# Valuation of Bonds — Formulae, Redeemable, Irredeemable [Worked Examples]

by **Jack Bodeley** on *October 05, 2021*

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## Classification of Bonds

Bonds are generally classified into two:

*Redeemable*bonds — bonds with defined maturity*Irredeemable*or*perpetual*or*console*bonds — bonds without defined maturity

## Valuation of Redeemable Bonds

### Formula (With Maturity)

$$B_0=\left[\frac{Int_1}{(1 + K_d)^1}+\frac{Int_2}{(1 + K_d)^2}+...\frac{Int_n}{(1 + K_d)^n}\right]+\frac{M_n}{(1 + K_d)^n}$$

Where;

- $B_0$ — Bond Value
- $Int_n$ — Annual Interest in year n
- $K_d$ — Cost of Debt
- $M_n$ — Maturity or Terminal Value of the bond
- $n$ — No of years to maturity

### Formula (Without Maturity)

$$B_0=\left[\frac{Int_1}{(1 + K_d)^1}+\frac{Int_2}{(1 + K_d)^2}+...\frac{Int_n}{(1 + K_d)^n}\right]+\frac{P_n}{(1 + K_d)^n}$$

Where;

- $B_0$ — Bond Value
- $Int_n$ — Annual Interest in year n
- $K_d$ — Cost of Debt
- $P_n$ — Par Value of the bond
- $n$ — No of years to maturity

## Valuation of Irredeemable or Perpetual Bonds

### Formula (Without Taxation)

$$B_0 = \frac{Int}{K_d}$$

### Formula (With Taxation)

$$B_0 = \frac{Int(1 - t)}{K_d}$$

Where;

- $t$ — Tax rate

## Worked Examples

### Example 1.1

A bond is to be issued for 5 years at par value of £3,000. The rate of interest on the bond it 10% p.a. and the amount to be redeemed on maturity is $3,300. If the required rate of return is 12%, what value should this bond be purchased at today?

#### Solution 1.1

This bond is redeemable with maturity value, $M_n$, of £3,300. Therefore;

$$Int = .1 \times 3,000 = £300$$

$$B_0=\left[\frac{300}{(1 + .12)^1}+\frac{300}{(1 + .12)^2}+\frac{300}{(1 + .12)^3}+\frac{300}{(1 + .12)^4}+\frac{300}{(1 + .12)^5}\right]+\frac{3,300}{(1 + .12)^5}$$

$$B_0 \approx 1,081.4329 + 1,872.5086$$

$$B_0 \approx £2,953.94$$

### Example 1.2

Suppose an investor in considering the purchase of a 5 year, £1,000 par value bond with a coupon rate of interest of 7%. If the investor's required rate of return is 8%, what should he be willing to pay now to purchase the bond if it matures at par?

#### Solution 1.2

This bond is redeemable which matures at par value, $P_n$, £1000. Therefore;

$$Int = .07 \times 1,000 = £70$$

$$B_0=\left[\frac{70}{1.08^1}+\frac{70}{1.08^2}+\frac{70}{1.08^3}+\frac{70}{1.08^4}+\frac{70}{1.08^5}\right]+\frac{1,000}{1.08^5}$$

$$B_0 \approx 279.4897 + 680.5832$$

$$B_0 \approx £960.07$$

### Example 2.1

A bond yields £300 annually. What is the value of the bond if the required rate of return is 12% (ignore taxation)?

#### Solution 2.1

This is a perpetual bond. Therefore;

$$B_0 = \frac{300}{.12} = £2,500$$

### Example 2.2

A bond yields £450 annually into perpetuity. What is the value of the bond if the required rate of return is 125% and the tax rate is 30%?

#### Solution 2.2

This is also perpetual bond. Therefore;

$$B_0 = \frac{450(1 - .3)}{.125} = £2,520$$