Chapter 6 Bonds and Stocks

Notations

• Price of bond \(P\).

• Number of payments (term of bond) \(n\).

• Yield-to-maturity (IRR, yield, interest rate per payment period) \(i\).

• Current yield \(i_c=\frac{rF}{P}\).

• Coupon rate per payment period \(r\).

• Modified coupon rate \(g=\frac{Fr}{C}\).

• Face value of bond \(F\).

• Redemption value of bond \(C\).

• Book value \(V_t\). Similar to outstanding loan balance.

Pricing formulas

• Basic formula: \[P=Fra_{\overline{n}\mid i} + Cv^n_i.\]

• Premium/discount formula: \[P=C+(Fr-Ci)a_{\overline{n}\mid i}=C\left[1+(g-i)a_{\overline{n}\mid i}\right].\]

  • \(P>C, g>i\) means the bond is purchased at a premium. Amortization of premium.

  • \(P<C, g<i\) means the bond is purchased at a discount. Accumulation of discount.

• Base amount formula: \[P=G+(C-G)v^n,\] where \(G=\frac{rF}{i}.\)

• Makeham formula: \[P=K+\frac{g}{i}(C-K),\] with \(K=Cv^n_i.\)

Price between coupon dates \[P_{t+k}=P_t(1+i)^k,\] with \(t\in\mathbf{N}^+, k\in(0,1).\)

Book value between coupon dates \[V_{t+k}=P_{t+k}-kFr=P_{t}(1+i)^k-\frac{rF}{i}\left[(1+i)^t-1\right] \approx P_{t}(1+i)^k-kFr,\] with \(t\in\mathbf{N}^+, k\in(0,1).\)

Price of callable bond

• \(g>i\) (sold at a premium): the call date will most likely be at the earliest date possible .

• \(g<i\) (sold at a discount): the call date will most likely be at the latest date possible.

Discounted dividend formula of stock \[P=\sum_{t=1}^\infty div_t v^t.\]

Price-to-earnings (P/E) ratio formula of stock \[\text{P/E ration}=\frac{\text{stock price per share}}{\text{earnings per share}}=\frac{P_0}{EPS},\] where \[EPS=\frac{\text{net income}}{\text{number of outstanding shares}}.\]

Profit of short sale \[\text{Profit}=(P_0-P_t) + \text{Margin deposit}\times i - \text{Div}.\]

Yield of short sale \[\text{Yield of short sale}=\frac{\text{Profit}}{\text{Margin deposit}}.\]