CHAPTER 2
Pricing of Financial Instruments
1. Introduction
In this chapter, we will introduce and explain the use of financial instruments involved in investing, lending, and borrowing money. In particular, we will focus on government and corporate bonds, interest, future and present value of financial instruments, mortgages, and stocks.
Marketable financial securities are contracts that are bought and sold by the general public in various venues. The two main examples of such securities are stocks and bonds. Securities are initially sold by investment banks such as Goldman Sachs. They can be bought and sold after the initial offering in organized exchanges, such as the New York Stock Exchange (NYSE), where stocks are traded. These instruments are also traded through associations like the National Association of Security Dealers (NASDAQ).
2. Bond pricing
The most basic tradable asset is a bond. When corporations, governments, or other entities would like to spend more money than they currently have, they issue bonds to raise capital. A purchaser of a bond is loaning the issuing institution money for a period of time to finance civil projects, corporate expansions, or other activities. To compensate the bond holder or creditor, the issuer or debtor will end up paying back more than was originally invested - through coupons or offering the bond at a discount. Coupon payments are periodic interest payments, paid to the bond holder, determined by the interest rate on the date of purchase. Large companies prefer bonds over bank loans since it is often a cheaper, more efficient, and easier alternative.
More specifically, a bond is a marketable financial security that gives the owner the right to a fixed payment at a predetermined date, called the maturity date. The amount loaned at time t = 0, or the amount the bond trades for, is the principal or the present value of the bond and is denoted P V (0). On the other hand, the amount the bond trades for at a later time t > 0, P V (t) or F V , is called the future
value of the bond. Notice that, in the case of the bond, the future value of the bond on the maturity date is predetermined when the bond is purchased.
There are several types of bonds classified by the borrowers of funds and the terms of maturity. The main two types of bonds we will consider are US government bonds and corporate bonds. The government issues bonds when their expenditures are greater than tax revenue. Government bonds include bills, notes, and bonds. Treasury bills, or T-Bills, have a time to maturity T of less than a year, T < 1, and do not issue coupon payments. Investors instead turn a profit by buying these bonds at a discount; that is, P V (0) < P V (T ). Notes and bonds, however, issue coupon payments every 6 months, where the last payment includes the coupon (interest) and the principal. Notes have a time to maturity between one and ten years 1 ≤ T ≤ 10, whereas bonds are more long term with T > 10 years.
Direct Financial
Instruments
Bonds Stocks Government Preferred Common Corporate Bills Notes Bonds Commercial Paper BondsFigure 1. Direct financial instruments
Corporations issue bonds to pay for large investments and large short term pay-ments. They issue two basic kinds of bonds - commercial paper and bonds. Com-mercial paper is a short term bond, T < 1, which is sold for immediate investments by companies and are sold at a discount from the principal price listed. Corporate bonds typically have time to maturity greater than a year T ≥ 1. Corporate bonds have higher interest rates and generally shorter terms for maturity than government bonds because they are riskier investments. Obviously, smaller firms generally are more likely to go bankrupt than larger, more established firms, but there is always a chance that any firm, even seemingly more stable ones, can unexpectedly go bank-rupt. A higher likelihood of bankruptcy means a firm’s bonds are sold with a higher interest rate. Because the government is much less likely to default on its bond pay-ments, government bonds are considered less risky and have lower interest rates and longer maturity terms.
3. FIXED INTEREST RATES 35
3. Fixed interest rates
In this section, we introduce the fundamental concepts of interest rates, present value, and future value of an investment. Let P V denote the initial value of the investment and F V denote the terminal value. We define the annual interest, or annual return, r of the investment to be
r =. F V − P V
P V . (3.1)
Therefore, the terminal value F V of an initial investment P V after 1 year is the value of the investment plus interest: F V = P V + rP V . Equivalently,
F V = (1 + r)P V or P V = F V
1 + r. (3.2)
The interest earned after the first year will be added to the principal and henceforth will also earn interest. As a result, how often the interest is paid affects the amount the investor earns. The more often interest is added, or compounded, the more money the investor makes.
Annual Compounding for t years:
If an initial amount PV is invested for t years at an interest rate r compounded annually, then applying formula (3.2) repeatedly we get
F V = P V (1 + r)t or P V = F V
(1 + r)t. (3.3)
Semiannual Compounding for t years:
Semiannual compounding means that every 6 months, the investment P V earns in-terest P V · (r/2) which is reinvested, where r is the annual inin-terest rate. Therefore, after one period of 6 months, we have F V = P V 1 + r2, and after two periods, or one year, we have the terminal value
F V = P V1 + r 2 ·1 + r 2 = P V1 + r 2 2 , which at the end of 2 years (four periods) becomes
F V = P V 1 + r 2 2 ·1 + r 2 2 = P V 1 + r 2 2·2 . Similarly, we find that at the end of 3 years, this investment becomes
F V = P V1 + r 2
2·3 .
After t years of semiannual compounding, we obtain the terminal value: F V = P V1 + r
2 2t
. Quarterly Compounding for t years:
Quarterly compounding means that every 3 months an investment P V earns in-terest P V · (r/4), which is reinvested. Following a similar process to the case of semi-annual compounding argument, we find that after t years of quarterly compounding we obtain the terminal value:
F V = P V1 + r 4
4t .
Furthermore, we obtain the similar formulas for monthly, weekly and daily compound-ing.
Compounding Frequency Future Value Annual F V = P V (1 + r)t Semiannual F V = P V (1 + r2)2t Quarterly F V = P V (1 + r 4) 4t Monthly F V = P V (1 +12r)12t Weekly F V = P V (1 +52r)52t Daily F V = P V (1 +365r )365t
Table 1. Formulas for interest compounded for t years at an annual rate r
The formulas in Table 1 are special cases of the following general case when the compounding frequency m takes the values: m = 1, 2, 4, 12, 52, 365.
Compounding with Frequency m times per year for t years:
This means that every 1/m portion of the year, an investment P V earns interest P V (r/m), which is reinvested. Therefore, after t years of such compounding, an initial investment P V reaches the terminal value F V given by
F V = P V1 + r m mt or P V = F V1 + r m −mt . (3.4)
3. FIXED INTEREST RATES 37
Solving for P V , as we have done, gives the companion discounting formula. Given a F V , it yields its present discounted worth of the future payment. The formulas in Table 1 are special cases of the general formula (3.4) when the compounding frequency m takes the values: m = 1, 2, 4, 12, 52, 365.
Example 1. The following table lists the value of $100 after 1 year of compound-ing with frequency m and annual interest rate r = 0.1.
m 100(1 + 0.1/m)m
(Compounding frequency) (Value of $100 in 1 year)
1 110.00 2 110.25 4 110.38129 12 110.47131 52 110.50648 365 110.51558 10,000 110.51704 1,000,000 110.51709
Exercise 1. Find the amount you need to deposit now into an account paying the annual interest rate of 5% compounded monthly so that it becomes $100,000 in 10 years.
Effective annual rate. If interest is reinvested with frequency m for t years at annual rate r, then the effective annual rate rm is the equivalent rate (the one that
gives the same terminal value) with compounding frequency 1. Thus, rm is found by
solving the equation
P V (1 + rm)t= P V 1 + r m mt , to get rm = 1 + r m m − 1. (3.5)
Example 2. If the compounding frequency m = 365 and the annual interest rate r = 10% = 0.1, then the effective annual interest r365 is equal to:
r365 = 1 + r 365 365 − 1 ≈ 0.10516.
Exercise 2. Given an annual interest rate (nominal interest rate) of 0.04, find the effective annual rate when interest is compounded every minute.
3.1. Finding the ask & bid price of a T-Bill. Here we illustrate how the formulas in the section above are used in a T-Bill situation.
Example 3 (Finding the Ask & Bid Price of a T-Bill). Dealers or market makers set a bid and ask price on financial securities. The bid price Pbid is what a dealer pays
when a customer wants to sell her security, while the ask price Pask is the price the
customer pays when they are buying the security from the dealer. The dealer makes a profit when you buy and then sell a security. The dealer’s profit is
Profit = Pask− Pbid.
The table provides a summary of who gets what from a trade.
Customer Dealer
Bid Price Receive Payment Pay
Ask Price Pay Receive Payment
Treasury bill, or T-Bill, security dealers quote a bid and ask yield on a discount basis rather than on a price basis for historical reasons. These discount quotes dictate the terms of the contractual agreement, and the ask yield gives the investor the literal annual yield from the bill. To determine the T-Bill’s bid and ask price, you can calculate it from the discounted yields. Suppose we have a T-Bill with the following information:
Time to Maturity: 34 days
Bid yield on discount basis: rdbid = 0.080%
Ask yield on discount basis: rdask = 0.060%
Ask yield: rask = 0.061%,
and we would like to determine how much we would pay (Pask) for a T-Bill with
future value F V = $1 million. Using the formula for P V (3.3), the actual ask yield on an annual basis for the T-Bill is defined by1
rask = 365 d F V − Pask Pask = 365 d 1 − Pask Pask .
To simplify calculations in the old days, the convention of multiplying by 360 rather than 365 was adopted to simplify computations. The same is true for the division
1Fabozzi and Mann (2010) refer to this as the bond equivalent yield when d ≤ 182. This yield is more complicated for d > 182 because the bond has semi-annual coupon rates. See Fabozzi and Mann (2010, p. 133).
3. FIXED INTEREST RATES 39
by 1 (FV) instead of Pask. This inspired the definition of the ask yield on a discount
basis rdask, an annual interest rate:
rdask . = 360 d F V − Pask F V = 360 d 1 − Pask 1 .
Now, to find the ask price of our T-Bill, we apply this formula with d = 34 days 0.00060 = 360
34
1 − Pask
1
to get Pask = 0.9999433333. Therefore, you pay $999,943.33 for a $1 million T-Bill
which expires in 34 days. Similarly, to find the bid price of the bill, we use the formula for the bid yield on a discount basis:
rdbid . = 360 d F V − Pbid F V = 360 d 1 − Pbid 1 .
Exercise 3. What is the broker’s profit in Example 3?
Example 4. The following is screenshot of a T-Bill quote from Wall Street Journal’s website. From this quote, we can read off all of the necessary
informa-Figure 2. Quote for a T-Bill purchased on January 29, 2012
tion needed to find the T-Bill’s price. The bond matures 4 days after its purchase date, and it has a discounted bid and ask yield of rdbid = 0.04% = 0.0004 and
rdask = 0.03% = 0.0003, respectively. The actual ask yield, or return, of this treasury
bill is rask = 0.03% = 0.0003. Using the formula for rdask, we can solve to find that
the price of a $1 million T-Bill is Pask = $999, 996.67.
Exercise 4. Find a T-Bill quote for today, and determine the bid and ask price of the bill.
Exercise 5. Find the ask and bid price for a $1 million T-Bill with the quote in Table 2.
This Table can be read as:
Purchase Date: January 20, 2012 Maturity Date: March 1, 2012 Bid yield on discount basis, rdbid = 0.03%
Ask yield on discount basis, rdask = 0.02%
Date Maturity Bid Asked Chg Asked Yield 1/20/2012 3/1/2012 0.030 0.020 0.01 0.0200
Table 2.
How much would you pay a dealer for this $1 million T-Bill? What is the dealer’s revenue if he buys and immediately sells this T-Bill?
Exercise 6. Find the ask and bid price for a $1 million T-Bill with the quote in Table 3.
Date Maturity Bid Asked Chg Asked Yield 1/20/2012 1/10/2013 0.11 0.090 0.01 0.0910
Table 3.
How much would a dealer pay you for this $1 million T-Bill? What is the dealer’s revenue if he buys and immediately sells this T-Bill?
Remark 3.1. T-Bill quotes can be found at http://online.wsj.com/ by clicking on Markets → Market Data Center ⇒ Bonds ⇒ Bonds, Rates, & Credit Markets ⇒ Treasury Quotes, or simply typing “T-Bill Quote WSJ” into Google and choosing the first link. To determine the number of days to maturity, you may use the website http://www.timeanddate.com/date/duration.html.
Alternatively, one can use the Excel function = DATEVALUE(”10 – Jan – 2013”) to find the excel number for the maturity date and subtract off the purchase date = DATEVALUE(”20 – Jan – 2012”).
4. Discrete income streams
In this section, we will consider two examples of discrete income streams: coupon bonds and mortgages. Using the simple notions of present value, future value, and interest rates, we will learn how to read coupon bond quotes and calculate mortgage payments.
4.1. Finding the yield of a coupon bond. A holder of a coupon bond receives scheduled payments Ct at the end of each period t (typically every 6 months). The
4. DISCRETE INCOME STREAMS 41
value, see Table 1, we can determine the current value of the discrete income stream produced by the coupons.
Example 5 (Finding the Yield of a Coupon Bond). Consider a $100 coupon bond which pays coupons every 6 months with the following information:
Time Elapsed in Period: 92 (out of 184) days Time to Maturity: 2.5 periods or 1 yr 92 days
Ask Price: $106.00 Bid Price: $105.97 Annual Coupon Rate: 6%
To determine the profitability of this investment, as compared with others, we would like to find the average annual rate of return, or the yield to maturity, of this coupon note. The yield to maturity is the interest rate, y, which makes the present value of the bond’s income stream, P V (y), equal the amount paid.
First, we notice that since the bond was purchased in the middle of a payment period, the buyer will owe the note’s seller the unpaid accrued interest in addition to the price of the bond. Since half of the period has elapsed, the buyer will pay
Ask Price + Accrued Interest = 106 + 1 2·
6
2 ≈ $107.50.
From the time line in Figure 3, we see that 2.5 six month periods remain. At the end of each period, the bond holder will be paid $3.00, and at the end of the last period, she will also be paid the principal.
t = 0 t = 1 t = 2 t = 3 $3.00 Time (in 6 mo periods) Payments $3.. $3.00 $103.00 t =1-92 180
Figure 3. Payment timeline for a coupon note
Using the formulas for present value, see Table 1, we have that the present value of this income stream is
P V (y) = $3.00 (1 + y/2)18492 + $3.00 (1 + y/2)1+18492 + $3.00 + $100 (1 + y/2)2+18492 . (4.6)
Thus, to find the yield y, we must solve $3.25 (1 + y/2)18092 + $3.25 (1 + y/2)1+18092 + $3.25 + $100 (1 + y/2)2+18092 = $107.5. for y.
∞ y 1 2 3 4 5 6 PV(y) 119 102 85 51 34 17 0 68 109.00 Figure 4. P V (y)
As we can see in Figure 4, the function P V (y) is $109.00 when y = 0 and goes to 0 as y → ∞. By the Intermediate Value Theorem, since 0 < 108.06 < 109.00, there must exist a solution y such that P V (y) = 107.5. We use Maple’s fsolve command to find that the average annual yield of this bond is y ≈ 0.01351 or 1.351%.
Example 6. Table 4 contains the information for a U. S. Treasury coupon note quote from Wall Street Journal’s website. This quote is for the purchase of the bond on July 23, 2012. From this quote, we can read off all of the pertinent information about the note. This note matures 22 days after its purchase date and for a $100 note, the bid and ask price are $104.2891 and $104.3047, respectively. The accrued interest is $4.252 × 1 − 22
182 = $1.868. 182 days are used for the length of the period
from February 15, 2012 to August 15, 2012. Using the present value calculations as in (4.6), the asked yield of the bond is y = 0.0018288 = 0.183%.
Purchase Date Maturity Coupon Bid Asked Chg Asked Yield 7/23/2012 8/15/2013 4.25 104.2891 104.3047 -0.0156 0.185
Table 4.
Remark 4.1. Notice that this result is different from the ask yield in the Table 4. Based on the yield in this Table P V (y) = $106.224376 rather than $104.3047 + $1.861 = $106.1728. You may think that 5 pennies on $100 is not that important but if you are trading $1 million at a time the difference is $515.76. The reason for this difference is an arbritrage opportunity, which investment banks like Goldman Sachs take advantage of. The arbitrage opportunity arises because the coupon bonds can be broken up into individual zero coupon bonds. In the case of Example 5 there are three separate payments which can be used to create three bonds. These three zero coupon bonds are called Treasury Strips. These bonds would have a different terms to maturity for example the first bond would have a term to maturity of 92 days.
4. DISCRETE INCOME STREAMS 43
Thus, you would use the ask yield to maturity on a 3 month T-Bill, while the second payment as a yield to maturity of 9 months. It turns out that the yield changes as the yield to maturity changes.2 In most cases, the yield increases as the maturity date increases. Thus, one can make money by buying the cheaper PV and selling short the more expensive PV.
Exercise 7. Find a US government note sold today, and determine its yield to maturity as in Example 5.
www.online.wsj.com: Home ⇒ Personal Finance ⇒ Investment ⇒ Data Center ⇒ Bond Rates ⇒ Treasury Quotes.
Exercise 8. Find the average annual rate of return, or yield to maturity, for the US government coupon note with a principal of $100 in Table 5.
Purchase Date Maturity Coupon Bid Asked Chg Asked Yield 1/16/2013 8/15/2013 0.750 100.3516 100.3594 0.0 0.129
Table 5.
If a dealer buys and sells this note, what will her revenue be per million dollars of this bond?
Exercise 9. Find the average annual rate of return, or yield to maturity, for the following US government coupon note with a principal of $100 in Table 6. If a dealer buys and sells this note, what will her revenue be per million dollars of this bond?
Purchase Date Maturity Coupon Bid Asked Chg Asked Yield 1/20/2012 12/15/2012 1.125 100.8594 100.9297 -0.0313 0.085
Table 6.
Remark 4.2. Such bond quotes can be found at http://online.wsj.com/ by click-ing on Markets ⇒ Market Data Center ⇒ Bonds ⇒ Bonds, Rates, & Credit Markets → Treasury Quotes, or simply typing “Coupon Bond Quote WSJ” into Google and choosing the first link. To determine the number of days to maturity, you may use the website http://www.timeanddate.com/date/duration.html.
2You can find these rates on the same website under Treasury Stips. For detailed description of the process see Fabozzi and Mann (2010).
4.2. Finding mortgage payments. The second example of a discrete income stream is a mortgage. Annuities, or mortgages, are used when an individual would like to borrow money and repay it over time in fixed payments. Each payment period, the borrower must pay a fixed amount C. If a loan of $P V is issued for a duration of n periods, then using the formulas for present value, we find that
P V = C 1 + r + C (1 + r)2 + · · · + C (1 + r)n = C n X k=1 1 (1 + r)k,
where r is the interest rate per period and C is the payment per period. Using the formula for a geometric series 1 + x + · · · + xn−1= 1−x1−xn, we can solve to find that
P V = C r 1 − 1 1 + r n . (4.7)
This formula also allows us to find the monthly payment C for an amount of money borrowed P V :
Example 7. Suppose you take out a 30 year mortgage for $200, 000 at an (annual) interest rate of 6%. What is your monthly payment?
Using (4.7), we can solve for the monthly payment. Here the monthly interest rate is r = 0.06
12 , the number of time periods is n = 12 · 30 = 360 months, and the
principal is P V = 200, 000. The monthly payment for this mortgage is
C = rP V 1 − (1+r1 )n = 0.06 12 200, 000 1 −1+10.06 12 360 = $1, 199.10.
Exercise 10. Look up the average mortgage rate on a 30 year loan and determine the amount you can borrow given that you can afford $1,500 per month.
5. Continuous compounding
In the previous sections, we have considered the value of investments for which interest is compounded yearly, monthly, or daily. Now we ask what happens as we continue this trend and compound all of the time or continuously. If we let the compounding frequency m go to infinity, we obtain what is known as continuous compounding. From the formula for compounding frequency m (3.4), the future
5. CONTINUOUS COMPOUNDING 45
value of an investment of P V at an annual interest rate r is F V = lim m→∞P V 1 + r m mt = P V lim m→∞ 1 + m1 r mr·rt = P V lim n→∞ 1 + 1 n n·rt = P Vh lim n→∞ 1 + 1 n nirt . Now, using the well-known definition of e:
e ˙= lim n→∞ 1 + 1 n n , (5.8)
we obtain the continuous compounding formula and the companion discounted value formula:
F V = P V ert and P V = F V e−rt, (5.9) respectively. The following table lists the terminal values of $1 after one year earning the annual interest rate of 100% with compounding frequency n = 1, 2, 3, . . . :
n (1 + 1/n)n
(Compounding frequency) (Value of $1 in 1 year)
1 2 2 2.25 4 2.44141 12 2.61304 24 2.66373 52 2.69260 365 2.71457 1,000 2.71692 10,000 2.71815 100,000 2.71827 1,000,000 2.71828
In mathematical terms, the sequence in the second column of the table is a se-quence which converges to e. This sese-quence converges since it increases and is bounded from above. The table suggests that the upper bound is a little above 2.718. Indeed, it can be shown that 1 + n1n< 3 for all n. The limit is an important irrational num-ber, which is denoted by the letter e in honor of the great mathematician Leonhard Euler (1707-1783). Its approximation to 40 decimal places is:
Example 8. Suppose you invest $10,000 in an account paying 4% annual interest, and you don’t make any further deposits or withdrawals. How much will you have at the end of 3 years if the interest is compounded continuously?
Use formula (5.9) with P V =10,000, r = 0.04 and t = 3 to get the amount F V = 10, 000e0.04·3 ≈ $11, 274.97.
Exercise 11. How much money must be put into a continuously compounded account paying 5% annual interest in order to have $10,000 at the end of 4 years? Effective annual rate for continuous compounding If r is a continuously com-pounded interest rate, then the effective annual rate reis found by solving the equation
P V (1 + re)t= P V ert;
i.e.,
re = er− 1. (5.10)
More generally, to find the relation between an interest rate rc with continuous
compounding and an interest rate rm with compounding frequency m, we must solve
the equation P V1 + rm m mt = P V erct or 1 + rm m m = erc. (5.11)
Example 9. Given a nominal interest rate of 0.04, find the effective annual rate if interest is compounded continuously.
Using formula (5.10), we find
re = e0.04− 1 = 0.0408107742.
Exercise 12. Assume that a bank offers you a savings account with the annual interest rate of 5% compounded daily. What is the equivalent rate with continuous compounding?
6. Variable interest rates
What if the interest rate is not fixed? Next, we discuss continuous compounding when the annual interest rate is variable, that is r = r(t). For this, let us assume that at t = 0 we are investing an amount A0, which at any subsequent time t is earning
7. CONTINUOUS INCOME STREAMS 47
of our investment at time t and by A(t + ∆t) the amount of our investment at time t + ∆t, then we have
A(t + ∆t) − A(t) ≈ r(t) · A(t) · ∆t, assuming that r(t) is continuous. Next, dividing by ∆t gives
A(t + ∆t) − A(t)
∆t ≈ r(t) · A(t).
Finally, letting ∆t go to zero yields the first order differential equation (DE) dA
dt = r(t)A(t). (6.12)
Solving it with initial data A(0) = A0, we obtain the formula
A(t) = A0e Rt
0r(s)ds,
which in the present/future value notation reads as follows:
F V = P V eR0tr(s)ds. (6.13)
Exercise 13. Solve DE (6.12) to produce formula (6.13). 7. Continuous income streams
Suppose that at t = 0, we open an investment account with an initial amount A0, and thereafter, we make continuous deposits at the rate of S(t) per year. If the
account pays an interest rate of r, compounded continuously, then we want to find the amount A(t) in the account at any subsequent time t.
We begin by modeling this situation with a differential equation. After we have a model, we will solve it to find the amount A(t). Observe that A(t) grows in two ways: by deposits and interest. The deposits increase the amount of money at a rate of A(t) per year, and the interest increases the amount at the interest rate r of the current balance.
To translate this information into the form of a differential equation, we let ∆t denote a very small amount of time, and we compare the balance at time t with the balance at time t + ∆t. During that time interval, the interest paid is approximately rA(t)∆t (which represents money growing at a rate of r of the current balance A(t) for a period of length ∆t). The new money added is approximately S(t)∆t (which represents money being added at a constant rate of S(t) for a period of length ∆t). Therefore, the approximate change in the balance is given by the formula
and we can make the approximation as good as we want by taking ∆t small enough. Dividing both sides by ∆t, we get
A(t + ∆t) − A(t)
∆t ≈ rA(t) + S(t).
Letting ∆t → 0, we obtain the first order linear differential equation dA
dt = rA(t) + S(t). (7.14)
Our continuous stream investment problem has now been modeled by equation (7.14) and the initial condition
A(0) = A0. (7.15)
Solving initial value problem (7.14)–(7.15) gives the formula:
A(t) = A0ert+
Z t
0
S(τ )er(t−τ )dτ (7.16)
Exercise 14. Solve initial value problem (7.14)– (7.15) to derive formula (7.16). Exercise 15. Verify that initial value problem (7.14)– (7.15) models continuous income streams even in the case that r = r(t) (variable) and solve it.
Exercise 16. Suppose you open a retirement account at age 30 with an initial amount of $5,000 and thereafter make continuous deposits at the rate of $10,000 per year. If the account pays an interest rate of 7%, compounded continuously, what is the balance in the account at any given time? In particular, what is the balance at the retirement age of 65?
Exercise 17. Suppose that a home buyer plans to take a 15-year mortgage at an interest rate of 5% and cannot spend more than $2,000 per month on payments.
(a) What is the maximum amount that she can afford to borrow?
(b) If the buyer borrows this maximum amount, how much total interest will she pay?
For simplicity, assume that the interest is compounded continuously and that the payments are made continuously at the rate of $24,000 per year. In practice, these assumptions may not be precisely satisfied, but they greatly simplify the problem and give close approximations to the actual solution.
8. STOCK PRICING 49
8. Stock pricing
A stock is a certificate of ownership of a company. A share is a unit of stock. At t = 0, the stock value, S0, is given by
S0 =
Market value of company Number of shares of company.
Example 10. If a company today has 40 million shares and the value of each share is $50, then the market value of the company is $50 · 40 million or $2 billion.
Stocks do not come with the predictability or security of bonds: (1) Dividends are unknown until payment.
(2) There is no maturity date for a stock.
(3) In the case of bankruptcy, bond holders are paid off before the shareholders. The residual of the company’s assets after meeting all other liabilities are divided among the shareholders.
Corporate stock offers a share of ownership in a company. There are two types of stocks: common and preferred. Common stock provides a dividend payment each quarter. Dividends, denoted Dt, are the share of the company’s profits that each
shareholder is entitled to. The company’s profits are divided so that dividends per share equal reported profits divided by number of shares. Common shareholders have the right to vote in corporate elections. Preferred stock, on the other hand, pays a specified dividend before the common shareholders receive dividends but after the bondholders are paid. Unlike common stock holders, preferred shareholders do not have voting rights.
The value of a stock is considered the sum of the dividends paid for a particular stock before the company goes bankrupt. The value or price of the stock is denoted S0: S0 = N X t=1 Dt (1 + rt)t .
Exercise 18. Consider a simple world where a stock gives the same dividend D = $10 at the end of each year. If the annual interest is 2% and compounded annually, what is the price of this stock?
The time N at which the company goes bankrupt, the dividends at each time of payment Dt, and the interest rate rt and any time t are all unknown. Therefore, the
price of a stock is based on the expectation of its future activity: S0 = E " N X t=1 Dt (1 + rt)t # .
The E [·] denotes the expected value.
The return on a stock is the profit a stockholder makes from selling the stock relative to the price at which it was purchased. The return R(t) is given by:
R(t) = Capital Gain + Dividend Yield = St− S0 S0
+ Dt S0
.
Note: The return on a stock R(t) is different from the interest rate rt although each
are dependent on the specific time t.
People make decisions regarding the trading of stocks in advance of knowing the outcome of the stock based on expected returns:
E [R(t)] = E St− S0 S0 + Dt S0 .
There is a risk involved in purchasing stocks that is not present while purchasing bonds. For a bond, the expected value of the bond is merely the bond’s face value plus interest - there is no uncertainty about the future value. The stock, however, might go up or down in the next period, so its future value is not determined. For this reason, we can only consider the “average” or expected future value of a stock.
For instance, if the “up value” of the stock S1(H) = uS0 occurs with probability
p and the “down value” S1(T ) = dS0 occurs with probability 1 − p, then the expected
payoff of a stock is p(uS0) + (1 − p)(dS0). Later, we shall go into more detail about
the risks involved in purchasing stocks compared to purchasing bonds.
Example 11. Suppose we have a S0 = $50 stock which can attain the value
S1(H) = 100 with probability p = 0.5 and the value S1(T ) = 25 with probability
1 − p = 0.5. If the stock dividends are 2% of S0, then what are the possible rates of
return for the stock?
For the higher stock price, the return is: R(H) = S1(H) − S0 S0 +Dt S0 = 100 − 50 50 + 0.02 = 1.02 or 102%. For the lower stock price, the return is:
R(T ) = S1(T ) − S0 S0 + Dt S0 = 25 − 50 50 + 0.02 = −0.48 or − 48%. The expected return is:
10. CHAPTER EXERCISES 51
9. Short selling
If an investor is confident that a stock’s price is going to decrease, he can make a profit by short selling the stock. The ideal scenario for someone engaging in this activity is as follows. The investor borrows the stock from a broker at time t = 0 under the agreement that the stock will be returned at t = T . He then sells the stock immediately for S0. After the price drops, the investor buys the stock at ST and
returns it, making a profit of
Profit = S(0) − S(T ) − Transaction Fees to Broker for Stock. Naturally, if the price goes up, the investor will lose money.
Example 12. Suppose you consider the stock price in Example 11, and you decide to short this stock since you think the probability of the low price is 1 − p = 0.9. How much would you gain when the price goes up and down? What is your expected gain?
If the price is high, your loss is
S0− S1(H) = 50 − 100 = −50.
If the price is low, your gain is
S0− S1(T ) = 50 − 25 = 25.
Your expected gain is
0.1(−50) + 0.9(25) = −5 + 22.5 = 17.5.
Exercise 19. Explain why you would have shorted Lehman Brothers stock in July 2007. What would have been the most beneficial term to maturity for you?
10. Chapter exercises
Exercise 20. If you would like to make a return of $10,000 on an investment in one year with a simple interest rate of r = 0.05, how much money should you plan to invest initially?
Exercise 21. If you would like to make a return of $10,000 on an investment in one year that grows by a compounded quarterly interest rate of rQ = 0.05, how much
should you plan to invest initially?
Exercise 22. Calculate the ask price Pask on a FV = $1,000,000 T-bill maturing
in 40 days with an actual ask yield of rask = 0.05%.
Exercise 23. Given a nominal interest rate of 0.05, find the effective annual rate given that the interest rate is compounded (a) daily, (b) hourly, and (c) continuously.
Exercise 24. Find the present value of a bond B0 with a continuous cash flow
B(t) = $5 and interest rate r = 0.05 held for 6 years.
Exercise 25. The holder of a coupon bond receives semi-annual coupon payments of $10 for 5 years while holding the bond at an interest rate of r = 0.05. At the time of the bonds purchase, what is the present value of those coupon payments?
Exercise 26. In order to purchase a new house, someone would like to borrow $400,000 at an interest rate of r = 5% annually and make constant monthly payments to repay the mortgage. What is the value of the monthly payments on a 30-year mortgage?
11. Glossary of terms
Definitions come from Bodie, Kane, and Marcus (2008), and Wikipedia.
• Bankruptcy – A legally declared inability or impairment of ability of an in-dividual or organization to pay their creditors.
• Bond – Marketable financial security that gives the owner the right to a fixed payment, at a predetermined future date.
• Borrower – An individual or organization who spends more than her current income.
• Corporate Bond – A bond issued by a corporation. The term is usually ap-plied to longer-term debt instruments, generally with a maturity date falling at least a year after their issue date. (The term “commercial paper” is some-times used for instruments with a shorter maturity.)
• Corporate Stock – A share, also referred to as equity, of stock means a share of ownership in a corporation (company).
Common – Pays a dividend payment each quarter. The company’s profits are divided so that dividends per share equal reported profits divided by number of shares. Common shareholders have the right to vote in corpo-rate elections.
Preferred – Pays a specified dividend before the common shareholders receive dividends but after the bondholders are paid. Preferred shareholders do not have voting rights.
11. GLOSSARY OF TERMS 53
• Coupon Payment – A periodic interest payment that the bondholder receives during the time between when the bond is issued and when it matures. • Creditor – An entity that is promised a payment in the future.
• Debtor – An entity that owes a debt to someone else; the entity could be an individual, a firm, a government, or an organization. The counterparty of this arrangement is called a creditor.
• Direct Financial Instrument – Contract between initial saver and borrower. • Dividends – Payments made by a corporation to its shareholder members.
When a corporation earns a profit or surplus, that money can be put to two uses: it can either be re-invested in the business (called retained earnings), or it can be paid to the shareholders as a dividend.
• Future Value– The nominal future sum of money that a given sum of money is “worth” at a specified time in the future assuming a certain interest rate, or more generally, rate of return.
• Liquidity – An asset’s ability to be easily converted through an act of buy-ing or sellbuy-ing without causbuy-ing a significant movement in the price and with minimum loss of value.
• Maturity – The future date at which a bond is repaid.
• Present Value – The value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk.
• Principal – Payment promised on a bond at its maturity.
• Saver – An individual or organization who spends less than her current in-come.
• Security – A contract between a borrower and a saver, which specifies the terms of the deal.
• Short Selling or “shorting” – The practice of selling a financial instrument that the seller does not own at the time of the sale. Short selling is done with intent of later purchasing the financial instrument at a lower price.
12. References
Bodie, Zvi, Kane, Alex, and Marcus, Alan J. Essentials of Investments edition 6 or 7, McGraw-Hill, New York, NY 2006 or 2008.
Cvitani´c, Jakˇsa, and Fernando Zapatero Introduction to the Economics and Math-ematics of Financial Markets , MIT press (2004).
Himonas, Alex, and Howard, Alan Calculus: Ideas & Applications, John Wiley & Sons , NJ 2003.
