Cost of Debt Bias
Derivation of “quick and dirty” approximation of bias formula
Thomas Noe
Balliol College/SBS
21st October, 2013
This note relates to the derivation of the “quick and dirty” formula for estimating the bias generated by using the YTM as an approximation of the expected return on debt. The assumptions:
1. Debt is perpetual
2. probability of default is δ in each period. The probability is the same in every period
3. If default occurs, bondholders receive ρ fraction of the face (principal) value of the bond plus accrued interest. 4. Bond is sold at par, i.e., the bonds initial price equals its principal value.
5. If the bond does not default, the bondholders receive the promised coupon payment.
6. Discount rates are constant over time.
At the start of each period in which the bond has yet to default, the bonds price must equal its initial price.
Why? At the start of period 1, the bond promises to pay a perpetual series of interest payments and with a δ probability of default and an a recovery rate of ρ; at the start of period 100, if the bond never defaulted in the previous 99 periods, the bond promises to pay a perpetual series of interest payments and with a δ probability of default and an a recovery rate of ρ. The same statement is true for any and all dates in the future. Thus, the price will be the same at all dates in the future. Thus, if the bond does not default at the end of the period, at the end of a period, it is worth P + rYTM P; if the bond defaults at the end of a period, it is worth γ(P + rYTM P). The expected value of the bond at the end of the period is thus δ (γ (P + rYTM P)) + (1 − δ ) (P + rYTM P)
(1)
The value of the bond at the start of any period equals the expected value at the end of the period discounted at the cost of capital r. So the value of the bond at the start of the each period (given no default in earlier periods) is δ (γ (P + rYTM P)) + (1 − δ ) (P + rYTM P)
1+r
(2)
{r → (1 − (1 − γ)δ )rYTM − (1 −
Thomas Noe
Balliol College/SBS
21st October, 2013
This note relates to the derivation of the “quick and dirty” formula for estimating the bias generated by using the YTM as an approximation of the expected return on debt. The assumptions:
1. Debt is perpetual
2. probability of default is δ in each period. The probability is the same in every period
3. If default occurs, bondholders receive ρ fraction of the face (principal) value of the bond plus accrued interest. 4. Bond is sold at par, i.e., the bonds initial price equals its principal value.
5. If the bond does not default, the bondholders receive the promised coupon payment.
6. Discount rates are constant over time.
At the start of each period in which the bond has yet to default, the bonds price must equal its initial price.
Why? At the start of period 1, the bond promises to pay a perpetual series of interest payments and with a δ probability of default and an a recovery rate of ρ; at the start of period 100, if the bond never defaulted in the previous 99 periods, the bond promises to pay a perpetual series of interest payments and with a δ probability of default and an a recovery rate of ρ. The same statement is true for any and all dates in the future. Thus, the price will be the same at all dates in the future. Thus, if the bond does not default at the end of the period, at the end of a period, it is worth P + rYTM P; if the bond defaults at the end of a period, it is worth γ(P + rYTM P). The expected value of the bond at the end of the period is thus δ (γ (P + rYTM P)) + (1 − δ ) (P + rYTM P)
(1)
The value of the bond at the start of any period equals the expected value at the end of the period discounted at the cost of capital r. So the value of the bond at the start of the each period (given no default in earlier periods) is δ (γ (P + rYTM P)) + (1 − δ ) (P + rYTM P)
1+r
(2)
{r → (1 − (1 − γ)δ )rYTM − (1 −