8 Interest

8.1 Introduction

Two questions structure this chapter. First, how does the passage of time affect the value of money, and what mathematical framework allows us to place cash flows occurring at different dates on a common footing? Second, how do the various conventions for quoting and compounding interest rates relate to one another, and which measure of interest — simple, compound, or continuously compounded — is the right one to use when pricing a financial security? These questions form the conceptual bedrock of fixed-income analysis and are prerequisites for virtually every quantitative application in finance.

The first question matters because almost every significant financial decision involves comparing cash flows that do not occur at the same time. A firm evaluating a capital investment must weigh an immediate outlay against revenues that arrive over years or decades. An individual choosing between a lump-sum retirement payout and a life annuity must compare a certain payment today against a stream of future payments that depends on longevity. A government issuing a bond must price the security so that the proceeds it receives today equal what the market regards as the present value of future coupon and principal payments. Without a rigorous discounting framework these comparisons collapse into apples-and-oranges arithmetic, and mispricing is inevitable. The chapter establishes this framework from first principles using the arbitrage principle: the present value of a future cash flow is precisely the amount one would need to invest today, at the prevailing interest rate, to replicate that payment. The resulting present-value formula extends naturally from a single payment to a finite stream of cash flows, and then — via the mathematics of geometric series — to infinite streams and fixed-horizon annuities.

The second question matters because the same nominal interest rate implies very different rates of wealth accumulation depending on how frequently interest is compounded. A stated annual rate of 12% compounded monthly leaves the investor meaningfully richer after one year than the same rate compounded annually, and continuous compounding — the limiting case as the compounding frequency tends to infinity — produces the largest accumulation of all. Financial contracts, regulatory disclosures, and security prospectuses employ all three conventions, and the ability to convert among them accurately is an essential practical skill. The limit of continuous compounding also connects interest-rate arithmetic to the exponential function via a calculus argument using L’Hôpital’s rule, illuminating why the natural exponential appears so pervasively in mathematical finance. The chapter closes by applying the full discounting framework to the pricing of canonical fixed-income instruments: zero-coupon Treasury bills, perpetuities (including the historically important British consol bond), and fixed-term annuities.

The goal of this lecture is to understand the pricing of risk free securities. All of the relationships presented rely on an understanding of the time value of money. The phrase time value of money refers to the fact that the preferences of most individuals are such that they are not indifferent between receving $1000 today and receiving $1000 in one year.

In the news

Interest rates are set in the headlines as well as the textbook. In June 2026 the Federal Reserve held its policy rate at 3.50–3.75% even as inflation climbed toward 3.6%, leaving the real return on cash slim or negative. The distinction between nominal and real rates, and the compounding that turns a quoted rate into actual growth, is the machinery this chapter builds. Read it at Fox Business.

8.2 Present value

Unless noted otherwise, suppose that an investor faces a risk-free interest rate of $r$ every period for the foreseeable future. This implies that the present value of $x$ dollars received one period from now is

\[PV_1 = \frac{1}{1+r}x.\]

What is the present value of $x$ dollars received in two periods? To answer this we consider exactly how much would have to be saved today, to obtain a payment of $x$ in two periods. By the arbitrage principle the present value of a risk free payment in two periods must be the same as the amount you would be required to save today in order to receive that payment in two periods. Let $PV_2$ be the required amount of saving today. If you save $PV_2$, then at the end of one period your total wealth from this investment will be $PV_2(1+r)$. Since you want to save just enough to receive the payment $x$ in two periods, you reinvest your principal and interest ($PV_2(1+r)$) at the market interest rate of $r$ and at the end of two periods you will have $PV_2(1+r)(1+r) = PV_2(1+r)^2$. Recall that you want $PV_2$ to have been just enough to yield $x$ at the end of two periods so you want $PV_2$ to solve

\[PV_2(1+r)^2 = x.\]

Solving this implies that $PV_2 = (1+r)^{-2} x$. In other words, the present value (or present discounted value) of the payment $x$ received in two periods is $(1+r)^{-2} x$.

Suppose that one were to receive the payment $x$ in $T$ years. The required amount of investment capital to generate the payment $x$ in $T$ years satisfies $PV_T(1+r)^T = x$ and so the present value of the payment $x$ to be received in $T$ years is $PV_T = (1+r)^{-T} x$.

Problem 1.: If the interest rate is $r = 0.10$ what is the present discounted value of $1000 received in one year? What is the PV of $1000 received in two years? In seven years? (See Answers.)

Now imagine that you own the right to receive the amount $Y_1$ one period from now and the amount $Y_2$ two periods from now? What is the present value of this asset? $PV = Y_1/(1+r) + Y_2/(1+r)^2$. Why? How much would you have to save in order to earn this payoff? You start by saving $PV$. At the end of the first period you now have $PV(1+r)$. From that, you remove the payment $Y_1$ that you are to receive and save $PV(1+r) - Y_1$ at interest. At the end of the second period you have $(PV(1+r) - Y_1)(1+r)$. By the arbitrage principle it must be that $(PV(1+r) - Y_1)(1+r) = Y_2$. Solving this equation for $PV$ gives $PV = Y_1/(1+r) + Y_2/(1+r)^2$.

In general, suppose that one owned the right to receive the sequence of payments $Y_1,Y_2,\ldots,Y_T$. In a manner analogous to the above, it can be shown that the present value of this sequence of payments is

\[PV(Y) = \sum_{t=1}^{T} \frac{Y_t}{(1+r)^t}\]

8.3 A digression on geometric series

An infinite geometric series is a function of the form

\[f(x) = \sum_{t=0}^{\infty} x^t\]

Notice that $f(x)$ is not defined for all $x \in \mathbb{R}$. If $x \ge 1$ then there is no real number $r$ such that $r = \sum_{t=0}^{\infty} x^t$. If $|x| < 1$ then it can be shown that there exists a number $r$ such that $r = \sum_{t=0}^{\infty} x^t$.¹

Infinite (and convergent) geometric series.

Let $y = \sum_{t=0}^{\infty} x^t = 1 + x + x^2 + \cdots$. Therefore, $yx = x + x^2 + x^3 + \cdots$. This implies that $y - yx = 1$ which in turn implies that

\[y = \frac{1}{1-x}\]

Finite geometric series.

Let $y = \sum_{t=0}^{T} x^t = 1 + x + \cdots + x^T$. Then $yx = x + x^2 + \cdots + x^{T+1}$. Therefore

\[y = \frac{ 1 - x^{T+1}}{1-x}\]

Notice that as $T \rightarrow \infty$ the value of the finite geometric series converges to the value of the infinite geometric series if $|x| < 1$.

Why is this useful? Consider an asset that makes the constant payment of $M$ once a year, forever. Assume that the first payment will be one year from now. What is the present value of such an asset if the interest rate is assumed to be the constant rate $r$ forever?

\[V = \sum_{t=1}^{\infty} \frac{M}{(1 + r)^{t}} = M\left ( \frac{1}{1 - \frac{1}{1+r}} - 1 \right ) = M\frac{1}{r}\]

To see the cancellation explicitly, set $x = \frac{1}{1+r}$, so that $|x| < 1$ whenever $r > 0$. The sum starts at $t = 1$ rather than $t = 0$, so we subtract the missing $t=0$ term:

\[V = M\sum_{t=1}^{\infty} x^{t} = M\left(\sum_{t=0}^{\infty} x^{t} - 1\right) = M\left(\frac{1}{1-x} - 1\right).\]

Now substitute $x = \frac{1}{1+r}$ into $\frac{1}{1-x}$:

\[\frac{1}{1-x} = \frac{1}{1 - \frac{1}{1+r}} = \frac{1+r}{(1+r) - 1} = \frac{1+r}{r},\]

so that

\[V = M\left(\frac{1+r}{r} - 1\right) = M\cdot\frac{(1+r) - r}{r} = M\frac{1}{r}.\]

The first equality follows from the rule on infinite and convergent geometric series. The quantity $\frac{1}{r}$ might be known as the perpetuity factor since a payment of $M$ made in perpetuity (i.e. forever) is worth $\frac{M}{r}$.

Now consider the case of an annuity.

Annuity.: A constant payment to be made regularly for a fixed period of time.
Example.: Consider an annuity that makes annual payments of $M$ for $T$ years. Assume that the first payment is to be made one year from now. If the interest rate over this period is to remain constant at $r$, what is the value (i.e. present value) of such an asset?

Using the rule on finite geometric series above, we see that

\[V = \sum_{t=1}^{T} \frac{M}{(1 + r)^{t}} = M\frac{1 - \left(\frac{1}{1+r}\right)^{T}}{r} = M \frac{1}{r} \left (1 - \frac{1}{(1+r)^T} \right )\]

Again writing $x = \frac{1}{1+r}$, the sum runs from $t=1$ to $T$, so factor out one power of $x$ and apply the finite-series rule to the remaining $T$ terms:

\[V = M\sum_{t=1}^{T} x^{t} = M\,x\sum_{t=0}^{T-1} x^{t} = M\,x\,\frac{1 - x^{T}}{1-x}.\]

The prefactor collapses just as in the perpetuity case, since $\dfrac{x}{1-x} = \dfrac{1/(1+r)}{\,r/(1+r)\,} = \dfrac{1}{r}$. Hence

\[V = M\,\frac{1}{r}\left(1 - x^{T}\right) = M\frac{1}{r}\left(1 - \frac{1}{(1+r)^{T}}\right).\]

The term $\frac{1}{r}(1 - \frac{1}{(1+r)^T} )$ is called the annuity factor at interest rate $r$.

Another way of thinking about this formula (using arbitrage) is the following. Consider purchasing a perpetuity today that makes constant payments of $M$ and selling the right to all payments after period $T$. The value of the perpetuity that you purchase originally is $M\frac{1}{r}$. The value at time $T$ (after the $T$-period payment) of the payments that you sell is $M\frac{1}{r}$, but the value of these payments today is $M\frac{1}{r(1+r)^T}$. Since the annuity makes exactly the same payments as purchasing a perpetuity today and selling all of the payments after time $T$, the prices of these assets must be the same. Therefore the value of the annuity is

\[V = M\frac{1}{r} - M\frac{1}{r(1+r)^T} = M \frac{1}{r} \left ( 1 - \frac{1}{(1 + r)^T} \right )\]

as above.

8.4 Simple and compound interest

Simple interest.

Interest that is calculated on the original principal only.

Example.: If the amount $x$ is deposited for two years in an account paying simple interest at the rate $r$ per year, at the end of the first year the account will have a balance of $B_1 = x + xr$ and at the end of two years the balance will be $B_2 = x + xr + xr = x(1 + 2r)$.

Compound interest.

Interest that is calculated on both the original principal and on previous interest payments.

Example.: If the amount $x$ is deposited for two years in an account paying compound interest at the rate $r$ per year, at the end of the first year the account will have a balance of $B_1 = x + xr = x(1 + r)$ and at the end of two years the balance will be $B_2 = x + xr + (x + xr)r = x(1+r)^2$.

Compound period.

The time beginning immediately after interest is calculated on an asset and ending with the next calculation of interest on the asset.

Example.: In the U.S. savings accounts often pay compound interest that is calculated monthly. Thus, every month the balance on the account is used to calculate the amount of interest that the bank will pay to the holder of the account. The compound period in this case is 1 month.

Periodic interest rate.

The rate of interest paid over one compound period.

Annual interest rate (annual rate).

The rate of interest paid on an asset annually without accounting for compound interest. This is calculated as the periodic interest rate multiplied by the number of compound periods in a year.

Example.: A savings account in the U.S. that pays a periodic interest rate of 1% pays an annual interest rate of 12%.

Effective annual rate.

The rate of interest paid on an asset annually when compound interest is taken into account. The effective annual rate is the percentage increase in funds invested over a 1-year horizon.

Example.

Consider depositing $x$ in the bank at a stated annual interest rate of $r$ that is compounded every 6 months. This means that every 6 months interest is calculated on the amount in the account. Thus the compound period is 6 months and the periodic interest rate is $\frac{r}{2}$. Therefore, at the end of six months (after interest has been paid) the balance on the account is

\[B_{6} = x + x\left ( \frac{r}{2} \right )\]

and at the end of the year the amount in the account is

\[\begin{aligned} B_{12} & = B_{6} + B_{6} \left ( \frac{r}{2} \right) \\ & = B_{6} \left ( 1 + \frac{r}{2} \right ) \\ & = x \left ( 1 + \frac{r}{2} \right )^2 \end{aligned}\]

If interest is compounded $n$ times during the year then at the end of the first period the balance in the account is $B_1 = x(1 + \frac{r}{n} )$. At the end of two periods the balance is $B_2 = x(1 + \frac{r}{n} )^2$ and after $m$ periods the balance is $B_m = x(1 + \frac{r}{n} )^m$. The effective annual rate is then

\[r^* = \left ( 1 + \frac{r}{n} \right )^n - 1.\]

Problem 1.

Consider a certificate of deposit that pays an annual interest rate of 1.99%. If interest is compounded monthly what is the compound period? What is the periodic interest rate? What is the effective annual interest rate?

If an asset with initial value $x$ is to be held for $t$ years at the annual interest rate $r$ and compounding occurs $n$ times a year, the balance on the account at the end of the $t$ years will be

\[B_t = x \left ( 1 + \frac{r}{n} \right ) ^{nt}\]

8.4.1 Continuous compounding

Consider what happens to the effective annual rate when the number of compounding periods in a year goes to $\infty$.

\[\lim_{n \rightarrow \infty} \left ( 1 + \frac{r}{n} \right ) ^{n}\]

This is the gross effective annual rate (i.e. 1 plus the effective annual interest rate) of an asset that is re-compounded at every instant of time. This type of compounding is known as continuous compounding. While very few assets are actually compounded continuously, mathematically this is a very interesting case.

Mathematical note.

There are a few rules from calculus that are needed here.

If $f$ and $g$ are continuous functions, $g$ has a continuous inverse and $\lim_{x \rightarrow \infty} f(x)$ exists then

\[g^{-1} \left (\lim_{x \rightarrow \infty} g(f(x)) \right ) = \lim_{x \rightarrow \infty} f(x).\]

This follows from the continuity of $g$ and its inverse.

L'Hôpital's rule. It says that if the functions $f$ and $g$ are differentiable and

\[\lim_{x \rightarrow \infty} \frac{f'(x)}{g'(x)} = L\]

then

\[\lim_{x \rightarrow \infty} \frac{f(x)}{g(x)} = L.\]

By rule 1 above, if

\[\lim_{n \rightarrow \infty} \log \left ( 1 + \frac{r}{n} \right ) ^{n} = L\]

then

\[\lim_{n \rightarrow \infty} \left ( 1 + \frac{r}{n} \right ) ^{n} = e^L.\]

We will find this first limit.

\[\begin{aligned} \lim_{n \rightarrow \infty} \log \left ( 1 + \frac{r}{n} \right ) ^{n} & = \lim_{n \rightarrow \infty} n \log \left ( 1 + \frac{r}{n} \right ) \\ & = \lim_{n \rightarrow \infty} \frac{ \log \left ( 1 + \frac{r}{n} \right )}{\frac{1}{n}} \end{aligned}\]

Notice that this bottom limit has the form $\frac{0}{0}$, a perfect candidate for L'Hôpital's rule. Finding the derivative of top and bottom

\[\begin{aligned} \lim_{n \rightarrow \infty} \frac{\left ( \frac{1}{ 1 + \frac{r}{n}} \right ) \frac{-r}{n^2}}{\frac{-1}{n^2}} & = \lim_{n \rightarrow \infty} \left ( \frac{1}{ 1 + \frac{r}{n}} \right )r \\ & = r \end{aligned}\]

By L'Hôpital's rule this implies that $\lim_{n \rightarrow \infty} \log \left ( 1 + \frac{r}{n} \right ) ^{n} = r$ which implies (by rule 1) that

\[\lim_{n \rightarrow \infty} \left ( 1 + \frac{r}{n} \right ) ^{n} = e^r.\]

Thus the value of assets that are compounded continously is easy to calculate.

Problem 2.: Suppose that a savings account pays annual interest rate $r = 0.05$ (5%) compounded continuously. If you deposit $1000 into the account today, how much will you have in one year? What will your effective annual rate have been? What would you have had in the account after 1 year if interest had been compounded daily? monthly? semiannually?

8.5 Pricing Securities

Problem 2 (Zero-coupon bonds).: The U.S. government issues short term debt (usually 28 days, 61 days or 182 days to maturity) that are referred to as T-Bills (Treasury Bills). These generally have face value of $1000 and do not pay any coupon payments. That is, the issuer of the bond agrees to pay the holder the face value of the bond ($1000 in this case) at maturity. If the government issues a six month T-Bill and the six month interest rate is $r$, what should be the price of a $1000 face value Bill?
Problem 3 (British Consol Bonds and perpetuities).: The British government has at times issued what is referred to as a consol bond (for consolidated fund stock--these were first issued in the 1750s). These securities pay the interest rate $r$ forever.² Consols pay dividends quarterly on the face value of the bond. What should be the price of a consol bond with face value $M$ paying quarterly interest payments with periodic rate $\frac{r}{4}$ if it is purchased immediately before an interest payment is made? What about if it is purchased immediately after an interest payment is made?
Problem 4 (Annuities).: An annuity is a security that pays a constant payment for a fixed period of time. Consider an
Problem 2 (The iPhone).: The original iPhone had an upfront cost of $399 plus a required monthly data cost of $20/month. This cost was on top of the monthly cost of the voice plan. The iPhone 3G introduced this year has an upfront cost of $199 and a required monthly data cost of $30. At a constant monthly interest rate of 0.407% (which is 0.004074 or about 5% annual interest with compounding) what is the cost (PV of all future payments) of owning the original iPhone for two years? What is the cost of owning the iPhone 3G for two years?. What if the monthly interest rate were 1%? 2%?

Problem 4 (

It turns out that for a lot of risk-free securities (like U.S. government issued bills, notes and bonds) one can calculate the correct price of the security using the methods just discussed.

Consider

8.6 Answers

The compound period is 1 month. The periodic interest rate is $r_p = 0.0199/12 = 0.0016583$. The effective annual rate is $r^* = (1 + \frac{0.0199}{12})^{12} - 1 = 0.02008$ or $2.008\%$.
The amount in the account in 1 year will be $B = \$1000e^{0.05} = \$1051.271$. The effective annual rate is $(1051.271 - 1000)/1000 = 0.051271$ or $5.1271\%$.
- Daily: $1000(1 + \frac{0.05}{365})^{365} = 1051.267$
- Monthly: $1000(1 + \frac{0.05}{12})^{12} = 1051.162$
- Semiannually: $1000(1 + \frac{0.05}{2})^{2} = 1050.625$.
- $PV_1 = \$1000/(1.10) = \$909.09$, - $PV_2 = \$1000/(1.10)^2 = \$826.45$, - $PV_7 = \$1000/(1.10)^7 = \$513.16$.
These calculations show that owning the iPhone 3G is more expensive than owning the original iPhone.

\[\begin{aligned} PV(Old) & = \$399 + \sum_{t=1}^{24}(1.004074)^{-t}(\$20) \\ & = \$399 + (1 - (1/1.004074)^{25})/(1/1.004074)(\$20) = \$399 + \$456.40 \\ & = \$855.40. \end{aligned}\]

\[\begin{aligned} PV(New) & = \$199 + (1 - (1/1.004074)^{25})/(1/1.004074)(\$30) \\ & = \$199 + \$684.59 \\ & = \$883.59 \end{aligned}\]
The payment $\frac{rM}{4}$ is received 4 times a year. The price is given by

\[P = \sum_{t=1}^{\infty} \frac{\frac{rM}{4}}{(1 + \frac{r}{4})^{t}} = \frac{rM}{4} \left ( \frac{1}{1 - \frac{1}{1 + \frac{r}{4}}} - 1 \right ) = \frac{rM}{4}\]

8.7 Application: A retiree weighing a CD

A saver approaching retirement is deciding whether to lock a year of living expenses into a certificate of deposit advertising an attractive rate. The tools of this chapter tell her what that rate actually means. Compounding determines how the quoted annual rate translates into the sum she will hold at maturity, and the Fisher relation — real return equals nominal return minus expected inflation — reveals the figure that truly matters: with inflation running above the rate the CD pays, her money is guaranteed to grow in dollars while quietly shrinking in purchasing power. Distinguishing nominal from real returns, and understanding how the time value of money links payments across dates, is what separates a decision that feels safe from one that actually preserves what she has earned.

Further listening

The Indicator from Planet Money (NPR) — “How the Fed’s interest rate hikes are like Ratatouille”: a short, vivid explanation of the two interest rates that really matter when the Fed sets policy.

8.8 Homework problems

8.8.1 Conceptual

INT-C1. A friend offers you a choice: receive $1000 today, or receive $1000 exactly one year from now. Using the idea of the time value of money, explain why most individuals strictly prefer the payment today even when they are certain the future payment will actually arrive. In your answer, connect the preference to the risk-free interest rate $r$ and to what the arbitrage principle says about the present value of the delayed payment.

INT-C2. Two risk-free assets, A and B, each promise a single payment of $5000. Asset A pays in 3 years and asset B pays in 8 years. Without doing any arithmetic, explain which asset has the larger present value and why. Then explain what happens to each asset’s present value, and to the gap between them, as the interest rate $r$ rises. Relate your reasoning to the present-value formula $PV_T = (1+r)^{-T}x$.

INT-C3. The chapter values a perpetuity paying $M$ forever as $V = M/r$ using the rule on infinite geometric series. Explain, without algebra, why the value is decreasing in $r$ and why a small perpetual payment can nonetheless be worth a large lump sum today. What feature of the arbitrage/replication argument guarantees that $M/r$ is exactly the amount you would need to invest today, at rate $r$, to reproduce the payment stream $M$ forever?

INT-C4. The chapter derives the annuity value $V = M\frac{1}{r}\left(1 - (1+r)^{-T}\right)$ two ways: from the finite geometric series, and from an arbitrage argument in which a perpetuity is bought today and the rights to all payments after period $T$ are sold. Explain in words why these two routes must give the same answer, and identify the role played by the “law of one price” (assets with identical cash flows have identical prices). As $T \to \infty$, what does the annuity value approach, and why should that be obvious from the arbitrage picture?

INT-C5. A student claims: “Whether the bank uses simple or compound interest doesn’t really matter — over a single year the balance is the same either way, and over longer horizons the difference is tiny.” Evaluate this claim. Explain the precise sense in which the student is right for a one-year horizon with interest paid once at year-end, and the sense in which they are wrong once interest is credited more than once per period or the money is left for many years.

INT-C6. Using the chapter’s two-year examples, explain why a balance earning compound interest at rate $r$ grows as $x(1+r)^2$ while the same principal earning simple interest grows only as $x(1+2r)$. Identify precisely the dollar term that compounding adds over the two years, explain in words what that term represents, and state whether the gap between the two balances widens or narrows as the horizon lengthens.

INT-C7. Two certificates of deposit both advertise an annual interest rate of 6%. The first compounds semiannually; the second compounds monthly. Explain why the second leaves the investor richer after one year even though the two quoted annual rates are identical. Define the effective annual rate $r^*$ in your answer and explain why $r^*$, rather than the quoted annual rate $r$, is the correct number to compare across the two CDs.

INT-C8. Explain, in words, why increasing the number of compounding periods $n$ within a year raises the effective annual rate $r^* = (1 + r/n)^n - 1$, yet the effective rate does not grow without bound as $n \to \infty$. What is the limiting gross return the chapter derives, and why does the natural exponential $e^r$ appear as the ceiling on how much more frequent compounding can do for you?

INT-C9. The chapter obtains continuous compounding by taking $n \to \infty$ in $(1+r/n)^n$ and, via L’Hôpital’s rule applied to $\log(1+r/n)^n$, finds the limit equals $e^r$. Explain in words why the logarithm is introduced before the limit is taken, and why the intermediate limit has the indeterminate form $0/0$ that makes L’Hôpital’s rule applicable. What does the resulting balance formula $B_t = xe^{rt}$ say about how a continuously compounded deposit grows through time?

INT-C10. In the chapter’s “In the news” callout, the Fed’s policy rate sits near 3.6% while inflation runs near 3.6%. Using the distinction between nominal and real returns (the Fisher relation, real return equals nominal return minus expected inflation), explain why a saver holding cash at this nominal rate may see her balance grow in dollars while her purchasing power stays flat or shrinks. Why does the compounding machinery of this chapter operate on the nominal rate, not the real rate?

INT-C11. A retiree can take her pension as a single lump sum today or as a level lifelong monthly payment. Explain how the Fisher relation and the annuity/perpetuity valuation of this chapter together shape the comparison: which quantity should she discount the payment stream at, and why does inflation erode a fixed nominal annuity in a way it does not erode a payment that is indexed to prices? Keep the discussion conceptual — no arithmetic required.

INT-C12. A T-Bill is a zero-coupon security: the issuer pays only the face value at maturity, with no intermediate coupons. Explain why the price of such a bill must be below its face value in a world with a positive interest rate, and describe how the bill’s price would move if the six-month interest rate $r$ suddenly rose. Relate your answer to the single-payment present value formula $PV_T = (1+r)^{-T}x$.

8.8.2 Quantitative

INT-Q1. An investor is entitled to a stream of risk-free payments: $400 one year from now, $600 two years from now, and $1000 three years from now. The risk-free rate is $r = 0.08$ every period. Using $PV(Y) = \sum_{t=1}^{T} \frac{Y_t}{(1+r)^t}$, compute the present value of this stream.

INT-Q2. You own the right to receive $2500 exactly $T = 4$ years from now, and the risk-free interest rate is $r = 0.07$. Using $PV_T = (1+r)^{-T}x$, find the present value. Then find the present value if the horizon is instead $T = 9$ years, holding $r$ fixed.

INT-Q3. A perpetuity pays a constant $750 at the end of each year, forever, with the first payment one year from today. The interest rate is $r = 0.05$ forever. Using $V = M/r$, compute the present value of the perpetuity. Then recompute it if the interest rate falls to $r = 0.04$, and comment on the direction of the change.

INT-Q4. A British consol has face value $M = \$1000$ and pays quarterly interest at the periodic rate $r/4$ where the annual rate is $r = 0.08$. The quarterly coupon is therefore $rM/4$. Using the perpetuity logic $P = \sum_{t=1}^{\infty} \frac{rM/4}{(1+r/4)^t} = \frac{rM/4}{\,r/4\,}$, compute the price immediately after a coupon has been paid, and then the price immediately before a coupon (when the buyer also collects the imminent payment).

INT-Q5. An annuity pays $M = \$1500$ at the end of each year for $T = 15$ years, with the first payment one year from now. The interest rate is $r = 0.06$. Using the annuity factor $V = M\frac{1}{r}\left(1 - \frac{1}{(1+r)^T}\right)$, compute the present value of the annuity.

INT-Q6. An annuity pays $M = \$800$ at the end of each year for $T = 25$ years, with the first payment one year from now, at interest rate $r = 0.09$. Using $V = M\frac{1}{r}\left(1 - \frac{1}{(1+r)^T}\right)$, compute the present value. As a check, confirm it is smaller than the value of the corresponding perpetuity $M/r$.

INT-Q7. You deposit $x = \$5000$ at an annual interest rate of $r = 0.10$ compounded $n = 4$ times per year, and leave it for $t = 8$ years. Using $B_t = x\left(1 + \frac{r}{n}\right)^{nt}$, compute the ending balance.

INT-Q8. A credit card quotes an annual rate of $r = 0.18$ compounded monthly ($n = 12$). Using $r^* = \left(1 + \frac{r}{n}\right)^n - 1$, compute the effective annual rate. Then compute the effective annual rate if the same 18% annual rate were compounded daily ($n = 365$).

INT-Q9. You deposit $3000 at an annual rate of $r = 0.05$ compounded continuously. Using $B_t = x e^{rt}$ (the continuous-compounding limit $\lim_{n\to\infty}(1+r/n)^n = e^r$), compute the balance after $t = 4$ years, and compute the effective annual rate $r^* = e^{r} - 1$ for this account.

INT-Q10. You deposit $10{,}000 at an annual rate of $r = 0.03$ compounded continuously and leave it for $t = 10$ years. Using $B_t = x e^{rt}$, compute the ending balance, and compute the effective annual rate $r^* = e^{r} - 1$.

This proof can be found in most calculus books. For example, Salas and Hille, 7th edition, pg 663.↩︎
Actually there is a call provision, but we will ignore that for now.↩︎