FIN 300

Discounted Cash Flows
Lecture 5a

Topics Covered



  • Multiple Cash Flows
  • Annuities
  • Perpetuities
  • Growing Annuities and Perpetuities
  • Different Types of Loans

Multiple Cash Flows


  • Investments involve cash flows that are separated in time
  • We can add up the value of a series of cash flows over time
  • Each cash flow must be correctly discounted

Example

$\$100$$+\$100$$=\$200$

Instead (Assume r = 1%)

$\$100$$+\dfrac{\$100}{(1+0.01)}$$=\$199.01$

What if we received $\$100$ for 6 years starting next year?

$PV = \dfrac{\$100}{(1+0.01)^1} + \dfrac{\$100}{(1+0.01)^2} + \dfrac{\$100}{(1+0.01)^3} \\ \space \space \space \space \space \space \space \space + \dfrac{\$100}{(1+0.01)^4} + \dfrac{\$100}{(1+0.01)^5} + \dfrac{\$100}{(1+0.01)^6} $

$=\$579.55$

This is an example of an annuity

Annuities


  • A stream of cash flows over a finite period
  • Fixed annuities have a level stream
      $CF_1=CF_2= \dots =CF_t$


$PV_{Annuity} = CF \times \left\lbrace \dfrac{1-[1/(1+r)^t]}{r} \right\rbrace$
Example

What if we received $\$100$ for 6 years starting next year?


$PV_{Annuity} = \$100 \times \left\lbrace \dfrac{1-[1/(1+0.01)^6]}{0.01} \right\rbrace$

$ = \$579.55$
Example

What if we received $\$100$ for 600 years starting next year?


$PV_{Annuity} = \$100 \times \left\lbrace \dfrac{1-[1/(1+0.01)^{600}]}{0.01} \right\rbrace$

$ = \$9,974.46$

What would the value be if this continued forever?

$$ \lim_{t \to \infty} CF \times \left\lbrace \dfrac{1-[1/(1+r)^t]}{r} \right\rbrace$$

$ = \dfrac{CF}{r}$

Perpetuities

  • Eternal annuities
  • Not typically observed in practice
  • An approximation for a long term annuity
  • The math will be reused for stock valuation


$PV_{Perpetuity} = \dfrac{CF}{r}$
Example

What is the present value of receiving $\$100$ every year forever, starting one year from today? Assume $r=1\%$.



$PV_{Perpetuity} = \dfrac{\$100}{0.01}$$=\$10,000$

Growing Annuities and Perpetuities


  • What happens if the payments increase each year?
  • We can adapt the formula to growth:

$PV_{\text{Growing Annuity}} = CF \times \left\lbrace \dfrac{1-\left(\dfrac{1+g}{1+r}\right)^t}{r-g} \right\rbrace$

  • This work for perpetuities as well:

    $PV_{\text{Growing Perpetuity}} = \dfrac{CF}{r-g}$


  • For this to work:   $r>g$

Different Types of Loans




  • Pure Discount
  • Interest Only
  • Amortized Loan

Summary



  • PV of a cash flow series
  • Annuities and Perpetuities
  • Growing Annuities and Perpetuities
  • Different Types of Loans




Next Topic:

Interest Rates and Inflation