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Rob's s1452 Notes

πŸ‘¨β€πŸ« Notes

Bond Pricing Review

General (Coupon): PB=Fc1+i+Fc(1+i)2+Fc(1+i)3+...+Fc+F(1+i)TP_B = \frac{Fc}{1+i} + \frac{Fc}{(1+i)^2} + \frac{Fc}{(1+i)^3} + ... + \frac{Fc+F}{(1+i)^T}
Zero Coupon: PZCB=F(1+i)TP_{ZCB} = \frac{F}{(1+i)^T}
Consol: PC=FciP_C = \frac{Fc}{i}

F is Face Value, c is the coupon rate, i is the YTM or the interest rate on comparable bonds, etc.

To calculate bond price or yield on a coupon bond or a zero coupon bond, just use a spreadsheet or a calculator. For a consol, you will need to use the formula.

✏️ A consol pays $10,000 every year. Its yield to maturity is 5%. What is its price?

βœ” Click here to view answer

PC=$10,0005%=$200,000P_C= \frac{\$10,000}{5\%} = \$200,000

✏️ A consol that pays $10,000 every year costs $180,000. What is its yield to maturity?

βœ” Click here to view answer

180,000=$10,000i180,000 = \frac{\$10,000}{i}
i=$10,000180,000=10/180=5.56%i = \frac{\$10,000}{180,000} = 10/180 = 5.56\%

Yield to Maturity (YTM): Interest rate that makes the present value of the bond’s payments equal to its price.

  • To find it, solve one of the above bond pricing formulas for i.
  • Pay attention to how returns are presented with bonds.

Yield to maturity = 3% per half year (periodic rate r(T))
Bond Equivalent Yield = 3%*2 = 6% (assumes no reinvestment of coupons/simple interest/it’s an APR)
Effective Annual Rate = (1+3%)2-1 = 6.09% (assumes reinvestment/compound interest)

CurrentΒ YieldΒ =AnnualΒ CouponΒ PaymentCurrentΒ Price\text{Current Yield }= \frac{\text{Annual Coupon Payment}}{\text{Current Price}}

Duration

Duration of a perpetuity is (1+y)/y

Duration of a Zero Coupon Bond is T

See the following page for how to calculate duration quickly in a spreadsheet: πŸ”Ž Quick duration calculations

What Determines Duration?

Rule 1: The duration of a zero-coupon bond equals its time to maturity

Rule 2: Holding maturity constant, a bond’s duration is higher when the coupon rate is lower

Rule 3: Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity

Rule 4: Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower

Rules 5: The duration of a level perpetuity is (1 + y) / y