Investment Science - Part I: Deterministic Cash Flow StreamsInvestment Science Part I: Deterministic...

Investment SciencePart I: Deterministic Cash Flow Streams

Dr. Xi CHEN

Department of Management Science and EngineeringInternational Business School

Beijing Foreign Studies University100089, Beijing, People’s Republic of China

Xi CHEN ([email protected]) Investment Science 1 / 174

Outline

1 Principal and Interest

2 Present Value

3 Present and Future Value of Streams

4 Internal Rate of Return

5 Evaluation Criteria

6 Applications and Extensions


Principal and Interest

Outline


2 Present Value







Example

If you invest $1.00 in a bank account that pays 8% interest per year, thenat the end of 1 year you will have in your account the principal (youroriginal amount) at $1.00 plus interest at $0.08 for a total of $1.08.

What if you invest a larger amount, say A dollars, in the bank?

What if the interest rate is r?



Simple Interest

Under a simple interest rule, money invested for a period different from 1year accumulates interest proportional to the total time of the investment.

If an amount A is left in an account at simple interest r , the totalvalue after n years is

V = (1 + rn)A.

If the proportional rule holds for fractional years, then after any timet (measured in years), the account value is

V = (1 + rt)A.

The account grows linearly with time. The account value at any time isjust the sum of the principal and the accumulated interest, which isproportional to time.



Compound Interest

Example

Consider an account that pays interest at a rate of r per year. If interest iscompounded yearly, then after 1 year, the first year’s interest is added tothe original principal to define a larger principal base for the second year.What is the account value after n years?

The account earns interest on interest!

Under yearly compounding, after n years, such an account will grow to

V = (1 + r)nA.

This is the analytic expression for the account growth under compoundinterest. This expression is said to exhibit geometric growth because ofits nth-power form.



Rule (The seven-ten rule)

Money invested at 7% per year doubles in approximately 10 years. Also,money invested at 10% per year doubles in approximately 7 years.



Exercise (The 72 rule)

The number of years n required for an investment at interest rate r todouble in value must satisfy

(1 + r)n = 2.

By Taylor series, we have

ln (1 + x) = x − x2

2+

x3

3− . . .+ (−1)n−1 x

n

n+ . . . (−1 < x ≤ 1).

Using ln 2 = 0.69 and the approximation ln(1 + r) ≈ r valid for smallr , show that n ≈ 69/i , where i is the interest rate percentage, i.e.,i = 100r .

Using the better approximation ln(1 + r) ≈ r − r2/2, show that forr ≈ 0.08, there holds n ≈ 72/i .



Compounding at Various Intervals

Example (Quarterly compounding)

Quarterly compounding at an interest rate of r per year means that aninterest rate of r/4 is applied every quarter. Hence, money left in the bankfor 1 quarter will grow by a factor of 1 + (r/4) during that quarter. If themoney is left in for another quarter, then that new amount will grow byanother factor of 1 + (r/4).

What is the account value after 1 year?

(1 +

r

4

)4> 1 + r , ∀r > 0.

Right or not, why?

What is the meaning?



Definition

Effective interest rate is the equivalent yearly interest rate that wouldproduce the same result after 1 year without compounding.

Example

An annual rate of 8% compounded quarterly will produce an increase of(1.02)4 ≈ 1.0824; hence the effective interest rate is 8.24%. The basicyearly rate (8% in this example) is termed the nominal rate.

Compounding can be carried out with any frequency.

The general method is that a year is divided into a fixed number of equallyspaced periods, say, m periods. The effective interest rate is the number r ′

that satisfies1 + r ′ =

(1 +

r

m

)m.



Continuous Compounding

Imagine dividing the year into infinitely small periods and use

limm→∞

(1 +

r

m

)m= er ,

where e = 2.7818... is the base of the natural logarithm. The effectiveinterest rate is the number r ′ that satisfies

1 + r ′ = er .



To calculate how much an account will have grown after any arbitrarylength of time t ≈ k/m (measured in years), we use(

1 +r

m

)k=(

1 +r

m

)mt⇒ lim

m→∞

[(1 +

r

m

)m]t= ert .

Continuous compounding leads to the familiar exponential growth curve.



Debt and Money Markets

Exactly the same thing happens to debt. If I borrow money from the bankat an interest rate r and make no payments to the bank, then my debtincreases according to the same formulas. Specifically, if my debt iscompounded monthly, then after k months my debt will have grown by afactor of [1 + (r/12)]k .


Present Value

Outline


2 Present Value






Present Value

Example

Consider two situations.

1 You will receive $110 in 1 year.

2 You receive $100 now and deposit it in a bank account for 1 year at10% interest.

Clearly, these situations are identical after 1 year!

We say that the $110 to be received in 1 year has a present value of$100. In general, $1 to be received a year in the future has a present valueof $1/(1 + r), where r is the interest rate.

The process of evaluating future obligations as an equivalent present valueis alternatively referred to as discounting. The factor by which the futurevalue must be discounted is called the discount factor. The 1-yeardiscount factor is

d1 =1

1 + r.


Present Value

The formula for present value depends on the interest rate that is availablefrom a bank or other source.

Example

Suppose that the annual interest rate r is compounded at the end of eachof m equal periods each year; and suppose that a cash payment of amountA will be received at the end of the kth period.

Then the appropriate discount factor is

dk =

[(1 +

r

m

)k]−1,

and thus the present value of a payment of A to be received k periods inthe future is

PV = dkA.


Present and Future Value of Streams

Outline


2 Present Value







The Ideal Bank

An ideal bank applies the same rate of interest to both deposit, andloans, and it has no service charges or transactions fees.

The interest rate applies equally to any size of principal.

Separate transactions in an account are completely additive in theireffect on future balances.

Interest rates for all transactions may not be identical.

Example

1 A 2-year CD might offer a higher rate than a 1-year CD.

2 A 2-year CD must offer the same rate as a loan that is payable in 2 years.

If an ideal bank has an interest rate that is independent of the lengthof time for which it applies, and that interest is compoundedaccording to normal rules, it is said to be a constant ideal bank.



Future Value

Theorem (Future value of a stream)

Given a cash flow (x0, x1, . . . , xn), and interest rate r each period, thefuture value of a stream is

FV =n∑

i=0

xi (1 + r)n−i = x0(1 + r)

n + x1(1 + r)n−1 + . . .+ xn.

Example

Consider the cash flow stream (−2, 1, 1, 1) when the periods are years andthe interest rate is 10%. The future value is 0.648.



Present Value

Theorem (Present value of a stream)

Given a cash flow (x0, x1, . . . , xn), and interest rate r each period, thepresent value of a stream is

PV =n∑

i=0

xi(1 + r)i

= x0 +x1

1 + r+

x2(1 + r)2

+ . . .+xn

(1 + r)n.

Example

Consider the cash flow stream (−2, 1, 1, 1). Using an interest rate of 10%,the present value is 0.487.

The relationship between PV and FV is

PV =FV

(1 + r)n.



Frequent and Continuous Compounding

Suppose that r is the nominal annual interest rate and interest iscompounded at m equally spaced periods per year. Furthermore, supposethat cash flows occur initially and at the end of each period for a total ofn periods, forming a stream (x0, x1, . . . , xn). Then

PV =n∑

k=0

xk[1 + (r/m)]k

.

Suppose now that the nominal interest rate r is compounded continuouslyand cash flows occur at times t0, t1, . . . , tn. Denote by x(tk) the cash flowat time tk . Then

PV =n∑

k=0

x(tk)e−rtk .



Present Value and an Ideal Bank

In general, if an ideal bank can transform the stream (x0, x1, . . . , xn) intothe stream (y0, y1, . . . , yn), it can also transform in the reverse direction.

Definition

Two streams that can be transformed into each other are said to beequivalent streams.

Theorem (Main theorem on present value)

The cash flow streams x = (x0, x1, . . . , xn) and y = (y0, y1, . . . , yn) areequivalent for a constant ideal bank with interest rate r if and only if thepresent values of two streams, evaluated at the bank’s interest rate, areequal.

Proof.

Since x⇔ (vx , 0, . . . , 0) and y⇔ (vy , 0, . . . , 0), the result is obvious.


Internal Rate of Return

Outline


2 Present Value







Definition

Let (x0, x1, . . . , xn) be a cash flow stream. Then the internal rate ofreturn of this stream is a number r satisfying the equation

0 =n∑

i=0

xi(1 + r)i

= x0 +x1

1 + r+

x2(1 + r)2

+ . . .+xn

(1 + r)n.

Equivalently, it is a number r satisfying 1/(1 + r) = c , where c satisfiesthe polynomial equation

0 =n∑

i=0

xici = x0 + x1c + x2c

2 + . . .+ xncn.

Example

Consider again the cash flow sequence (−2, 1, 1, 1). Its internal rate ofreturn is r = 0.23 with c = 0.81.



Exercise (Newton’s method)

Suppose that we define f (λ) = −a0 + a1λ+ a2λ2 + . . .+ anλn, where allai ’s are positive and n > 1. Here is an iterative technique that generates asequence λ0, λ1, λ2, . . . , λk , . . . of estimates that converges to the rootλ > 0, solving f (λ) = 0. Start with any λ0 > 0 close to the solution.Assuming λk has been calculated, we use

f ′(λk) = a1 + 2a2λk + 3a3λ2k + . . .+ nanλ

n−1k , λk+1 = λk −

f (λk)

f ′(λk).

Try the above procedure on the function f (λ) = −1 + λ+ λ2 with thestarting points λ0 = ±1, ±10, . . .



Theorem (Main theorem of internal rate of return)

Suppose the cash flow stream (x0, x1, . . . , xn) has x0 < 0 and xk ≥ 0 for allk , k = 1, 2, . . . , n, with at least one term being strictly positive. Thenthere is a unique root to the equation

0 =n∑

i=0

xici = x0 + x1c + x2c

2 + . . .+ xncn.

Furthermore, if∑n

k=0 xk > 0 (meaning that the total amount returnedexceeds the initial investment), then the corresponding IRR is positive.


Evaluation Criteria

Outline


2 Present Value






Evaluation Criteria

Net Present Value

Definition

Net present value is the present value of the benefits minus the presentvalue of the costs.

Example

Suppose that you have the opportunity to plant trees that later can be soldfor lumber. This project requires an initial outlay of money to purchaseand plant the seedlings. No other cash flow occurs until the trees areharvested. However, you have a choice as to when to harvest: after 1 yearor after 2 years. If you harvest after 1 year, you get your return quickly; butif you wait an additional year, the trees will have additional growth and therevenue generated from the sale of the trees will be greater. (r = 10%)

1 (−1, 2), cut early, NPV = 0.82.2 (−1, 0, 3), cut later, NPV = 1.48. X


Evaluation Criteria


Provided that it is greater than the prevailing interest rate, the higher theinternal rate of return, the more desirable the investment.

Example (contd.)

Let us use the internal rate of return method to evaluate the two treeharvesting proposals. The equations for the internal rate of return in thetwo cases are

1

−1 + 2c = 0⇒ c = 12

=1

1 + r⇒ r = 1.0. X

2

−1 + 3c2 = 0⇒ c =√

3

3=

1

1 + r⇒ r = 0.7.

In other words, for cut early, the internal rate of return is 100%, whereasfor cut late, it is about 70%. Hence under the internal rate of returncriterion, the best alternative is to cut early.


Evaluation Criteria

Discussion of the Criteria

There is considerable debate as to which of the two criteria, NPV or IRR,is the most appropriate for investment evaluation. Both have attractivefeatures, and both have limitations.

Net present value

is simplest to compute;

does not have the ambiguity associated with the several possible rootsof the internal rate of return equation; and

can be broken into component pieces, unlike internal rate of return.

However, internal rate of return has the advantage that it depends only onthe properties of the cash flow stream, and not on the prevailing interestrate (which in practice may not be easily defined).


Evaluation Criteria

Which criterion to choose?1 In the situation where the proceeds of the investment can be

repeatedly invested in the same type of project but scaled in size, itmakes sense to select the project with the highest internal rate ofreturn in order to get the greatest growth of capital.

2 In one-time opportunity, the net present value method is theappropriate criterion, since it compares the investment with whatcould be obtained through normal channels (which offer the prevailingrate of interest).

Many other factors, e.g., “risk-free” interest rate, rates for borrowing andlending, cost of capital, and rate of return, can influence NPV analysis.

Definition

In business decisions, it is common to use a figure called the cost ofcapital as the baseline rate. This figure is the rate of return that thecompany must offer to potential investors in the company; that is, it is thecost the company must pay to get additional funds. Or sometimes it istaken to be the rate of return expected on alternative desirable projects.


Evaluation Criteria

However, some of these cost of capital figures are derived from uncertaincash flow streams and are not really appropriate measures of a risk-freeinterest rate. For NPV analysis, it is best to use rates that represent trueinterest rates, since we assume that the cash flows are certain.

Another factor to consider is that NPV by itself does not reveal muchabout the rate of return.

Example (NPV and rate of return)

Two alternative investments might each have a net present value of $100,but one might require an investment of $100, whereas the other requires$1,000,000. Clearly these two alternatives should be viewed differently.


Applications and Extensions

Outline


2 Present Value







Net Flows

Example (SimpIico gold mine)

The Simplico gold mine has a great deal of remaining gold deposits, andyou are part of a team that is considering leasing the mine from its ownersfor a period of 10 years. Gold can be extracted from this mine at a rate ofup to 10,000 ounces per year at a cost of $200 per ounce. This cost is thetotal operating cost of mining and refining, exclusive of the cost of thelease. Currently the market price of gold is $400 per ounce. Assume thatthe price of gold, the operating cost, and the interest rate r = 10% remainconstant over the 10-year period.

What is the present value of the lease?

PV =10∑k=1

10000× ($400− $200)1.1k

= $12.29M.



Cycle Problems

Example (Automobile purchase)

You are contemplating the purchase of an automobile and have narrowedthe field down to two choices. Car A costs $20,000, is expected to have alow maintenance cost of $1,000 per year (payable at the beginning of eachyear after the first year), but has a useful mileage life that for youtranslates into 4 years. Car B costs $30,000 and has an expectedmaintenance cost of $2,000 per year (after the first year) and a useful lifeof 6 years. Neither car has a salvage value and r = 10%.

Which car should you buy?

Assume that similar alternatives will be available in the future, and assumea planning period of 12 years, corresponding to three cycles of car A andtwo of car B.

PVA =?⇒ PVA3 = PVA( ? ), X PVB =?⇒ PVB2 = PVB( ? ).



Example (Machine replacement)

A specialized machine essential for a company’s operations costs $10,000and has operating costs of $2,000 the first year. The operating costincreases by $1,000 each year thereafter. Assume that these operatingcosts occur at the end of each year, the machine has no salvage value, andr = 10%.

How long should the machine be kept until it is replaced by a newidentical machine?

Suppose that the machine is replaced every year. Then we have

PV = 10 +2

1.1+

PV

1.1,

since after the first machine is replaced, the stream from that point looksidentical to the original one, except that this continuing stream starts 1year later and hence must be discounted by the effect of 1 year’s interest.



Example (contd.)

We may do the same thing assuming 2-year replacement, then 3 years, andso forth. The general approach is based on the following equation

PVtotal = PV1cycle +

(1

1.1

)kPVtotal ,

where k is the length of the basic cycle.



Taxes

Example (Depreciation)

Suppose a firm purchases a machine for $10,000. This machine has auseful life of 4 years and its use generates a cash flow of $3,000 each year.The machine has a salvage value of $2,000 at the end of 4 years.

The government does not allow the full cost of the machine to be reportedas an expense the first year, but instead requires that the cost of themachine be depreciated over its useful life.

There are several depreciation methods, each applicable under variouscircumstances, but for simplicity we shall assume the straight-line method.In this method a fixed portion of the cost is reported as depreciation eachyear. Hence corresponding to a 4-year life, one-fourth of the cost (minusthe estimated salvage value) is reported as an expense deductible fromrevenue each year.



Example (contd.)

If we assume a combined federal and state tax rate of 43%, we obtain thecash flows, before and after tax, shown in the following table. The salvagevalue is not taxed (since it was not depreciated). The present values forthe two cash flows with r = 10% are also shown.

Tax rules convert an otherwise profitable operation into anunprofitable one!



Inflation

Definition

Inflation rate f : Prices 1 year from now will on average be equal totoday’s prices multiplied by 1 + f . Inflation compounds much likeinterest does.

Constant (real) dollars is defined relative to a given reference year.These are the (hypothetical) dollars that continue to have the samepurchasing power as dollars did in the reference year.

Actual (nominal) dollars are what we really use in transactions.

Real interest rate r0 is the rate at which real dollars increase if leftin a bank that pays the nominal rate.

1 + r0 =1 + r

1 + f⇒ r0 =

r − f1 + f

.



Example (Inflation)

Suppose that inflation is 4%, the nominal interest rate is 10%, and we havea cash flow of real (or constant) dollars as shown in the second column ofthe following table (it is common to estimate cash flows in constantdollars, relative to the present, because “ordinary” price increases can thenbe neglected in a simple estimation of cash flows). To determine thepresent value in real terms, we use the real rate of interest, r0 = 5.77%.


Outline

7 The Market for Future Cash

8 Value Formulas

9 Bond details

10 Yield

11 Duration

12 Immunization

13 Convexity


Outline

An interest rate is a price, or rent, for the most popular of all tradedcommodities – money. The market interest rate provides a readycomparison for investment alternatives that produce cash flows arisingfrom transactions between individuals, associated with business projects, orgenerated by investments in securities.

Definition

Financial instruments: Vast assortments of bills, notes, bonds,annuities, futures contracts, and mortgages are part of thewell-developed markets for money. They are traded only as pieces ofpaper, or as entries in a computer database rather than real goods inthe sense of having intrinsic value.

If there is a well-developed market for an instrument, so that it canbe traded freely and easily, then that instrument is termed a security.

Fixed-income securities are financial instruments that are traded inwell-developed markets and promise a fixed (i.e., definite) income tothe holder over a span of time.


Outline

Fixed-income securities are

important to an investor because they define the market for money,and most investors participate in this market;

important as additional comparison points when conducting analysesof investment opportunities that are not traded in markets, such as afirm’s research projects, oil leases, and royalty fights.

The only uncertainties about the promised stream from a fixed-incomesecurity were associated with whether the issuer of the security mightdefault in which case the income would be discontinued or delayed.

However, some “fixed-income” securities promise cash flows whosemagnitudes are tied to various contingencies or fluctuating indices.

payment levels on an adjustable-rate mortgage may be tied to aninterest rate index; or

corporate bond payments may in part be governed by a stock price.


The Market for Future Cash

Outline


8 Value Formulas

9 Bond details

10 Yield

11 Duration

12 Immunization

13 Convexity



Savings Deposits: Demand deposit pays a rate of interest thatvaries with market conditions. A time deposit must be maintainedfor a given length of time, or else a penalty for early withdrawal isassessed. A similar instrument is a certificate of deposit (CD).Money Market Instruments

Definition

Money market refers to the market for short-term (1 year or less) loansby corporations and financial intermediaries, including, for example, banks.

Definition

Commercial paper is used to describe unsecured loans (that is, loanswithout collateral) to corporations. Eurodollar deposits; Eurodollar CDs.

A banker’s acceptance. If company A sells goods to company B, companyB might send a written promise to company A that it will pay for the goodswithin a fixed time. Some bank accepts the promise by promising to pay thebill on behalf of company B. Company A can then sell the banker’sacceptance to someone else at a discount before the time has expired.



U.S. Government Securities1 U.S. Treasury bills are issued in denominations of $10,000 or more

with fixed terms to maturity of 13, 26, and 52 weeks, and are sold on adiscount basis.

2 U.S. Treasury notes have maturities of 1 to 10 years and are sold indenominations as small as $1,000. The owner of such a note receives acoupon payment every 6 months until maturity.

3 U.S. Treasury bonds are issued with maturities of more than 10 yearsand make coupon payments.

Definition

Some Treasury bonds are callable, meaning that at some scheduled couponpayment date, the Treasury can force the bond holder to redeem the bond at thattime for its face (par) value.

4 U.S. Treasury strips are bonds the U.S. Treasury issue in strippedform, offering minimal risk and some tax benefits in certain states.Each interest payment and the principal payment becomes a separatezero-coupon security. Each component has its own identifying numberand can be held or traded separately.



Definition

A security generates a single cash flow with no intermediate couponpayments is called a zero-coupon bond.

Other Bonds1 Municipal bonds are issued by agencies of state and local

governments. Two main types are general obligation bonds andrevenue bonds.

2 Corporate bonds are issued by corporations for the purpose of raisingcapital for operations and new ventures.

A bond carries with it an indenture. Some of the features might be:1 Callable bonds: A bond is callable if the issuer has the right to

repurchase the bond at a specified price.2 Sinking funds: Rather than incur the obligation to pay the entire

face value of a bond issue at maturity, the issuer may establish asinking fund to spread this obligation out over time.

3 Debt Subordination: To protect bond holders, limits may be set onthe amount of additional borrowing by the issuer.



Mortgages1 The standard mortgage is structured so that equal monthly payments

are made throughout its term, which contrasts to most bonds thathave a final payment equal to the face value at maturity.

2 There may be modest sized periodic payments for several yearsfollowed by a final balloon payment that completes the contract.

3 Adjustable-rate mortgages adjust the effective interest rateperiodically according to an interest rate index.

Mortgages are not usually thought of as securities since theyare written as contracts between two parties!

However, mortgages are typically “bundled” into large packages andtraded among financial institutions. These mortgage-backedsecurities are quite liquid.



Annuities

Definition

An annuity is a contract that pays the holder (the annuitant) moneyperiodically, according to a predetermined schedule or formula, over aperiod of time, e.g., pension.

There are numerous variations, for example,1 Sometimes the level of the annuity payments is tied to the earnings of

a large pool of funds from which the annuity is paid.2 Sometimes the payments vary with time.

Annuities are not really securities since they are not traded!

Annuities are, however, considered to be investment opportunitiesthat are available at standardized rates. Hence from an investor’sviewpoint, they serve the same role as other fixed-income instruments.


Value Formulas

Outline


8 Value Formulas

9 Bond details

10 Yield

11 Duration

12 Immunization

13 Convexity


Value Formulas

Perpetual Annuities

Definition

A perpetual annuity, or perpetuity, pays a fixed sum periodically forever.

Suppose an amount A is paid at the end of each period, starting at theend of the first period, and suppose the per-period interest rate is r . Then

P =∞∑k=1

A

(1 + r)k=

A

1 + r+∞∑k=2

A

(1 + r)k=

A

1 + r+

P

1 + r.

Formula (Perpetual annuity formula)

The present value P of a perpetual annuity that pays an amount A everyperiod, beginning one period from the present, is

P =A

r,

where r is the one-period interest rate.


Value Formulas

Finite-Life Streams

Suppose that the stream consists of n periodic payments of amount A,starting at the end of the current period and ending at period n.

Figure: Time indexing.

The present value of the finite stream relative to the interest rate r perperiod is

P =n∑

k=1

A

(1 + r)k=

A

r− A

r(1 + r)n.


Value Formulas

Formula (Annuity formulas)

Consider an annuity that begins payment one period from the present,paying an amount A each period for a total of n periods. The presentvalue P, the one-period annuity amount A, the one-period interest rate r ,and the number of periods n of the annuity are related by

P =A

r

[1− 1

(1 + r)n

], or equivalently, A =

r(1 + r)nP

(1 + r)n − 1.

Figure: Finite stream from two perpetual annuities.


Value Formulas

Definition

The annuity formula is frequently used in the reverse direction; that is, Aas a function of P. This determines the periodic payment that isequivalent (under the assumed interest rate) to an initial payment of P.This process of substituting periodic payments for a current obligation isreferred to as amortization.

Example (Loan calculation)

Suppose you have borrowed $1,000 from a credit union. The terms of theloan are that the yearly interest is 12% compounded monthly. You are tomake equal monthly payments of such magnitude as to repay (amortize)this loan over 5 years. How much are the monthly payments?

Five years is 60 months, and 12% a year compounded monthly is 1% permonth. Hence we use the formula for n = 60, r = 1%, and P = $1, 000.We find that the payments A are $22.20 per month.


Value Formulas

Definition

The annual percentage rate (APR) is the rate of interest that, ifapplied to the loan amount without fees and expenses, would result in amonthly payment of A.

Example (APR)

Consider a mortgage corresponding to the first listing and calculate thetotal fees and expenses. Using the APR of 7.883%, a loan amount of$203,150, and a 30-year term, A = $1, 474. Using an interest rate of7.625% and the monthly payment calculated, the total initial balance is$208,267. Total of fees and expenses is $208, 267− $203, 150 = $5, 117.The loan fee is 1 point, or $2,032. Hence other expenses are $3, 085.


Value Formulas

Running Amortization

Example

Consider the loan of $1,000, which you will repay over 5 years at 12%interest (compounded monthly). Suppose you took out the loan onJanuary 1, and the first payment is due February 1. The running balanceaccount procedure is consistent with reamortizing the loan each month.

What will happen if one needs to amortize the balance of $937.76at 12% on June 1 over a period of 55 months?


Value Formulas

Annual Worth

The value A below is the annual worth (over n years) of the project.

(x0, x1, x2, . . . , xn)→ (v , 0, 0, . . . , 0)→ (0,A,A, . . . ,A).

Example (A capital cost)

The purchase of a new machine for $100,000 (at time zero) is expected togenerate additional revenues of $25,000 for the next 10 years starting atyear 1. Suppose the interest rate r = 16%.

Is this a profitable investment?

We simply need to determine how to amortize the initial cost uniformlyover 10 years, that is, we need to find the annual payments at 16% thatare equivalent to the original cost. Using the annuity formula, thiscorresponds to $20,690 per year. Hence the annual worth of the project is$25, 000− $20, 690 = $4, 310, and thus the investment is profitable.


Bond details

Outline


8 Value Formulas

9 Bond details

10 Yield

11 Duration

12 Immunization

13 Convexity


Bond details

Definition

A bond is an obligation by the bond issuer to pay money to the bondholder according to rules specified at the time the bond is issued.

A bond pays a specific amount, its face value or, equivalently, its parvalue such as $1,000 or $10,000 at the date of maturity.

Most bonds pay periodic coupon payments. The last coupon datecorresponds to the maturity date, so the last payment is equal to theface value plus the coupon value. The coupon amount is described asa percentage of the face value.

The bid price is the price dealers are willing to pay for the bond.

The ask price is the price at which dealers are willing to sell the bond.

The issuer of a bond initially sells the bonds to raise capital immediately,and then is obligated to make the prescribed payments. Usually bonds areissued with coupon rates close to the prevailing general rate of interest sothat they will sell at close to their face value.


Bond details


Bond details

Definition

The accrued interest (AI) is

AI =number of days since last coupon

number of days in current coupon period× coupon amount.

Example (Accrued interest calculation)

Suppose we purchase on May 8 a U.S. Treasury bond that matures onAugust 15 in some distant year. The coupon rate is 9%. Couponpayments are made every February 15 and August 15. The accruedinterest is computed by noting that there have been 83 days since the lastcoupon and 99 days until the next coupon payment. Hence, AI = 2.05.

This 2.05 would be added to the quoted price, expressed as a percentageof the face value. For example, $20.50 would be added to the bond if itsface value were $1,000.


Bond details

Quality Ratings

Although bonds offer a supposedly fixed-income stream, they are subjectto default if the issuer has financial difficulties or falls into bankruptcy. Tocharacterize the nature of this risk, bonds are rated by rating organizations.

Bonds that are either high or medium grade are considered to beinvestment grade. Bonds that are in or below the speculative categoryare often termed junk bonds.


Yield

Outline


8 Value Formulas

9 Bond details

10 Yield

11 Duration

12 Immunization

13 Convexity


Yield

Yield to Maturity (YTM): The interest rate at which the present valueof the stream of payments (consisting of the coupon payments and thefinal face-value redemption payment) is exactly equal to the current price.

Yield to maturity is just the internal rate of return of the bond atthe current price!

Definition

Suppose that a bond with face value F makes m coupon payments ofC/m each year and there are n periods remaining. The coupon paymentssum to C within a year. Suppose also that the current price of the bond isP. Then the yield to maturity is the value of λ such that

P =F

[1 + (λ/m)]n+

n∑k=1

C/m

[1 + (λ/m)]k.


Yield

Formula (Bond price formula)

The price of a bond, having exactly n coupon periods remaining tomaturity and a yield to maturity of λ, satisfies

P =F

[1 + (λ/m)]n+

C

λ

{1− 1

[1 + (λ/m)]n

},

where F is the face value of the bond, C is the yearly coupon payment,and m is the number of coupon payments per year.

As in any calculation of internal rate of return, to derive λ generallyrequires an iterative procedure, easily carried out by a computer!


Yield

Qualitative Nature of Price-Yield Curves

Focus on the bond with 10% coupon rate!

Figure: Price-yield curves and coupon rate (30 years).

1 What are the prices at YTM = 0 or YTM = 0.10 (par bond)?2 Price and yield have an inverse relation. The price of the bond must

tend toward zero as the yield increases. The shape is convex.3 With a fixed maturity date, the price-yield curve rises as the coupon

rate increases.Xi CHEN ([email protected]) Investment Science 67 / 174

Yield

Figure: Price-yield curves and maturity (10% coupon).

1 All of these bonds are at par when the yield is 10%; hence the threecurves all pass through the common par point.

2 As the maturity is increased, the price-yield curve becomes steeper,which indicates that longer maturities imply greater sensitivity ofprice to yield.


Yield

Exercise

What will happen to the price if the yield of a 30-year, 10% par bondincreases from 10% to 11%? What if it is a 3-year, 10% par bond?

Bond holders are subject to yield risk in the sense that if yields change,bond prices also change.

The bond with 30-year maturity is much more sensitive to yieldchanges than the bond with 1-year maturity!


Yield

Other Yield Measures

Definition

Current Yield (CY) is defined as

CY =annual coupon payment

bond price× 100.

which gives a measure of the annual return of the bond.

Example

Consider a 10%, 30-year bond. If it is selling at par (that is, at 100), thenthe current yield is 10, which is identical to the coupon rate and to YTM.If the same bond were selling for 90, then CY = 11.11, YTM = 11.16.

Definition

Yield to Call (YTC) is defined as the internal rate of return calculatedassuming that the bond is in fact called at the earliest possible date.


Duration

Outline


8 Value Formulas

9 Bond details

10 Yield

11 Duration

12 Immunization

13 Convexity


Duration

Rule of thumb: The prices of long bonds are more sensitive to interestrate changes than those of short bonds.

Definition

The duration of a fixed-income instrument is a weighted average of thetimes that payments (cash flows) are made. The weighting coefficientsare the present values of the individual cash flows.

D =n∑

k=0

PV (tk)tkPV

=PV (t0)t0 + PV (t1)t1 + PV (t2)t2 + . . .+ PV (tn)tn

PV,

where PV (tk) denotes the present value at the cash flow that occurs attime tk ; PV in the denominator is the total present value, which is thesum of the individual PV (tk) values.

Duration is a time intermediate between the first and last cash flows, i.e.,

t0 ≤ D ≤ tn.


Duration

Macaulay Duration

Definition

Suppose a financial instrument makes payments m times per year, withthe payment in period k being ck , and there are n periods remaining. TheMacaulay duration D is defined as

D =1

PV

n∑k=1

(k

m

)PVk =

1

PV

n∑k=1

(k

m

)ck

[1 + (λ/m)]k,

where λ is the yield to maturity and

PV =n∑

k=1

PVk =n∑

k=1

ck[1 + (λ/m)]k

.

The factor k/m in the formula for D is time, measured in years!


Duration

Example (A short bond)

Consider a 7% bond with 3 years to maturity. Assume that the bond isselling at 8% yield. We can find the value and the Macaulay duration bythe simple spreadsheet layout as shown below. The duration is 2.753 years.

Figure: Layout for calculating duration.


Duration

Explicit Formula

Formula (Macaulay duration formula)

The Macaulay duration for a bond with a coupon rate c per period, yieldy per period, m periods per year, and exactly n periods remaining, is

D =1 + y

my− 1 + y + n(c − y)

mc[(1 + y)n − 1] + my.

Proof.

PV =n∑

k=1

c

(1 + y)k+

1

(1 + y)n=

c

y

[1− 1

(1 + y)n

]+

1

(1 + y)n,

D =1

PV

[n∑

k=1

k

m

c

(1 + y)k+

n

m

1

(1 + y)n

]=

1

PV

[c

mS +

n

m

1

(1 + y)n

].


Duration

contd.

S =n∑

k=1

k

(1 + y)k, (1 + y)S =

n∑k=1

k

(1 + y)k−1,

yS =n∑

k=1

1

(1 + y)k−1− n

(1 + y)n,

y

1 + yS =

n∑k=1

1

(1 + y)k− n

(1 + y)n+1

=1

c

[PV − 1

(1 + y)n

]− n

(1 + y)n+1,

S =1

c

[PV (1 + y)

y− 1

y(1 + y)n−1

]− n

y(1 + y)n.


Duration

contd.

c

mS =

PV (1 + y)

my− 1

my(1 + y)n−1− nc

my(1 + y)n,

D =1

PV

[PV (1 + y)

my− 1 + y + n(c − y)

my(1 + y)n

]=

1 + y

my− 1 + y + n(c − y)

mc[(1 + y)n − 1] + my.

Example (Duration of a 30-year par bond)

Consider the 10%, 30-year bond. Assume that it is at par; that is, theyield is 10%, At par, c = y , and

D =1 + y

my

[1− 1

(1 + y)n

]⇒ D = 9.938.


Duration

Qualitative Properties of Duration

1 As the time to maturity increases to infinity, the durations do not alsoincrease to infinity, but instead tend to a finite limit that isindependent of the coupon rate.

2 The durations do not vary rapidly with respect to the coupon rate.The fact that the yield is held constant tends to cancel out theinfluence of the coupons.


Duration

A general conclusion: Very long durations (of, say, 20 years or more) areachieved only by bonds that have both very long maturities and very lowcoupon rates.

Exercise (Duration limit)

For Macaulay duration formula, show that the limiting value of duration asmaturity is increased to infinity is

D → 1 + (λ/m)λ

.

For the bonds in the previous table (where λ = 0.05 and m = 2), weobtain D → 20.5. Note that for large λ, this limiting value approaches1/m, and hence the duration for large yields tends to be relatively short.


Duration

Duration and Sensitivity

In the case where payments are made m times per year and yield is basedon those same periods, we have

PVk =ck

[1 + (λ/m)]k.

The derivative with respect to λ is

dPVkdλ

=−(k/m)ck

[1 + (λ/m)]k+1= − 1

1 + (λ/m)

(k

m

)PVk .

Now apply this to the expression for price,

P = PV =n∑

k=1

PVk ⇒dP

dλ=

n∑k=1

dPVkdλ

= − 11 + (λ/m)

D · P = −DMP,

where DM is called the modified duration.


Duration

Formula (Price sensitivity formula)

The derivative of price P with respect to λ of a fixed-income security is

dP

dλ= −DMP, or equivalently,

1

P

dP

dλ= −DM ,

where DM = D/[1 + (λ/m)] is the modified duration.

By dP/dλ ≈ ∆P/∆λ, the price sensitivity formula can be used toestimate the change in price due to a small change in yield (or vice versa),

∆P ≈ −DMP∆λ.

Example (A zero-coupon bond)

Consider a 30-year zero-coupon bond. Suppose its current yield is 10%.Then we have D = 30 and DM ≈ 27. Suppose that yields increase to 11%.According to the price sensitivity formula, the relative price change isapproximately equal to 27%. This is a very large loss in value. Because oftheir long durations, zero-coupon bonds have very high interest rate risk.


Duration

Example (A 10% bond)

Consider the price-yield curve for a 30-year, 10% coupon bond, whoseduration at the par point (where price is 100) is D = 9.94. Therefore,DM = 9.94/(1 + 0.10/2) = 9.47. The slope of the price-yield curve at thatpoint is equal to dP/dλ = −947. If the yield changes to 11%, thenP ≈ 90.53 by estimating the change in price as

∆P ≈ −DMP∆λ = −9.47.


Duration

Duration of a Portfolio

Consider a portfolio that is the sum of two bonds A and B.

DA =

∑nk=0 tkPV

Ak

PA

DB =

∑nk=0 tkPV

Bk

PB

⇒ D =PADA + PBDB

PA + PB.

Formula (Duration of a portfolio)

Suppose there are m fixed-income securities with prices and durations ofPi and Di , respectively, i = 1, 2, . . . ,m, all computed at a common yield.The portfolio consisting of the aggregate of these securities has price Pand duration D given by

P =m∑i=1

Pi , D =m∑i=1

wiDi =m∑i=1

PiDiP

, i = 1, 2, . . . ,m.


Immunization

Outline


8 Value Formulas

9 Bond details

10 Yield

11 Duration

12 Immunization

13 Convexity


Immunization

Example

Suppose that you wish to invest money now that will be used next year fora major household expense.

If you invest in 1-year Treasury bills, you know exactly how muchmoney these bills will be worth in a year (little risk).

If, on the other hand, you invested in a 30-year zero-coupon bond, thevalue of this bond a year from now would be quite variable, dependingon what happens to interest rates during the year (high risk).

Suppose that you are saving money to pay off an obligation that is due in10 years (reverse situation).

The 10-year zero-coupon bond will provide completely predictableresults (little risk).

The 1-year Treasury bill will impose reinvestment risk since theproceeds will have to be reinvested after 1 year at the then prevailingrate (high risk).


Immunization

Example

Suppose now that you face a series of cash obligations and you wish toacquire a portfolio that will be used to pay these obligations as they arise.

1 One way is to purchase a set of zero-coupon bonds that havematurities and face values exactly matching the separate obligations.However, this simple technique may be infeasible if corporate bondsare used since there are few corporate zero-coupon bonds.

2 You may instead acquire a portfolio having a value equal to thepresent value of the stream of obligations.

You can sell some of your portfolio whenever cash is needed to meet aparticular obligation.If your portfolio delivers more cash than needed at a given time (fromcoupon or face value payments), you can buy more bonds.

Provided the yield does not change, the value of your portfoliowill, throughout this process, continue to match the present

value of the remaining obligations.


Immunization

A problem with this present-value-matching technique arises if the yieldschange, the value of your portfolio and the present value of the obligationstream will both change in response, but probably by amounts that differfrom one another.

Your portfolio will no longer be matched!

Immunization “immunizes” the portfolio value against interest ratechanges, at least approximately, by matching durations and presentvalues.

If yields increase, the present value of the asset portfolio will decrease,but the present value of the obligation will decrease by approximatelythe same amount.

If yields decrease, the present value of the asset portfolio will increase,but the present value of the obligation will increase by approximatelythe same amount.


Immunization

Example (The X Corporation)

The X Corporation has an obligation to pay $1 million in 10 years. Itwishes to invest money now that will be sufficient to meet this obligation.The X Corporation is planning to select from the three corporate bonds,whose durations are D1 = 11.44, D2 = 6.54, and D3 = 9.61, respectively.

The present value of the obligation PV = $414, 643 is computed at 9%interest compounded every six months. The immunized portfolio is foundby solving the two equations

PV1 + PV2 = PV , D1PV1 + D2PV2 = 10PV .


Immunization

Example (contd.)

The solution is PV1 = $292, 788.73 and PV2 = $121, 854.27. The numberof bonds to be purchased is then found by dividing each value by therespective bond price (par value is $100). These numbers are thenrounded to integers to define the portfolio.


Immunization

Exercise

Suppose that an obligation occurring at a single time period is immunizedagainst interest rate changes with bonds that have only nonnegative cashflows (as in the X Corporation example).

Let P(λ) be the value of the resulting portfolio, including the obligation,when the interest rate is r + λ, where r is the current interest rate. Byconstruction, P(0) = 0 and P ′(0) = 0.

We show that P(0) is a local minimum; that is, P ′′(0) ≥ 0!

Assume a yearly compounding convention. The discount factor for time tis dt(λ) = (1 + r + λ)

−t . Let dt = dt(0). For convenience, assume thatthe obligation has a magnitude 1 and is due at time t.


Immunization

Exercise (contd.)

The conditions for immunization are then

P(0) =∑t

ctdt − dt = 0,

P ′(0)(1 + r) =∑t

tctdt − tdt = 0.

1 Show that for all values of α and β there holds

P ′′(0)(1 + r)2 =∑t

(t2 + αt + β)ctdt − (t2 + αt + β)dt .

2 Show that α and β can be selected so that the function t2 + αt + βhas a minimum at t and has a value 1.

3 Use these values to conclude that P ′′(0) ≥ 0, and explain the resultsin the previous example.


Convexity

Outline


8 Value Formulas

9 Bond details

10 Yield

11 Duration

12 Immunization

13 Convexity


Convexity

An even better approximation can be obtained by including a second-order(or quadratic) term. This second-order term is based on convexity, whichis the relative curvature at a given point on the price-yield curve.

Definition

Convexity is the value of C defined as

C =1

P

d2P

dλ2=

1

P

n∑k=1

d2PVkdλ2

.

Formula

Assuming m coupons (and m compounding periods) per year, we have

C =1

P[1 + (λ/m)]2

n∑k=1

k(k + 1)

m2ck

[1 + (λ/m)]k.

Convexity has units of time squared!


Convexity

Convexity is the weighted average of tktk+1, where, like for duration, theweights are proportional to the present values of the corresponding cashflows. Then the result is modified by a factor 1/[1 + (λ/m)]2.

Exercise

Find the convexity of a zero-coupon bond maturing at time T undercontinuous compounding (that is, when m→∞).

Formula

Suppose that at a price P and a corresponding yield λ, the modifiedduration DM and the convexity C are calculated. Then if ∆λ is a smallchange in λ and ∆P is the corresponding change in P, we have

∆P ≈ −DMP∆λ+PC

2(∆λ)2.

To account for convexity in immunization, one structures a portfolio ofbonds such that its present value, duration, and convexity match thoseof the obligation.Xi CHEN ([email protected]) Investment Science 94 / 174

Outline

14 The Yield Curve

15 The Term Structure

16 Forward Rates

17 Term Structure Explanations (Self-learning)

18 Expectation Dynamics

19 Running Present Value

20 Floating Rate Bonds

21 Duration

22 Immunization


The Yield Curve

Outline

14 The Yield Curve


16 Forward Rates





21 Duration

22 Immunization


The Yield Curve

All yields tend to move together in this market. However, all bond yieldsare not exactly the same because

1 bonds have various quality ratings (a high rated bond costs more);

2 the time to maturity varies (a long bond offers high yield).

If long bonds happen to have lower yields than short bonds, the result issaid to be an inverted yield curve, which tends to occur when short-termrates increase rapidly, and investors believe that the rise is temporary, sothat long-term rates remain near their previous levels.Xi CHEN ([email protected]) Investment Science 97 / 174

The Term Structure

Outline

14 The Yield Curve


16 Forward Rates





21 Duration

22 Immunization


The Term Structure

Spot Rates

Definition

The spot rate st is the rate of interest, expressed in yearly terms, chargedfor money held from the present time (t = 0) until time t.

Various possibilities of compounding convention.

1 Yearly: The spot rate st is defined so that (1 + st)t is the factor bywhich a deposit held t years will grow.

2 m periods per year: The spot rate st is defined so that(1 +

stm

)mtis the corresponding factor, where mt must be an integer, so t mustbe an integral multiple of 1/m.

3 Continuous: The spot rate st is defined so that est t is thecorresponding growth factor, which applies directly to all values of t.


The Term Structure

Spot rates measured by zero-coupon bondsSince a zero-coupon bond promises to pay a fixed amount at a fixed datein the future, the ratio of the payment amount to the current price definesthe spot rate for the maturity date of the bond. By this measurementprocess we can develop a spot rate curve.


The Term Structure

Discount Factors and Present Value

Definition

Discount factors are the factors by which future cash flows must bemultiplied to obtain an equivalent present value.

For the various compounding conventions, we have1 Yearly

dk =1

(1 + sk)k.

2 m periods per year

dk =1

[1 + (sk/m)]mk.

3 Continuous dt = e−st t .

Given (x0, x1, x2, . . . , xn), PV, relative to the prevailing spot rates, is

PV =n∑

k=0

dkxk = x0 + d1x1 + d2x2 + . . .+ dnxn.


The Term Structure

Example (Price of a 10-year bond)

Using the spot rate curve in the previous figure, let us find the value of an8% bond maturing in 10 years. Normally, for bonds we would use the ratesand formulas for 6-month compounding; but for this example let usassume that coupons are paid only at the end of each year, starting a yearfrom now, and that 1-year compounding is consistent with our generalevaluation method.

Example (Simplico gold mine)

Consider the lease of the Simplico gold mine discussed in Chapter 2, butnow let us assume that interest rates follow the term structure pattern inthe previous figure. What is the present value of the lease now?


The Term Structure

Determining the Spot Rate

To find the prices of a series of zero-coupon bonds with variousmaturity dates (However, there were essentially no “zero” couponbonds available with long maturities).

To be determined from the prices of coupon-bearing bonds bybeginning with short maturities and working forward toward longermaturities. X

Example

1-year Treasury bill rate⇒ s1

P =C

1 + s1+

C + F

(1 + s2)2⇒ s2

P =C

1 + s1+

C

(1 + s2)2+

C + F

(1 + s3)3⇒ s3

...


The Term Structure

To be determined by a subtraction process. Two bonds of differentcoupon rates but identical maturity dates can be used to constructthe equivalent of a zero-coupon bond. X

Example (Construction of a zero)

Bond A is a 10-year bond with a 10% coupon. Its price is PA = 98.72.Bond B is a 10-year bond with an 8% coupon. Its price is PB = 85.89.Both bonds have the same face value, normalized to 100.

Consider a portfolio with -0.8 unit of bond A and 1 unit at bond B. Thisportfolio will have a face value at 20 and a price of

P = PB − 0.8PA = 6.914.

The coupon payments cancel, so this is a zero-coupon portfolio. The10-year spot rate s10 must satisfy

(1 + s10)10P = 20⇒ s10 = 11.2%.


Forward Rates

Outline

14 The Yield Curve


16 Forward Rates





21 Duration

22 Immunization


Forward Rates

Forward rates f are interest rates for money to be borrowed between twodates in the future, but under terms agreed upon today.

(1 + s2)2 = (1 + s1)(1 + f )⇒ f =

(1 + s2)2

1 + s1− 1.

Example

Suppose that the spot rates for 1 and 2 years are, respectively, s1 = 7%and s2 = 8%. We then find that the forward rate is

f =1.082

1.07− 1 = 9.01%.

Therefore, the 2-year 8% rate can be obtained either as a direct 2-yearinvestment, or by investing for 1 year at 7% followed by a second year at9.01%.


Forward Rates

Definition

The forward rate between times t1 and t2 with t1 < t2 is denoted byft1,t2 . It is the rate of interest charged for borrowing money at time t1which is to be repaid (with interest) at time t2.

Formula (Forward rate formulas)

The implied forward rate between times t1 and t2 with t2 > t1 is the rateof interest between those times that is consistent with a given spot ratecurve. Under various compounding conventions the forward rates arespecified as follows.

Yearly. For yearly compounding, the forward rates satisfy, for j > i ,

(1 + sj)j = (1 + si )

i (1 + fi ,j)j−i ⇒ fi ,j =

[(1 + sj)

j

(1 + si )i

]1/(j−i)− 1.


Forward Rates

Formula (contd.)

m periods per year. For m period-per-year compounding, theforward rates satisfy, for j > i , expressed in periods,(

1 +sjm

)j=(

1 +sim

)i (1 +

fi ,jm

)j−i.

Hence

fi ,j = m

[(1 + sj/m)

j

(1 + si/m)i

]1/(j−i)−m.

Continuous. For continuous compounding, the forward rates ft1,t2 aredefined for all t1 and t2, with t2 > t1, and satisfy

es2t2 = es1t1eft1,t2 (t2−t1) ⇒ ft1,t2 =st2t2 − st1t1

t2 − t1.


Term Structure Explanations (Self-learning)

Outline

14 The Yield Curve


16 Forward Rates





21 Duration

22 Immunization


Term Structure Explanations (Self-learning)

Why are the yield curve and spot rate curve not just flat at a commoninterest rate?

1 Expectations Theory.

Definition

The spot rate curve slopes upward, with rates increasing for longermaturities. The 2-year rate is greater than the 1-year rate. It is arguedthat this is so because the market (that is, the collective of all people whotrade in the interest rate market) believes that the 1-year rate will mostlikely go up next year. (This belief may, for example, be because mostpeople believe inflation will rise, and thus to maintain the same real rate ofinterest, the nominal rate must increase). This majority belief that theinterest rate will rise translates into a market expectation.

2 Liquidity Preference.

3 Market Segmentation.


Expectation Dynamics

Outline

14 The Yield Curve


16 Forward Rates





21 Duration

22 Immunization



Spot Rate Forecasts

Assume that the expectations implied by the current spot rate curve willactually be fulfilled.

Begin with the current spot rate curve s1, s2, . . . , sn, and we wish toestimate next year’s spot rate curve s ′1, s

′2, . . . , s

′n−1.

(1 + sj)j = (1 + f1,j)

j−1(1 + s1)

⇒ s ′j−1 = f1,j =[

(1 + sj)j

1 + s1

]1/(j−1)− 1, 1 < j ≤ n.

We term this transformation expectations dynamics, since it gives anexplicit characterization of the dynamics of the spot rate curve based onthe expectations assumption.



Example (A simple forecast)

Let us take as given the spot rate curve shown in the first row of the table.The second row is then the forecast of next year’s spot rate curve underexpectations dynamics.

Future spot rate curves implied by an initial spot rate curve can be shownby listing the forward rates associated with the initial spot rate curve.

f0,1 f0,2 f0,3 . . . f0,n−2 f0,n−1 f0,nf1,2 f1,3 f1,4 . . . f1,n−1 f1,nf2,3 f2,4 f2,5 . . . f2,n...

...fn−2,n−1 fn−2,nfn−1,n



Discount Factors

Definition

Let dj ,k denote the discount factor used to discount cash received at timek back to an equivalent amount of cash at time j . The normal, time zero,discount factors are d1 = d0,1, d2 = d0,2, . . . , dn = d0,n. Then

dj ,k =

(1

1 + fj ,k

)k−j.

Formula (Discount factor relation)

The discount factor between periods i and j is defined as

di ,j =

(1

1 + fi ,j

)j−i.

These factors satisfy the compounding rule di ,k = di ,jdj ,k for i < j < k.



Short Rates

Definition

Short rates, rk , are the forward rates spanning a single time period, i.e.,at time k , rk = fk,k+1, the forward rate from k to k + 1. For sk and fi ,j ,

(1 + sk)k = (1 + r0)(1 + r1) . . . (1 + rk−1),

(1 + fi ,j)j−i = (1 + ri )(1 + ri+1) . . . (1 + rj−1).



Theorem (Invariance theorem)

Suppose that interest rates evolve according to expectations dynamics.Then (assuming a yearly compounding convention) a sum of moneyinvested in the interest rate market for n years will grow by a factor of(1 + sn)

n independent of the investment and reinvestment strategy (solong as all funds are fully invested).

Proof.

By induction. See for example n = 2.

Every investment earns the relevant short rates over its duration.

1 A 10-year zero coupon bond earns the 10 short rates that are definedinitially.

2 An investment rolled over year by year for 10 years earns the 10 shortrates that happen to occur.

Under expectations dynamics, the rate initially implied for a specifiedperiod in the future will be realized when that period arrives.Xi CHEN ([email protected]) Investment Science 116 / 174

Running Present Value

Outline

14 The Yield Curve


16 Forward Rates





21 Duration

22 Immunization



Suppose (x0, x1, x2, . . . , xn) is a cash flow stream. Denote the presentvalue of this stream PV (0) and the discount factors at time zero dk ’s.

PV (0) =n∑

k=0

dkxk = x0 + d1

[n∑

k=1

(dkd1

)xk

]= x0 + d1PV (1).

In general, for arbitrary time points, with double-indexing system fordiscount factors, the present value at time k is

PV (k) = xk + dk,k+1xk+1 + dk,k+2xk+2 + . . .+ dk,nxn.

Using the discount compounding formula, dk,k+j = dk,k+1dk+1,k+j holds.

Formula (Present value updating)

The running present values satisfy the recursion

PV (k) = xk + dk,k+1PV (k + 1), where dk,k+1 =1

1 + fk,k+1.



Exercise

Show explicitly that if the spot rate curve is flat,with sk = r for allk = 1, 2, . . . , n, then all forward rates also equal r .

Example (Constant running rate)

Suppose that the spot rate curve is flat, with sk = r for all k = 1, 2, . . . , n.Let (x0, x1, x2, . . . , xn) be a cash flow stream. In the flat case, all forwardrates are also equal to r . Hence the present value can be calculated as

PV (n) = xn,

PV (k) = xk +1

1 + rPV (k + 1), k = 0, 1, 2, . . . , n − 1.

This recursion is run from the terminal time backward to k = 0.



Example (General running)

A sample present value calculation is shown in the following table. Thebasic cash flow stream is the first row. The appropriate one-perioddiscount rates are listed in the second row.

The present value at any year k is computed by multiplying the discountfactor listed under that year times the present value of the next year, andthen adding the cash flow for year k . This is done by beginning with thefinal year and working backward to time zero. Hence PV (0) = 168.95.


Floating Rate Bonds

Outline

14 The Yield Curve


16 Forward Rates





21 Duration

22 Immunization


Floating Rate Bonds

Definition

A floating rate note or bond has a fixed face value and fixed maturity,but its coupon payments are tied to current (short) rates of interest.

Example

Consider a floating rate bond that makes coupon payments every 6months.

When the bond is issued, the coupon rate for the first 6 months is setequal to the current 6-month interest rate. At the end of 6 months, acoupon payment at that rate is paid; specifically, the coupon is the ratetimes the face value divided by 2 (because of the 6-month schedule).

Then, after that payment, the rate is reset: the rate for the next 6 monthsis set equal to the then current 6-month (short) rate.

The process continues until maturity.


Floating Rate Bonds

Theorem (Floating rate value)

When coupon payments are tied to current (short) rates of interest, thevalue of a floating rate bond is equal to par at any reset point.

Proof.

Work backward using a running present value argument.

1 Look at the last reset point, 6 months before maturity. The finalpayment, in 6 months, will be the face value plus the 6-month rate ofinterest on this amount. The present value at the last reset point isobtained by discounting the total final payment at the 6-month rate,leading to the face value. So the present value is par at that point.

2 Move back another 6 months to the previous reset point. The presentvalue there is found by discounting the sum of the next present valueand the next coupon payment, again leading to a value of par.

3 We can continue this argument back to time zero.


Duration

Outline

14 The Yield Curve


16 Forward Rates





21 Duration

22 Immunization


Duration

Duration is a measure of interest rate sensitivity with respect to yielddeveloped earlier. In the term structure framework, a different, yet similar,measure of risk can be constructed.

Given the spot rates s1, s2, . . . , sn, we imagine that these rates all changetogether by an additive amount λ, e.g., s1 + λ, s2 + λ, . . . , sn + λ.

This is a hypothetical instantaneous change, for the new spot rates arefor the same periods as before. It is a generalization of a change in yield.


Duration

Fisher-Weil Duration

Given a cash flow sequence xt0 , xt1 , xt2 , . . . , xtn and the spot rate curve st ,t0 ≤ t ≤ tn, the present value is

PV =n∑

i=0

PV (ti ) =n∑

i=0

xti e−sti ti .

Definition (Fisher-Weil duration)

The Fisher-Weil duration is defined as

DFW =1

PV

n∑i=0

tiPV (ti ) =1

PV

n∑i=0

tixti e−sti ti .

This corresponds exactly to the general definition of duration as apresent-value-weighted average of the cash flow times.


Duration

For Fisher-Weil duration, it is clear that given all xti ≥ 0, we have

t0 ≤ DFW ≤ tn.

For an arbitrary λ, the price is

P(λ) =n∑

i=0

xti e−(sti+λ)ti .

We then differentiate to find

P ′(0) =dP(λ)

dλ

∣∣∣∣λ=0

= −n∑

i=0

tixti e−sti ti ,

so immediately we find that the relative price sensitivity as

1

P(0)

dP(0)

dλ= −DFW .


Duration

Formula (Fisher-Weil formulas)

Under continuous compounding, the Fisher-Weil duration of a cash flowstream (x0, xt1 , xt2 , . . . , xtn) is

DFW =1

PV

n∑i=0

tiPV (ti ) =1

PV

n∑i=0

tixti e−sti ti .

where PV denotes the present value of the stream. If all spot rates changeto sti + λ, i = 0, 1, 2, . . . , n, the corresponding present value function P(λ)satisfies

1

P(0)

dP(0)

dλ= −DFW .


Duration

Discrete-Time Compounding

We have a cash flow stream (x0, x1, x2, . . . , xn) (where the indexing is byperiod). The price is

P(λ) =n∑

k=0

xk

(1 +

sk + λ

m

)−k.

We then find that

P ′(0) =dP(λ)

dλ

∣∣∣∣λ=0

=n∑

k=1

−(

k

m

)xk

(1 +

skm

)−(k+1).

Definition (Quasi-modified duration)

DQ = −1

P(0)

dP(0)

dλ=

∑nk=1(k/m)xk [1 + (sk/m)]

−(k+1)∑nk=0 xk [1 + (sk/m)]

−k .


Duration

DQ does have the units of time; however, it is not exactly an average ofthe cash flow times because in the numerator, it is

1

[1 + (sk/m)](k+1)instead of

1

[1 + (sk/m)]k.

Formula (Quasi-modified duration)

Under compounding m times per year, the quasi-modified duration of acash flow stream (x0, x1, x2, . . . , xn) is

DQ =1

PV

n∑k=1

(k

m

)xk

(1 +

skm

)−(k+1),

where PV denotes the present value of the stream. If all spot rates changeto sk + λ, k = 1, 2, . . . , n, the corresponding present value function P(λ)satisfies

1

P(0)

dP(0)

dλ= −DQ .


Immunization

Outline

14 The Yield Curve


16 Forward Rates





21 Duration

22 Immunization


Immunization

Example (A million dollar obligation)

Suppose that we have a $1 million obligation payable at the end of 5years, and we wish to invest enough money today to meet this futureobligation. We wish to do this in a way that provides a measure ofprotection against interest rate risk. To solve this problem, we firstdetermine the current spot rate curve. A hypothetical spot rate curve sk isshown as the column labeled spot in the table.


Immunization

Example (contd.)

We use a yearly compounding convention and decide to invest in twobonds described as follows:

B1 is a 12-year 6% bond with price 65.95; and

B2 is a 5-year 10% bond with price 101.66.

We decide to immunize against a parallel shift in the spot rate curve.

We calculate dP/dλ, denoted by −PV ′, by multiplying each cash flow byt and by (1 + st)

−(t+1) and then summing these. The quasi-modifiedduration is then the quotient of these two numbers. For example, thequasi-modified duration of bond B1 is 466/65.95 = 7.07. We also find thepresent value of the obligation to be $627, 903.01 and the correspondingquasi-modified duration is

5

(1 + s5)= 4.56.


Immunization

Example (contd.)

To determine the appropriate portfolio, let x1 and x2 denote the number ofshares of bonds 1 and 2, respectively, in the portfolio (assuming, forsimplicity, face values of $100). We solve the following two equations{

P1x1 + P2x2 = PV ,

P1D1x1 + P2D2x2 = PV · D,⇒

{x1 = 2, 208.17,

x2 = 4, 744.03.


Outline

23 Capital Budgeting

24 Optimal Portfolios

25 Dynamic Cash Flow Processes

26 Optimal Management

27 The Harmony Theorem

28 Valuation of a Firm


Capital Budgeting

Outline



25 Dynamic Cash Flow Processes

26 Optimal Management

27 The Harmony Theorem

28 Valuation of a Firm


Capital Budgeting

Definition

Capital budgeting typically refers to allocation among projects orinvestments for which there are not well-established markets and where theprojects are lumpy in that they each require discrete lumps of cash.

Capital budgeting problems often arise in a firm where several proposedprojects compete for funding.

The projects may differ considerably in their scale, their cashrequirements, and their benefits.

Even if all proposed projects offer attractive benefits, they cannot allbe funded because of a budget limitation.


Capital Budgeting

Independent Projects

The projects are independent in the sense that it is reasonable to selectany combination from the list.

Definition (Zero-one programming problem)

Suppose that there are m potential projects. Let bi be the total benefit(usually the net present value) of the ith project, and let ci denote itsinitial cost. Finally, let C be the total capital available-the budget. Foreach i , i = 1, 2, . . . ,m, we introduce the zero-one variable xi , which iszero if the project is rejected and one if it is accepted.

maxm∑i=1

bixi ,

s.t.m∑i=1

cixi ≤ C , xi = 0 or 1, i = 1, 2, . . . ,m.


Capital Budgeting

A good approximate solution to zero-one programming problem is thebenefit-cost ratio method.

Definition

The benefit-cost ratio is defined as the ratio of the present worth of thebenefits to the magnitude of the initial cost.

Example (A selection problem)

During its annual budget planning meeting, a small computer companyhas identified several proposals for independent projects that could beinitiated in the forthcoming year.

These projects include the purchase of equipment, the design of newproducts, the lease of new facilities, and so forth. The projects all requirean initial capital outlay in the coming year.

The company management believes that it can make available up to$500,000 for these projects.


Capital Budgeting

Example (contd.)

The projects are already listed in order of decreasing benefit-cost ratio.

According to the approximate method, the company would select project1, 2, 3, 4, and 5 for a total expenditure of $370,000 and a total netpresent value of $540,000.


Capital Budgeting

Example (contd.)

However, the solution derived by the approximate method is not optimal.

According to the zero-one programming problem, the solution should be toselect projects 1, 3, 4, 5, and 6 for a total expenditure of $500,000 and atotal net present value of $610,000.


Capital Budgeting

Interdependent Projects

Sometimes various projects are interdependent, the feasibility or onebeing dependent on whether others are undertaken.

Assume that there are m goals and that associated with the ith goal, thereare ni possible projects. Only one project can be selected for any goal. Asbefore, there is a fixed available budget.

We formulate this problem by introducing the zero-one variables xij fori = 1, 2, . . . ,m and j = 1, 2, . . . , ni .

maxm∑i=1

ni∑j=1

bijxij ,

s.t.m∑i=1

ni∑j=1

cijxij ≤ C ,ni∑j=1

xij = 1, i = 1, 2, . . . ,m,

xij = 0 or 1, ∀i , j .


Capital Budgeting

Example (County transportation choices)

Suppose that the goals and specific projects shown in the following tableare being considered by the County Transportation Authority.


Capital Budgeting

Example (contd.)

To formulate this problem we introduce a zero-one variable for eachproject (However, for simplicity we index these variables consecutivelyfrom 1 through 10, rather than using the double indexing procedure of thegeneral formulation presented earlier).


Optimal Portfolios

Outline



25 Dynamic Cash Flow Process

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