The price of a security is generalized as follows
Where P is the price, x is the future payout and m is the stochastic discount factor. What this means that the price of the security is dependent on what I will receive in the future as a payout. For the annuity, the payout (payouts) are the series of payments that I will receive in the future, but remember money in the future isn’t worth the same as money today, that’s where m comes in.
m in the case of the simple annuity represents the ratio of the value of money today over the value of money tomorrow. Basically, what is the least amount of money I want to receive sometime in the future for me to part with 1 dollar?
Let’s say the prevailing interest rate is 10% annually and I will receive a payment of $100 in one year if I bought that single payment ordinary annuity. This means that I will only part with $1 if I know it’s worth at least $1.10 in a year. Thus:
Which means for that $100 payment I will only buy it for at most $90.91. At the same time, if I sell this security, the buyer will also only buy for at most $90.91.
Of course, an annuity with more than 1 payments has more steps in derivation but I’ll get to it in a later post. The idea, however, will be used to price bonds, stocks, options and many different types of securities.
Thank you for reading my article, if you found it useful please like and share and if you have any questions please feel free to leave them in the comments – I may use your question for a further post.