Bonds as a source of financing

It is possible to get an appreciable sum of money through a large number of small investors, by issuing long-term obligations, the issuance of bonds are a clear example of this

The issue of financing through bond issuance has already been discussed in previous articles in its theoretical form, the objective that we now propose is to apply this theory to practice by explaining the mathematical foundations and a series of applications that can be usefulness when you decide to look for short-term resources through this financing method.

Returning to the subject, it must be taken into account that the financing of the company through bonds must be taken in accordance with the support they offer to buyers, these classified as:

Mortgage bonds

Which are those that are backed by a mortgage on a specific asset. As the risk of these bonds is minimal, they provide the least return.

Bonds without backup

That they are those that do not have specific support, they are only endorsed by the good name that the company has vis-à-vis third parties, they are almost always convertible into shares. These are the most profitable since their risk is very high.

State bonds

That they are those issued by the state, their profitability is almost always too low, but they are imposed by the central government as forced investments for companies.

Central theme

Bond issuance is one of the most widely used ways to raise money from large companies in the short term

Preliminary considerations

To clarify the terminology and the theoretical basis of the valuation of the bonds issued by a company, it is necessary to be very clear about the following concepts:

Nominal value: It is the value that is written on the bond at the time it is issued. It is symbolized by the letter (F) Redemption value: It is the value that will be paid when the bond matures. It is represented by the letter (V). When the nominal value is equal to the redemption value, the bond is said to be redeemable at par, but if this value is different, the redemption value is a percentage of the nominal value. Interest rate of the bond: The amount of interest (R) that periodically pays a bond is calculated, applying the rate (K) to the nominal value (F), then we have that the periodic interest is R = K * F where (K) is the interest rate of the bond. Internal rate of return: Symbolized by the acronym (IRR) and is the interest rate at which each of the payments made on the bond is transported, up to the date of purchase. of purchase of the bond: Equivalent to the current value on the date of acquisition. It is represented by the letter (P). Bond periods: It is the number of interest payments, from the date of purchase, until its redemption. It is represented by (n) and indicates the number of periods.

With the above aspects clear, a mathematical and applicative analysis of the valuation of a bond issue within the company can be started with a series of examples that will clarify this issue.

Purchase price calculation

The purchase price is calculated using the following expression:

P = Ran¬i + V (1 + i) ^-n

Application examples

Example 1

A bond has a nominal value of $ 1,000, its redemption is agreed in 120 in 5 years. if you pay 20% cash per year, find the purchase price to rent 30% CT.

Solution

1. The interest to be paid during the four quarters is found.

R = 1,000 * 0.2 / 100 = $ 50

2. We find the redemption value.

V = 1,000 * 120/100 = $ 1,200

3. Taking into account that it is expected to rent quarterly, the number of quarters in 5 years is calculated and the purchase price formula is applied.

P = Ran¬i + V (1 + i) ^-n

P = 50a20¬7.5 + 1200 (1 + 0.075) ^-20

P = $ 792.22

The bond at the time of issue must have a retail price of $ 792.22

The people or entities that buy the bonds issued by the company are considered as small lenders who seek to obtain the greatest benefit with minimal risk.

Example 2

Find the internal rate of return of a bond of nominal value $ 10,000 redeemable at par in 5 years, which pays 18% semi-annually and whose market price is 90%.

Solution

1. Find the percentage applicable to the nominal value.

P = 10,000 * 90/100 = $ 9,000

2. Interest is calculated for the semester.

R = 10,000 * (0.09) = $ 900

3. The semesters that are in the 5 years are calculated and replaced in the formula and you have it.

9000 = 900a10¬i + 10,000 (1 + i) ^-10

Since there are two unknowns, the interpolation method is used to find the interest rate 0 obtaining that

i = 10.6843%

It must be taken into account that the methodology described above can be used when calculating the value of the issue as long as the payment date coincides with that of the interest, that is, in full periods.

General bond price

When the bond issue does not coincide with the period of the interest, but the sale is made at any time of the validity of the documents, a different analysis must be carried out since the purchase price must be found on the date of the last interest caused and this value is increased by calculating the simple amount on the transaction date and using the internal rate of return (IRR).

Example 1

A $ 2,000 face value bond is purchased on November 20, 1994; pays 18% interest payable monthly and is redeemed to 110 on the first of 2001. calculate the purchase price, if you expect to get a return of 24% CM.

Solution

1. Find the value of the bond's interest rate.

R = 2,000 * 0.18 / 12 = 30

2. The redemption value is calculated.

V = 2,000 * 110/100 = $ 2,200

3. Since the last payment was made on November 1, 1994, then the price must be found on that date.

P = 30a74¬2 + 2,200 (1 + 0.02) ^-74

P = 1,661.69

4. The previous value corresponds to November 1, 1994, but since the value as of November 20, 1994 needs to be found, the difference between this time (19 days) must be found. Using the simple interest method, this amount can be easily found.

1,661.69 * (1 + 0.24 * 19/360) = $ 1,682.74

$ 1,682.74 is the value of the purchase price of the bond on November 20, 1994 to obtain the expected return.

Bond table

A bond table presents all the changes that occur in the book value of a bond, from the date of its acquisition, to the date of sale or, from the date of issue, to the date of redemption. The book value represents the amount invested in each period.

Premium-Discount

When the purchase price of a bond is greater than the redemption value, the bond is said to be purchased at a premium and if it is less, it is said to be purchased at a discount

Example 1

Prepare an investment table for a bond with a nominal value of $ 1,000, redeemable at 120 in 3 years, if you pay interest of 25% CS and an IRR is 15% CS.

Solution

1. The purchase price is found

P = Ran¬i + V (1 + i) ^-n

P = 125a6¬7.5% + 1,200 (1 + 0.075) ^-6

P = $ 1,364.28

2. The initial book value is the purchase price and the interest is calculated, applying the IRR to the book value. The change in book value is the difference between the interest and the coupon value. This difference can be positive or negative.

n	Value in books	Interest	Coupon value	Change in book value
one	$ 1,364.28	102.32	125	- 22.68
two	$ 1,341.60	100.62	125	- 24.38
3	$ 1,317.22	98.79	125	- 26.21
4	$ 1,291.01	96.83	125	- 28.17
5	$ 1,262.84	94.71	125	- 30.29
6	$ 1,232.55	92.44	125	- 32.55
	$ 1,000

Example 2

A $ 1,000 par value bond will be redeemed at par on November 1, 2001, it pays interest of 10% MN and its IRR is 20% CS. If purchased on June 20, 1999, find your purchase price and prepare the bond table.

Solution

1. Find the purchase price for May 1, 1999.

P = Ran¬i + V (1 + i) ^-n

P = 50a5¬10% + 1,000 (1 + 0.1) ^-5

P = $ 810.46

2. The price is calculated between May 1, 1999 to June 20, 2001. Between the two dates there are 50 days.

810.46 * (1 + 0.2 * 50/360) = $ 832.97

3. The fraction to which the seller is entitled for having had the bonus for 50 days is calculated. It is calculated proportionally.

X = 50 * 50/180 = $ 13.89

4. The book value is calculated, which is the value on the transaction date, less the seller's commission.

832.97 - 13.89 = 819.08

5. The table is prepared taking into account that for the first period, the book value is $ 819.08 and that the interest is 819.08 X 0.1 = 81.91, which is for the entire period as until the moment of the transaction only 50 have passed days and 130 days to go, then the interest on the book value should be calculated proportionally.

81.91 * 130/180 = $ 59.16

6. The value of the coupon is calculated proportionally for the 130 days.

50 * 130/180 = $ 36.11

When the IRR is less than the K rate used to calculate the interest, the bond will be issued with a premium, and if it is the opposite, it will be a discount

7. The table is prepared in this way.

n	Value in books	Interest	Coupon value	Change in book value
one	$ 819.08	59.16	36.11	- 23.05
two	$ 842.13	84.21	fifty	- 34.21
3	$ 876.34	87.63	fifty	- 37.63
4	$ 913.97	91.40	fifty	- 41.40
5	$ 955.37	95.54	fifty	- 45.54
	$ 1000.91

Note: The difference is an error of approximation in the first period evaluated.

Bonds with increasing interest

This type of bond is presented in a model that in inflationary times, can be well received by investors.

Example 1

A $ 1,000 par bond, redeemable for $ 5,000 over 10 years, pays quarterly interest. These interests follow the law of formation of a stepped geometric gradient, so during the first year you pay interest of 24% quarterly and each following year, the value of the interest increases by 15%. What should be the issue value, if you want the investment to earn 30% interest, compounded quarterly.

Solution

1. The quarterly interest corresponding to the first year is calculated.

24% / 4 = 6%

0.06 * 1000 = $ 60

2. Then they are calculated for the following periods.

3. The present value of the bond is found by means of the stepped gradients methodology.

60s4¬7.5% = 268.38

4. Calculate the rate for the period of the gradient, finding the annual effective rate.

(1 + i) ¹ = (1 + 0.075) ⁴

i = 33.54%

5. Finally, the present value formula for the bond is calculated

P = 268.38 + 5,000 (1 + 0.3354) ^-10