FIN 300


Time and Money

Lecture 4

Topics Covered



See Textbook Chapter 5
  • Present Value, Future Value and Interest
  • Compounding
  • Compound and Simple Interest
  • Discount Rates

Present Value, Future Value and Interest


  • Fundamental trade-off
  • The interest rate is the price of borrowing
  • For loans, present and future values can be related through the interest rate
  • One period loan:

$\text{Present Value} \times (\text{1 + Interest Rate})= \text{Future Value}$

$ \$100 \times (\text{1 + 10%})= \$110 $
  • We can rearrange our equation to isolate present value
$\text{PV} = \dfrac{\text{FV}}{\text{1 + r}}$

$\$100 = \dfrac{\$110}{(1+ 10 \% )} $

Compounding

  • What happens if the loan continues for multiple periods?
  • Imagine the first year has passed and we already have $\$110$.
  • What would the value be after one more year?

$ \text{V}_2 = \text{V}_1 \times (1+r)$

$ \text{V}_2 = 110 \times (1+0.1)$

$=\$121$
We could find the third year by multiplying 121 by (1+r)
  • Year by year calculations are not efficient
  • We can make this easier by spotting a pattern
  • $ \text{V}_1 = \text{V}_0 \times (1+r)$
  • $ \text{V}_2 = \text{V}_1 \times (1+r) $ $= \text{V}_0 \times (1+r) \times (1+r) $
  • $ \text{V}_2 = \text{V}_0 \times (1+r)^2 $

$ \text{FV}_t = \text{PV}_0 \times (1+r)^t $

Compound and Simple Interest


  • As the investment grows, the total interest payment increases
  • We can dissect the total (compounded) interest:

      Compound Interest = Simple Interest + Interest on Interest

  • Simple Interest: amount paid on the original principal
  • Interest on Interest: amount paid on accrued interest
  • Total interest payment: Future Value - Principal
  • $ \text{FV}_t =$ Principal $\times (1+r)^t $
  • $ \text{FV}_t - $Principal $= $Principal$ \times (1+r)^t -$ Principal
  • Total Interest $=$ Principal $\times[(\text{1+r})^t-1]$

  • Simple interest is based only on the principal
      Simple Interest = Principal $\times rt$
How do we calculate interest on interest?

  • We know:
    $\text{Total Interest} = \text{Principal}\times[(\text{1+r})^t-1]$
    and
    $\text{Simple Interest} = \text{Principal}\times[rt] $
  • Interest on Interest = Total Interest - Simple Interest
  • Interest on Interest = $\text{Principal}\times[(\text{1+r})^t-1] - \text{Principal}\times[rt] $

  • $\text{Interest on Interest} = \text{Principal} \times [(\text{1+r})^t-1-rt] $

Discount Rates

  • Sometimes we may know the present and future values
  • From this, we can figure out an implied rate of return
  • This is known as the discount rate
  • Calculating discount rates requires solving for $r$:

$ \dfrac{\text{FV}}{(1 + r)^t}=\text{PV} $

To get the discount rate, isolate $r$

$ \dfrac{\text{FV}}{(1 + r)^t}= \text{PV} $ [multiply by $(1+r)^t$ and divide by PV]

$\dfrac{\text{FV}}{\text{PV}} = (\text{1 + r})^t$ [raise to exponent $1/t$]

$\left(\dfrac{\text{FV}}{\text{PV}}\right)^{1/t} = (\text{1 + r})^{t/t}$ $= 1+r$ [subtract $1$]



$ \left(\dfrac{\text{FV}}{\text{PV}}\right)^{1/t}-1 = r$
Example

    Suppose: $PV=10$, $FV=45$, and $t=10$
    $r=?$
$\left(\dfrac{\text{FV}}{\text{PV}}\right)^{1/t}-1 = r$

$\left(\dfrac{45}{10}\right)^{1/10}-1 = r$$=0.1623$

Summary



$\text{PV} = \dfrac{\text{FV}}{(1 + r)^t}$