Interest-Discount Relationships¶

The relationship between interest rates and discount rates can be expressed with a variety of equations. One thing to keep in mind is that if we borrow a dollar at time $t_1$ a discount rate of $d$ , we will receive $(1-d)$ dollars today.

If we were to hold invest that dollar for a year at the interest rate $i$ , it would grow to:

$(1 -d)(1 + i) = 1.$

This relationship can be generalized to apply between two time periods, $t_1$ and $t_2$ :

$1 = (1 + i_{[t_1, t_2]})(1 - d_{[t_1, t_2]}).$

In the age of hand calculations, several other equations have been useful:

$i_{[t_1, t_2]} &= \frac{d_{[t_1, t_2]}}{1-d_{[t_1, t_2]}}\\ d_{[t_1, t_2]} &= \frac{i_{[t_1, t_2]}}{1 + i_{[t_1, t_2]}}\\ 1 &= (1 + i_n)(1 - d_n)\\ i_n &= \frac{d_n}{1 - d_n}\\ d_n &= \frac{i_n}{1 + i_n} \\ i &= \frac{d}{1-d} \\ i &= \frac{1}{1-d} - 1 \\ d &= \frac{i}{1 + i} \\ d &= 1 - \frac{1}{1 + i}$

Examples¶

TmVal’s Rate class provides a built-in method to convert interest rates to discount rates and vice-versa. These are simple functions, but are very useful as they tend to be embedded in more complex financial instruments.

Suppose the interest rate is 5%, what is the discount rate?

First, we define a compound effective rate using the Rate class. Then, we use the method convert_rate() to convert the rate to a discount rate:

In [1]: from tmval import Rate

In [2]: i = Rate(.05)

In [3]: d = i.convert_rate(
   ...:    pattern='Effective Discount',
   ...:    interval=1
   ...: )
   ...: 

In [4]: print(d)
Pattern: Effective Discount
Rate: 0.04761904761904767
Unit of time: 1 year

Again using convert_rate(), we can convert the discount rate back to an interest rate:

In [5]: from tmval import Rate

In [6]: i = d.convert_rate(
   ...:    pattern='Effective Interest',
   ...:    interval=1
   ...: )
   ...: 

In [7]: print(i)
Pattern: Effective Interest
Rate: 0.050000000000000044
Unit of time: 1 year