What I love about bond mathematics is that it touches on the corner stone of all financial valuations.
The value of any and all financial assets is the sum of its expected future cashflow, discounted by an interest rate, that represents opportunity cost.
A bond is a loan that is tradeable, or transferrable, also known as an “I owe you” note. When I borrow $100 from Bob, I also give Bob a note, that says “Whomever holds this note shall receive $100 from me at maturity” Maturity is some stated time in the future.
The value or price of a bond is the net present value (NPV) of its expected future cashflow, also known as Discounted Cash Flow (DCF).
Finance is funny in that there’s all these various terms that just mean the same thing.
$$ Value = \frac{C_1}{(1+i)} + \frac{C_2}{(1+i)^2} + \frac{C_3}{(1+i)^3} +... + \frac{C_n}{(1+i)^n} + \frac{P}{(1+i)^n} $$
Where C is the period coupon payment, i is the discount rate or yield, n is the number of periods, and P is the final principal or sometimes known as “par value”.
Using some numbers, let’s say the coupon was $5, the loan balance was $100, the loan was for 10 years, and the discount rate or required yield is 6%
$$ \frac{\$5}{(1.06)}+\frac{\$5}{(1.06)^2}+\frac{\$5}{(1.06)^3}+...\frac{\$5}{(1.06)^{10}}+\frac{\$100}{(1.06)^{10}} = \$92.64 $$
Bonds are often quoted or priced per dollar, so everything is normalized to $100, which is simply$Price = Value/Par$
Mental shortcuts as to the relationships.