If F = Face value of a bond c = Maturity value of a bond R = Periodic…

The correct answer is Option (3) → \( V=R\left[\displaystyle\frac{(1+i)^n-1}{i(1+i)^n}\right]+c(1+i)^{-n} \)

The price of a bond is the present value, at the market rate i, of everything the holder receives:

• n coupon payments of R each, an ordinary annuity with present value \( R\cdot\frac{1-(1+i)^{-n}}{i} \);

• the maturity value c after n periods, with present value \( c(1+i)^{-n} \).

    \( \displaystyle V=R\left[\frac{1-(1+i)^{-n}}{i}\right]+c(1+i)^{-n} \)

Multiplying the numerator and denominator of the bracket by \( (1+i)^n \):

    \( \displaystyle \frac{1-(1+i)^{-n}}{i}=\frac{(1+i)^n-1}{i(1+i)^n} \)

    \( \displaystyle V=R\left[\frac{(1+i)^n-1}{i(1+i)^n}\right]+c(1+i)^{-n} \)

This is Option (3).

Why the other options are incorrect

• Option (1) \( V=R\left[\displaystyle\frac{1-(1+i)^n}{i}\right]+c(1+i)^n \) — Incorrect. Both terms use \( (1+i)^n \); the bracket is then negative, and \( c(1+i)^n \) is the accumulated value of c, not its present value.

• Option (2) \( V=R\left[\displaystyle\frac{(1+i)^{-n}-1}{i}\right]+c(1+i)^{-n} \) — Incorrect. The bracket \( \frac{(1+i)^{-n}-1}{i} \) is negative, because \( (1+i)^{-n}\lt 1 \); it is the annuity factor with the wrong sign.

• Option (4) \( V=R\left[\displaystyle\frac{1-(1+i)^{-n}}{i}\right]+c(1+i)^n \) — Incorrect. The annuity part is correct, but the maturity value must be discounted to the present, \( c(1+i)^{-n} \), not accumulated as \( c(1+i)^n \).