This is a foundational piece about the time value of money. It will feel a bit more like accountancy rather than mathematical finance, but it’s the absolute bedrock of what we do so definitely worth spending a bit of time on!
$100 today is worth more than $100 in a year’s time. This should be obvious – if I have $100 today, I can put it into a bank account and earn (back in the good old days!) perhaps 3% on it – so in a year’s time I have $103.
For the rest of this post (and in much of finance), I will assume that I can put money on deposit at a risk-free rate r(t), and that it will grow according to the following p.d.e.:
note that this has no stochastic component – it’s solution is risk-free, exponential growth so that if I start with at time 0, by time T it has grown to – compound interest with ‘short rate’ r(t).
Time varying interest rates can be troublesome to work with, a more convenient – but closely related – concept is the Zero Coupon Bond (ZCB). This is a bond that I buy today with a specific maturity date, at which I will get paid exactly $1. It doesn’t pay any interest in between, and since I’ve said that money is more valuable now than later, I expect to pay less than $1 for the coupon now to reflect this. How much less? Well, thinking about the bank account example above, I locked away my $100 to receive $103 a year later – this is just the same as buying 103 ZCBs maturing in a year – so the price of each bond must be 100/103 = $97.09 or there is an arbitrage opportunity. As we can see, there is a 1-to-1 correspondence between interest rate curves and ZCB prices over a period, knowing one allows us to calculate the other in a rather straightforward way,
where is the price of a ZCB bought now with maturity at T [the above formula is only strictly true in the context of deterministic rates – more on this later]. I’ve added a little doodle here to allow conversion between a (constant) interest rate and time period and a discount factor and also on the PRICERS page – I’ll add functionality for time-varying rates another time.