}$

$d = 1- (1- \frac{d^{(m)}}{m})^{m}$

$i^{(m)}= \frac{d^{(m)}}{1- \frac{d^{(m)}}{m}}$

$i = (1+ \frac{i^{(m)}}{m})^{m}-1$

2nd-ICONV to calculate NOM and EFF

1.11 Friendly competition (constant force of interest)

1. force of interest:
   and , and
   

1.12 force of interest

1. force of interest