Future value & compounding frequencies

Bond Valuation and Analysis in Python

Joshua Mayhew

Options Trader

Compound interest with multiple cash flows

USD 1,000 deposit
3% interest rate paid monthly
USD 100 top ups (extra deposits) at the end of each month
How much do we have after 3 months?

Compound interest with multiple cash flows

deposit_fv = 1000 * (1 + 0.03) ^ 3

topup_1_fv = 100 * (1 + 0.03) ^ 2

topup_2_fv = 100 * (1 + 0.03) ^ 1

topup_3_fv = 100

print(deposit_fv + topup_1_fv + topup_2_fv + topup_3_fv)

1401.82

print(deposit_fv + topup_1_fv + topup_2_fv + topup_3_fv - 1000 - 100 - 100 - 100)

101.82

The future value function

Previous approach can get very repetitive
NumPy Financial can help simplify these calculations

import numpy_financial as npf

?npf.fv

Signature: npf.fv(rate, nper, pmt, pv)

Given:
 * an interest `rate` compounded once per period, of which there are

 * `nper` total

 * a (fixed) payment, `pmt`

 *  a present value, `pv`


Return:
   the value at the end of the `nper` periods

The future value function

Rate: 3% per period (per month)
Number of periods: 3 months
Payment: USD -100 top ups at the end of each month
PV: USD -1,000 deposit

The future value function

npf.fv(rate=0.03, nper=3, pmt=-100, pv=-1000)

1401.82

Compounding frequencies

How much do we have after 10 years investing $1,000 (no top-ups) at:

5% annual interest paid annually
5% annual interest paid monthly
5% annual interest paid daily

Compounding frequencies

The rate is divided by the frequency and the number of periods multiplied by the frequency

# Using annual compounding frequency
npf.fv(rate=0.05, nper=10, pmt=0, pv=-1000)

1628.89

# Using monthly compounding frequency
npf.fv(rate=0.05/12, nper=10*12, pmt=0, pv=-1000)

1647.01

# Using daily compounding frequency
npf.fv(rate=0.05/365, nper=10*365, pmt=0, pv=-1000)

1648.66

Let's practice!

Bond Valuation and Analysis in Python