When you earn interest on an amount, the accumulated total (initial capital + interest earned) is called the future value whilst the initial amount is called the present value. Moving a payment backwards in time means that the value is discounted. This shows the time value of money: the value of money depends on time.
Question: How much capital should a person invest to obtain a capital, $C$, after $n$ time units?
Answer: A person should invest $Cv^n$ where $v=\frac{1}{(1+i)}$ (the discount factor).
What is the present value of a payment of $£70$ in two years’ time if the interest rate is $7\%$?
The discount factor is $v=\frac{1}{(1+i)}=\frac{1}{1.07}=0.93457943925$, so the present value is given by \[Cv^n=70 \times 0.93457943925^2 \approx £61.14\].
What is the present value of a payment of $£350$ in three months’ time if the interest rate is $3.25\%$?
The discount factor is $v=\frac{1}{(1+i)}=\frac{1}{1.0325}=0.9685230024$, so the present value is given by \[Cv^n=350 \times 0.9685230024^{\frac{3}{12}~} \approx £347.21\]. Here $n=\frac{3}{12}=0.25$ as $n$ is given in years and $3$ months is a quarter of a year.
The rate of discount, $d$, is the interest paid at the beginning of a time unit divided by the capital at the end of the time unit. \[d=1-v=1-\frac{1}{(1+i)}=\frac{i}{(1+i)}=iv\]
How much interest would you have to pay at the start of the year if you borrow $£1000$ for one year at a rate of $7\%$?
a) If the interest was paid in arrears (end of the year) then the interest to be paid would be $0.07\times1000=£70$. To pay this interest at the start of the year, we must discount this value, so the interest paid at the start of the year is $70v=\frac{70}{1.07}\approx£65.42$.
b) At time $t=0$, you receive $£1000$ but as you have to pay interest immediately, you in fact get less from the lender. You pay back the $£1000$ at the end of the year, so the amount received now (at $t=0$) and the $£1000$ you pay back must be equivalent.
The discount factor is $v=\frac{1}{1.07}=0.93457943925$, so the present value of the $£1000$ is $£934.58$. Thus the interest you have to pay is $1000-934.58=£65.42$.