4 Bonds

4.1 Supplementary Concepts

**Notation**
\(P\)	Price (i.e., present value of cash flows)
\(F\)	Face amount or par value
\(C\)	Redemption amount
\(r\)	coupon rate
\(Fr\)	Coupon amount
\(g=\large{\frac{Fr}{C}}\)	Adjusted coupon rate
\(i\)	Yield rate (or internal rate of return)
\(n\)	Number of coupon payment periods
\(P-C\)	Premium (if \(P>C \Leftrightarrow i<g)\)
\(C-P\)	Discount (if \(C>P \Leftrightarrow i>g)\)

Bonds are loans that investors make to corporations or governments, where the borrower (issuer) promises to repay the principal (face value) of the loan at a specified future date (maturity) and also pay periodic interest payments (coupons) to the bondholder. Bonds are a fixed-income investment, meaning they offer a predictable stream of income through interest payments.
The following notation will be used in this unit:
The main pricing formulas are: \[ \boxed{ \begin{array}{rl} P &= Fr\,a_{\enclose{actuarial}{n}}+Cv^n & \text{(basic} \,\, \text{formula)}\\ \\ P &= C+(Fr-Ci)\,a_{\enclose{actuarial}{n}} & \text{(premium-discount formula)}\\ \end{array} } \]
Callable bonds are bonds in which the borrower has an option to redeem prior to its maturity date. If \(C_t=C\) , \(\forall t\), then the following rules apply:

If \(i<g\) (i.e., the bond sells at a premium), assume that the redemption date will be the earliest possible date
If \(i>g\) (i.e., the bond sells at a discount), assume that the redemption date will be the latest possible date

Preferred stock is a fixed-income security without a redemption date. In this case, \(P=\large{\frac{Fr}{i}}\).
Common stock is a variable-income security. In theory, its price should represent the present value of future dividends. If \(D\) is a dividend that is projected to change geometrically with common ratio \(1+k\), then \(P=D \, \large{\frac{1}{i-k}}\)

4.2 Exercises

Find the price of a \(1\,000\) par value \(10\)-year bond with coupons at \(8\%\) convertible semiannually, which will be redeemed at \(1\,050\).

The bond is bought to yield \(10\%\) convertible semiannually.

A \(1\,000\) par value one-year \(10\%\) bond with semiannual coupons is bought to yield \(i^{(2)}=6\%\).

Calculate the book value at the end of the first six months, immediately after the coupon has been paid.

Find the price of a \(10\)-year \(1\,000\) bond bearing a \(10\%\) coupon rate payable semiannually and redeemable at \(1\,050\) is bought to yield \(8\%\) convertible semiannually.

\(A\) is a bond with a price of \(1\,136.778\) and coupon rate of \(5\%\) payable semiannually.

\(B\) is a bond with a coupon rate of \(2.5\%\) payable semiannually.

Both bonds have par value of \(1\,000\), are redeemable at par, have the same period, and are bought to yield \(4\%\) convertible semiannually.

Find the ratio of \(A\) to \(B\).

A \(1\,000\) par value 12-year bond with \(10\%\) semiannual coupons is selling for \(1\,100\).

Find the yield rate convertible semiannually.

A \(1\,000\) bond with a coupon rate of \(9\%\) payable semiannually is redeemable after an unspecified number of years at \(1\,125\).

The bond is bought to yield \(10\%\) convertible semiannually.

If the present value of the redemption amount is \(224.86\), find the purchase price.

A \(1\,000\) par value \(𝑛\)-year bond maturing at par with \(100\) annual coupons is purchased for \(1\,110\).

If \(Cv^{n}=450\), find \(\frac{Fr}{i}\).

A \(1\,000\) par value \(n\)-year bond maturing at par has a coupon rate of \(12\%\) convertible semiannually.

It is bought at a price to yield \(10\%\) convertible semiannually.

If the term of the bond is doubled, the price will increase by \(50\).

Find the price of the \(n\)-year bond.

A company issues an inflation-adjusted bond with par value of \(1\,000\) and annual coupons at the end of each year for \(20\) years.

The initial coupon rate is \(5\%\) and each coupon will be \(10\%\) greater than the preceding one.

The bond is redeemed for \(1\,800\) at the end of \(20\) years.

Find the price that produces a yield rate of \(8\%\) effective.

A bond with annual coupons in the amount of \(160\) is purchased at a discount to yield \(15\%\).

The write-up for the first year is \(44\).

What was the purchase price?

Consider a \(1\,000\) par value two-year \(8\%\) bond with semiannual coupons bought to yield \(6\%\) convertible semiannually.

Calculate the interest earned at the time the third coupon is paid.

Consider a \(1\,000\) par value two-year \(8\%\) bond with semiannual coupons bought to yield \(10\%\) convertible semiannually.

Calculate the amount of accumulation of discount at the time the third coupon is paid.

A \(1\,000\) par value \(6\%\) bond with semiannual coupons is callable at par five years after issue.

It is sold to yield \(7\%\) under the assumption that the bond will be called.

The bond is not called and it matures at the end of \(10\) years.

The bond issuer redeems the bond at \(1\,000 + X\) without altering the investor’s yield rate of \(7\%\) convertible semiannually.

Find \(X\)

Consider a \(1\,000\) par value bond with a coupon rate of \(4\%\) per annum paid semiannually.

The spot rates of interest in the market are given by

\(\\\)

Investment Horizon	Spot Rate
0.5	3.00%
1.0	3.00%
1.5	3.50%
2.0	3.50%
2.5	4.02%
3.0	4.02%

Find the price of the bond if it matures in \(3\) years.

A \(1\,000\) par value \(16\)-year bond with \(m\) coupons per annum has a price of \(910.63\).

If the total write-up value in the book value of the bond in the first eight years is \(54\), and the yield rate of the bond is \(5.0945\%\) effective per annum, find the redemption value of the bond.

Two \(1\,000\) bonds redeemable on the same date at par to yield \(4.5\%\) have annual coupon rates of \(r\) and \(2r\), respectively.

If the total purchase price of the two bonds is \(2\,153.52\), and the difference of the bond prices is \(307.04\), find \(r\).

A \(1\,000\) par value \(10\)-year bond with annual coupons is redeemable at \(1\,055\), and has a purchase price of \(986\) at a yield rate of \(4\%\) per annum.

The coupons increase at a rate of \(3\%\) per year.

Find the amount amortized or discounted for the fifth coupon.

An investor purchases a \(10\)-year \(1\,000\) par value bond bearing a \(10\%\) coupon rate payable semiannually and redeemable at \(1\,050\), yielding \(8\%\) convertible semiannually.

The investor reinvests the coupons in a fund earning \(9\%\) convertible semiannually.

Find the overall yield convertible semiannually to the investor.

A preferred stock (i.e., a bond without a redemption date) pays a \(100\) dividend (i.e., coupon) at the end of the first year, with each successive annual dividend being \(5\%\) greater than the preceding one.

What level annual dividend would be equivalent if \(i=12\%\)?

A \(1\,000\) par value \(12\)-year bond has coupons at the annual rate of \(9\%\) payable continuously.

If the bond is bought to yield rate \(i\) which is equivalent to \(\delta=3\%\), find the price of the bond.

The interest paid during the \(20^{th}\) year on a \(20\)-year \(1\,000\) bond with annual coupons to be redeemed at par is equal to \(70\%\) of the principal adjustment during the same year.

If \(r=i+0.03\), where \(r\) is the coupon rate and \(i\) is the yield rate, find the original price of the bond.

A common stock is currently earning \(4\) per share and will pay \(2\) per share in dividends at the end of the current year.

Assuming that earnings increase \(5\%\) per year and that \(50\%\) of the earnings will be paid as dividends, find the theoretical price to earn an investor an annual effective yield rate of \(10\%\).

A common stock is purchased at a price equal to \(10\) times current earnings.

During the next six years, the stock pays no dividends, but earnings increase \(60\%\).

At the end of six years, the stock is sold at a price equal to \(15\) times earnings.

Find the effective annual yield rate earned on this investment.

A twelve-year bond with semiannual level coupons is bought at a premium to yield \(7.5\%\) convertible semiannually.

If the amount for amortization of premium in the fourth to the last payment is \(8.02\), find the total amount for amortization of premium in the first three years.

A \(100\,000\) par-value twenty-year \(14%\) bond with annual coupons is bought for \(95\,626.25\).

Find the amount for accumulation of discount in the third coupon.
Find the amount of interest for the third year.

A student bought a newly-issued \(2,000\) \(20\%\) ten-year bond, redeemable at \(2,100\) and having yearly coupons.

It was bought at a premium at a price of \(2,800.03\).

The student immediately took a constant amount \(k\) from each coupon and deposited it in a savings account earning \(6%\) effective annual interest, so as to accumulate the full amount of the premium a moment after the final deposit.

At the end of the ten years, the student closed out her savings account.

Find the yearly effective yield rate earned by the student for her combined ten-year investment.

A \(1\,000\) par value \(4\%\) bond with seminannual coupons is callable at \(1\,090\) on any coupon date starting five years after issue for the next five years, at \(1,045\) starting \(10\) years after issue for the next five years, and maturing at \(1\,000\) at the end of \(15\) years.

What is the highest price which an investor can pay and still be certain of a yield of \(5\%\) convertible semiannually?

Consider a \(1\,000\) face value \(15\)-year bond with coupon rate of \(4\%\) convertible semiannually.

The bond is callable and the first call date is the date immediately after the \(15^{th}\) coupon payment.

Assume that the issuer will only call the bond at a date immediately after the \(n^{th}\) coupon and the redemption value is

\[ C = \left\{ \begin{array}{ll} 1\,000, & 15 \leq n \leq 20 \\ 1\,000 + 10(n-20), & 20 \leq n \leq 30 \\ \end{array} \right. \] Find the price of the bond if the investor wants to achieve a yield of at least \(5\%\) compounded semiannually.

A \(1\,000\) par value \(4\%\) bond with seminannual coupons is callable at \(1\,090\) on any coupon date starting five years after issue for the next five years, at \(1\,045\) starting 10 years after issue for the next five years, and maturing at \(1\,000\) at the end of \(15\) years.

What is the highest price which an investor can pay and still be certain of a yield of \(3\%\) convertible semiannually?

A \(1\,000\) par value bond has \(8\%\) seminannual coupons and is callable at the end of the \(10^{th}\) to \(15^{th}\) years at par.

Find the price that yields \(6\%\) convertible semiannually.

A \(1\,000\) par value \(6\%\) bond with semiannual coupons is callable at par five years after issue.

It is sold to yield \(7\%\) convertible seminannually under the assumption that it will not be called.

The bond is not called and it matures at the end of \(10\) years.

The bond issuer redeems the bond at \(1000 + X\) without altering the investor’s yield rate of \(7\%\) convertible semiannually.

Find \(X\)

A \(1\,000\) par value \(8\%\) bond with semiannual coupons is callable at par five years after issue.

The bond matures at \(1\,000\) at the end of ten years and is sold to yield a nominal rate of \(6\%\) convertible quarterly under the assumption that the bond will not be called.

Find the redemption value at the end of five years that will provide the purchaser the same yield rate.

A \(1\,000\) par value bond has \(8\%\) semiannual coupons and is callable at the end of the \(10^{th}\) to \(15^{th}\) years at par.

Find the price to yield \(6\%\) convertible semiannually.

An insurance company issues bonds of \(10\,000\) par value with annual \(10\%\) coupons maturing at par in five years, and yielding \(12\%\) annual effective.

The company’s CFO wants to replace these bonds with bonds that have annual \(11\%\) coupons.

How long must the time to maturity be if the yield rate is still\(12\%\) annual effective?

[Round answer to nearest year]

A \(10-\)year \(1\,000\) par value bond bearing a \(10\%\) coupon rate payable semiannually and redeemable at \(1\,050\) is bought to yield \(8\%\) convertible semiannually.

Find the value of \(\frac{dP}{dg}\).

Two \(1\,000\) par value bonds both with \(8\%\) coupon rate payable semiannually are currently selling at par.

Bond \(A\) matures in five years at par, while Bond \(B\) matures in \(10\) years at par.

If prevailing market rates of interest suddenly go to \(6\%\) convertible semiannually, find the basis point difference between the percentage change in the price of \(A\) and the percentage change in the price of \(B\).

A \(10\)-year bond has annual coupons which vary \(10\), \(9\), \(8\), \(\cdots\), \(1\), and matures for \(95\).

If the bond is bought to yield \(1\%\) effective, find the amount of the fifth coupon used to amortize the book value.

A \(1\,000\) par value \(3\%\) bond with semiannual coupons on which \(500\) matures in nine years at \(510\) and the other \(500\) matures in \(10\) years at \(500\).

Calculate the price of the bond if the yield rate is \(4\%\) convertible semiannually.

A \(1\,000\) bond providing annual coupons at \(9\%\) is redeemable at par on 1 November, 2030.

The write-down in the first year was \(5.63\), while the write-down in the eleventh year was \(19.08\).

Determine the book value of the bond on 1 November, 2026.

A \(1\,000\) bond with semiannual coupons at \(10\%\) convertible semiannually is redeemable for \(1\,100\).

If the amount for the \(16^{th}\) write-up is \(5\), calculate the purchase price to yield \(12\%\) convertible semiannually.

A \(10\,000\) \(15-\)year bond is priced to yield \(i^{(4)}=12\%\).

It has quarterly coupons of \(200\) each the first year, \(215\) each the second year, \(230\) each the third year, \(\cdots\), \(410\) each the fifteenth year.

Find the book value after \(14\) years.

4.3 Code snippets

Newton-Raphson example to calculate the interest rate of a bond:

Let \(f(j)\) represent the price of a bond with \(F=1\,000\), \(C=1\,100\), \(n=24\), and \(r=0.05\).

The following algorithm finds the root of \(f(j)\) with \(│f(x)│ \le 10^{-8}\)

x0      <- 0.06
epsilon <- 10e-08

nr <- function(ini, eps){
  
  f   <- function(j) 1100 - 50 * ((1 - (1 + j)^-24) / j) - 1000 * (1 + j)^-24  # original function
  fpr <- function(j) -50 * ((j * (24 * (1 + j)^-25) - (1 + (1 + j)^-24)) / j^2) + 20000 * (1 + j)^-25   # first derivative
  
  xn1 <- ini - (f(ini) / fpr(ini))
  print(xn1)
  if(abs(xn1 - ini) > eps) nr(xn1, eps)
  
}

nr(x0, epsilon)

[1] 0.04697952
[1] 0.04497165
[1] 0.04410052
[1] 0.04367711
[1] 0.04346146
[1] 0.04334916
[1] 0.04329002
[1] 0.0432587
[1] 0.04324206
[1] 0.0432332
[1] 0.04322849
[1] 0.04322597
[1] 0.04322463
[1] 0.04322392
[1] 0.04322354
[1] 0.04322334
[1] 0.04322323
[1] 0.04322317