CFA Level 1 · Module 02 Quantitative Methods · Chapter 4
The first three readings of this module measured returns. This one turns the question around. If you know the cash flows an asset will pay and the return investors require, what is the asset worth today? That single question, discounting expected cash flows back to the present, is the engine behind almost every valuation in finance, and this reading applies it to bonds, to shares, and then to the elegant idea that ties markets together: that identical cash flows must carry identical prices, or a riskless profit appears.
Nothing here is new arithmetic; it is the present-value machinery from the time-value-of-money basics, put to work on real instruments. What matters is seeing that a bond price, a share price, a forward interest rate, a forward exchange rate, and an option value are all the same calculation wearing different clothes. Learn the pattern once and each application becomes a variation, not a fresh formula to memorize.
Work the examples with a calculator. The exam rewards candidates who can set up the cash flows correctly and discount them cleanly, and who can run the calculation backward to solve for an implied rate or an implied growth when the price is given.
An investment is a claim on future cash. A bond promises coupons and a final repayment; a share promises dividends and, ultimately, its future sale value; a rental property promises rent. The value of any such claim today is the present value of those expected cash flows, each discounted at a rate that reflects the return investors require for the risk and the wait.
where PV is the value today, CFt is the expected cash flow at time t, r is the required return per period, and n is the number of periods. Every valuation in this reading is a special case of this one equation.
The required return r is the same required return built up in the first reading: a real rate, plus expected inflation, plus a risk premium for the asset. A higher required return discounts future cash flows more heavily and lowers the value today, which is the seesaw at the heart of all valuation: when required returns rise, prices fall.
Two special cases are worth naming because they recur throughout finance. A single cash flow far in the future is valued simply by dividing it by one plus the rate raised to the number of periods. A level stream that continues forever, a perpetuity, has an even simpler value: the periodic cash flow divided by the rate. That perpetuity result, value equals cash flow divided by r, is the seed of the equity growth model later in this reading, which is just a perpetuity whose payments grow. Recognizing these shapes lets you value a complicated instrument by breaking it into pieces you already know how to price.
A standard coupon bond is the cleanest case, because its cash flows are contractual. It pays a fixed coupon each period and repays its face value at maturity. Its value is the present value of that stream at the required return, which for a bond is called its yield.
where P is the bond price, C is the periodic coupon, FV is the face value repaid at maturity, r is the required yield per period, N is the number of periods, and the sum runs over every coupon date from 1 to N.
Setup. Kaveri Cement issues a three-year bond with a face value of 1,000 rupees and an annual coupon of 8%, so it pays 80 each year plus 1,000 at maturity. Investors require a yield of 10%. Find the price.
Answer: the bond is worth about 950.27 rupees. It sells below its 1,000 face value because its 8% coupon is lower than the 10% yield investors demand, so the price falls until the bond offers a competitive return.
| Year | Cash flow | Discount factor at 10% | Present value |
|---|---|---|---|
| 1 | 80 | 0.9091 | 72.73 |
| 2 | 80 | 0.8264 | 66.12 |
| 3 | 1,080 | 0.7513 | 811.42 |
| Total | – | – | 950.27 |
Most real bonds pay coupons twice a year rather than once, and the valuation adjusts in the obvious way: halve the annual coupon, halve the annual yield, and double the number of periods, so a ten-year bond becomes twenty semiannual periods. The principle does not change, only the size and count of the cash flows. Whatever the schedule, the price is always the sum of the present values of the individual payments, which is why laying the cash flows out on a timeline before discounting is a habit worth keeping under exam pressure.
The relationship between a bond’s coupon rate and its required yield decides whether it trades below, at, or above face value. If the coupon rate is below the yield, as in the example, the bond is worth less than face value and trades at a discount. If the coupon equals the yield, the bond is worth exactly its face value and trades at par. If the coupon exceeds the yield, the bond trades at a premium. This is not a rule to memorize but a consequence of discounting: a coupon that undershoots what investors demand can only compete if the buyer pays less up front.
| Condition | Price relative to face | Called |
|---|---|---|
| Coupon rate below yield | Below face value | Discount |
| Coupon rate equals yield | Equal to face value | Par |
| Coupon rate above yield | Above face value | Premium |
Price and yield move in opposite directions. Anything that raises the required yield, such as higher interest rates or greater perceived risk, lowers the present value of the fixed coupons and pushes the bond price down. This inverse relationship is the foundation of the entire fixed-income module.
Turn the bond calculation around. If you know the market price of a bond and its cash flows, the implied return, or yield to maturity, is the single discount rate that makes the present value of the cash flows equal the price. It is the internal rate of return on the bond, the exact idea met as the money-weighted return in the previous reading, now applied to a bond.
Setup. A three-year bond from Northbank pays no coupon and repays 1,000 rupees at maturity. It trades today at 751.31 rupees. Find its implied annual return.
Answer: the implied return is 10%. For a zero-coupon bond the yield falls straight out of the price and the face value; for a coupon bond the same logic holds, but the rate is found with the internal-rate-of-return function rather than a single root.
For a bond that pays coupons, the implied return cannot be isolated with a single root, because the price is a sum of several discounted cash flows. In practice it is found with the internal-rate-of-return function, which searches for the one yield that makes the discounted coupons and face value add back to the market price. It is worth distinguishing this yield to maturity from the simpler current yield, which is just the annual coupon divided by the price. The current yield ignores the pull of the price back toward face value over the bond’s life, while the yield to maturity captures the whole return, both coupon income and any price convergence, and it assumes coupons are reinvested at that same yield.
Equity is harder than a bond because its cash flows are not contractual, but the principle is identical: a share is worth the present value of the dividends it is expected to pay, discounted at the required return on equity. This is the dividend discount model. Over a finite horizon you discount each expected dividend and the expected sale price; over an infinite horizon you discount the whole future dividend stream.
where P0 is the value of the share today, Dt is the expected dividend at time t, and r is the required return on equity. Because the dividends extend indefinitely, a workable version needs an assumption about how they grow.
Over a finite horizon the model is more intuitive still. An investor who plans to hold a share for three years values it as the present value of the three dividends plus the present value of the price at which the share is sold at the end. The catch is that the future sale price is itself the present value of all the dividends after that point, so the finite view and the infinite view are two descriptions of the same thing. This is why equity valuation is harder than bond valuation: the cash flows are neither fixed in size nor guaranteed in timing, and the terminal value hides an assumption about everything that happens beyond the horizon. Every equity model is an attempt to make that assumption manageable.
The most useful special case assumes dividends grow at a constant rate forever. With that single assumption the infinite sum collapses into one compact expression, the constant growth model, often called the Gordon growth model. It values the share as next year’s dividend divided by the required return minus the growth rate.
where P0 is the share value today, D1 is the dividend expected one period from now, r is the required return on equity, and g is the constant growth rate of dividends. The model requires r to exceed g, or the value is meaningless.
Setup. Corvus Logistics is expected to pay a dividend of 4 rupees next year, growing at 4% a year forever. Investors require a 12% return on the shares. Find the value today.
Answer: the share is worth 50 rupees. Notice how sensitive this is to the gap between r and g: if growth rose to 5%, the denominator would shrink to 0.07 and the value would jump to about 57.14, so small changes in the growth assumption move the price a lot.
Using the current dividend instead of next year’s dividend in the numerator. The constant growth model discounts D1, the dividend one period ahead. If you are given the dividend just paid, D0, grow it first: D1 = D0 × (1 + g). Forgetting this understates the value.
The constant growth model is really the perpetuity result with growth added, which is why it is so compact and why it is so sensitive. Because the value is next year’s dividend divided by the small difference between the required return and the growth rate, a tiny change in either input can swing the price dramatically. That sensitivity is a strength when a company genuinely has a stable, predictable dividend, and a weakness when growth is uncertain or the firm pays little or nothing, in which case the model gives a false sense of precision. The model fits mature, steadily growing dividend payers; it should not be forced onto young, fast-changing companies where the constant growth assumption simply does not hold.
The growth model magnifies its inputs. Since it divides by the gap between the required return and growth, values explode as that gap narrows. Always sanity-check the growth rate you feed it: a long-run growth rate above the economy’s growth is not sustainable, and a growth rate near the required return produces a price that is not to be trusted.
Just as a bond price implies a yield, a share price implies a return. Rearranging the constant growth model, the required return equals the dividend yield plus the growth rate. That decomposition is worth remembering on its own: the return on an equity comes partly from the dividend it pays and partly from the growth of that dividend.
where D1 ÷ P0 is the forward dividend yield, g is the dividend growth rate, and r is the required return. Given any two of price, return, and growth, the third follows.
Setup. Corvus trades at 50 rupees and is expected to pay a 4-rupee dividend next year. First, if dividends grow at 4%, what return is implied? Second, if investors require 12%, what growth is the price implying?
Answer: the implied return is 12% and the implied growth is 4%. When a share looks expensive, this is how analysts check what growth the market must be assuming to justify the price, and whether that assumption is realistic.
Reading a price backward like this is one of the most practical uses of the whole reading. Rather than argue about whether a share is cheap or dear in the abstract, an analyst can ask what the current price must be assuming, then judge whether that assumption is believable. If a mature company’s price only makes sense with double-digit dividend growth forever, the market is probably too optimistic. If a solid business is priced for almost no growth, it may be a bargain. The valuation model becomes a lens for testing expectations, not just a machine for producing a single number, and that habit of asking what is priced in carries all the way through equity analysis at the higher levels.
Here is the principle that unifies everything that follows. The cash flow additivity principle says that the value of a set of cash flows equals the sum of the values of its parts. It sounds obvious, and it is, but its power comes from the flip side: if two investments produce exactly the same cash flows, they must have the same price. If they did not, an investor could buy the cheap one, sell the expensive one, and pocket a riskless profit, an arbitrage. Because such free money is competed away almost instantly, prices settle where no arbitrage remains.
No arbitrage is not a gentle tendency; it is a hard constraint that pins down forward interest rates, forward exchange rates, and option values without anyone needing to forecast the future. Each of the next three sections is just additivity applied: build a package of known cash flows that must equal another, and solve for the one price that removes any free profit.
What makes the principle so reliable is speed. Professional traders watch for price gaps constantly, and the moment two identical cash-flow packages diverge in price, they trade until the gap closes. An individual candidate will never see these opportunities, because they vanish in fractions of a second, but that is exactly the point: prices in liquid markets already reflect no arbitrage, so the formulas built on it describe the world as it actually is, not an idealization. When a question says to value something under no arbitrage, it is inviting you to assume the profit-seekers have already done their work.
Additivity lets you value something unknown by replicating it with things you can already price. If a combination of a bond and a loan reproduces a forward contract’s cash flows exactly, the forward must cost the same as that combination, or arbitrage appears. Valuation by replication is the single most powerful tool in this reading.
To see replication concretely, picture two ways to hold a bond for one year and then have cash. You could buy the bond and sell it in a year, or you could enter a forward agreement to sell it at a set price. Because both routes deliver a known amount of cash at the same future date, additivity says the forward price cannot be arbitrary: it must equal the bond’s price grown at the risk-free rate, adjusted for any coupons received in between. If it did not, a trader would buy the cheaper route and sell the dearer one for a certain profit. Every forward and option formula in the curriculum is built from exactly this move, replicate the unknown with known pieces, then set the prices equal.
A forward interest rate is a rate agreed today for borrowing or lending over a future period. No arbitrage fixes it, because two strategies must give the same result: investing for two years at the two-year spot rate, or investing for one year at the one-year spot rate and then reinvesting for a second year at the forward rate. If those two paths did not end at the same wealth, an arbitrage would exist.
where z1 is the one-year spot rate, z2 is the two-year spot rate, and f is the implied one-year forward rate beginning one year from now. The two-year growth on the left must equal the one-year growth followed by the forward-year growth on the right.
Setup. The one-year spot rate is 4% and the two-year spot rate is 5%. Find the implied one-year forward rate starting one year from now.
Answer: the implied forward rate is about 6.01%. Because the yield curve slopes up, the market is effectively pricing a higher rate for the second year, and no arbitrage forces that rate to exactly 6.01% given the two spot rates.
The same logic sets a forward exchange rate, the rate agreed today to exchange currencies at a future date. An investor with home-currency cash has two ways to end up with foreign currency in a year: convert now and earn the foreign interest rate, or stay home, earn the home rate, and convert later at the forward rate. No arbitrage forces the two routes to match, which fixes the forward rate relative to the spot rate and the two interest rates. This relationship is known as covered interest parity.
where F is the forward rate, S is the spot rate, both quoted as units of the price currency per one unit of the base currency, rprice is the interest rate of the price currency, and rbase is the interest rate of the base currency.
Setup. The spot rate is 80 rupees per US dollar. The rupee interest rate is 7% and the dollar interest rate is 3%. Find the one-year forward rate in rupees per dollar.
Answer: the forward rate is about 83.11 rupees per dollar. The dollar buys more rupees forward than today, because the rupee carries the higher interest rate, and the currency with the higher rate trades at a forward discount. No arbitrage, not a forecast, sets this rate.
Forward-rate questions live or die on the quote convention. Fix which currency is the price currency and which is the base before plugging in, and put the price currency’s interest rate on top. A sign or ratio flipped here is the single most common way candidates lose an otherwise easy mark.
This relationship also explains the well-known carry trade, where investors borrow a low-interest-rate currency and invest in a high-interest-rate one. Covered interest parity says that if the position is fully hedged with a forward, the forward discount on the high-rate currency exactly cancels the interest advantage, leaving no riskless gain. Any profit a carry trade earns therefore comes from taking on unhedged currency risk, that is, from betting the currency will not move as the forward implies. The forward rate is the market’s no-arbitrage anchor, and stepping away from it means accepting risk, not finding free money.
Options round out the pattern. An option’s payoff at expiry can be reproduced by a carefully chosen package of the underlying asset and borrowing or lending. Because that replicating package has known cash flows, additivity and no arbitrage say the option must cost exactly what the package costs. You do not need to predict whether the option finishes in the money; you need only find the portfolio that mimics it and price that portfolio.
This reading only asks you to understand that principle, not to run the full calculation, which the Derivatives module develops with the one-period binomial model. The takeaway is that option pricing is one more application of the same rule: value the unknown by replicating it with the known, and let no arbitrage supply the price.
You are expected to explain, not compute, the option connection at this level. If a question describes replicating an option’s payoff with the underlying and a loan, the point being tested is that no arbitrage forces the option price to equal the cost of the replicating portfolio.
It is worth pausing on why this is remarkable. An option has an asymmetric payoff, all upside beyond a strike with the downside capped, which feels impossible to value without predicting where the underlying will end up. Replication dissolves that difficulty. By holding a shifting mix of the underlying and a loan that always matches the option’s payoff, an investor can manufacture the option synthetically, and the cost of manufacturing it is its fair price. The forecast of the underlying never enters, because both the option and its replica rise and fall together whatever happens. That is the deepest lesson of this reading, and it is why the time value of money, extended by additivity, reaches all the way from a simple bond to the most intricate derivative.
A two-year bond pays no coupon and repays 1,000 rupees. It trades at 826.45. What is its implied annual return?
(1 + r)2 = 1,000 ÷ 826.45 = 1.2100. Taking the square root, 1 + r = 1.10, so r = 10%.
A share is expected to pay a 3-rupee dividend next year, growing at 5% forever, and investors require 11%. What is the value today?
P0 = 3 ÷ (0.11 − 0.05) = 3 ÷ 0.06 = 50 rupees.
The one-year spot rate is 3% and the two-year spot rate is 4%. What is the implied one-year forward rate one year from now?
(1.04)2 = (1.03)(1 + f). 1.0816 ÷ 1.03 = 1.0501, so f = 5.01%.
A share trades at 40, is expected to pay a 2-rupee dividend next year, and investors require 10%. What growth rate is the price implying?
g = r − (D1 ÷ P0) = 0.10 − (2 ÷ 40) = 0.10 − 0.05 = 5%.
Because the coupons and face value are fixed, so raising the discount rate lowers the present value of every one of those cash flows. The only way a bond with a fixed coupon can offer a higher required return is for its price to fall, which is why price and yield always move in opposite directions.
They are the same idea for a bond. The yield to maturity is the discount rate that equates the present value of the bond’s cash flows with its market price, which is exactly the return the market currently requires to hold it.
Because the model divides by the difference between the two. If growth equaled or exceeded the required return, the denominator would be zero or negative and the value would be infinite or meaningless. Sustained growth above the required return is also economically impossible in the long run.
Next year’s, the dividend one period ahead. If you are given the dividend just paid, grow it by one period first before placing it in the numerator, or you will understate the value.
It means there is no way to make a certain profit with no investment and no risk. If two packages of identical cash flows had different prices, traders would buy the cheap one and sell the dear one until the prices met. Valuation formulas for forwards and options are built on the assumption that this has already happened.
No. A forward rate is fixed today by the current spot rates and the no-arbitrage condition, not by anyone’s prediction. It is the rate that removes any riskless profit between investing long and rolling short investments forward, whatever the future turns out to hold.
Because covered interest parity must hold. If a currency paid a higher interest rate and also held its value forward, investors could borrow the low-rate currency, invest in the high-rate one, and lock in a riskless gain. The forward rate adjusts so that the interest advantage is exactly offset, leaving no free profit.
Through direct valuation: pricing a bond, finding its implied yield, valuing a share with the growth model, backing out an implied return or growth, and computing forward interest and exchange rates. Set the cash flows up correctly, discount cleanly, and mind the quote convention on currencies.
Loading comments...
Add your Thoughts: