Valuation of Bonds

Present Value 

Present Value Analysis tells how much money required to be invested in a risk-free assets so that it would generate the specified value. The rate of return is called discount rate and the calculation of present value of future value is called discounting.

Future Value (or compound value) = Present Value x ( 1 + r ) ^ t

PV = FV / ( 1 + r) ^ t

where,
PV = Present Value
FV = Future Value
r = Discount rate in a compounding period
t = Number of compounding period

Net Present Value for an investment that generates cash outflow (C₀) at date 0 and cash inflows (C_i) at regular intervals in future dates would be

NPV = - C₀+ C₁ / ( 1 + r) + C₂ / ( 1 + r) ^2 + .... + C_t / ( 1 + r) ^t

Pure Discount Bond

Pure discount bond promises single payment at a fixed future date i.e. maturity date of the bond. The payment at maturity is termed as face or par value of the bond. The present value of pure discount bond is given by

PV = FV / ( 1 + r)^t

where r = discount rate in a time period
t = number of time period

Level Coupon Bonds

Along with maturity date, typically bonds also offer cash payments at regular intervals in between. These payments are termed as coupons of the bond. Such bonds are called level coupon bonds.

PV = C / (1+r) + C / (1+r)^2 + .... C  / (1+r)^t + FV / ( 1 + r)^t

 PV = C A + FV / ( 1 + r)^t

where A denotes the PV of Annuity of 1 unit,

A = 1/r - 1/r * 1/(1+r)^t

 

Yield To Maturity (YTM)

The discount rate that equates the price of the bond with the discounted value of the coupons and the face value is termed as yield. Yield indicates the expectation of the market. The yield to maturity could be different from coupon discount rate and difference between yield and discount rate depends present value of the coupons payment and par value.

Price of Bond

The quoted price is not the same as the cash price that is paid by the purchaser. Quoted price is referred to as clean price and cash price as dirty price by the traders. Price quoted is different from actual cash price, it leave the accrued interest after last coupon. This way every day discounting of the interest is not required to be done in quoting.

Cash price of bond = quoted price + accrued interest

 

In India, price is quoted in decimals whereas in USA it is quoted (in 1/32) as e.g. 90-05 which means 90 + 05 / 32

Points to remember: 

1. Interest rate are inversely proportional to present value. Bond prices and market interest rates move in opposite directions
When coupon rate = YTM, price = par value
When coupon rate > YTM, price > par value (premium bond)
When coupon rate < YTM, price < par value (discount bond) 

2. Government bonds are risk-free only in local currency. In foreign currency, bonds are not risk-free. e.g. Greece..

3. Consols are the perpetual bonds i.e. bond with no maturity date. The present value for such bond is given by following equation

PV = C / (1+r) + C / (1+r)^2 + ....  = C / r

EXCEL provides specific formulas for finding bond prices and yields: PRICE(Settlement,Maturity,Rate,Yld,Redemption,Frequency,Basis) YIELD(Settlement,Maturity,Rate,Pr,Redemption,Frequency,Basis) Settlement and maturity need to be actual dates The redemption and Pr need to be given as % of par value