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Time Value of Money – CFA Level 1 Study Guide

This guide covers the core TVM concepts tested on the CFA Level 1 exam: present and future value calculations, how to price fixed-income and equity instruments, implied returns, and the cash flow additivity principle that enforces no-arbitrage pricing.

Present Value and Future Value: The Core Relationship

The entire TVM framework comes down to one equation. If you invest a present value (PV) at a discount rate r for t periods, the future value (FV) is:

Flip it around and you get the present value — what a future cash flow is worth today:

When compounding happens continuously rather than at discrete intervals, the formulas use the exponential function:

The key intuition: a higher discount rate or longer time horizon means a larger gap between PV and FV. This underpins everything that follows — bond pricing, equity valuation, and arbitrage-free pricing.

Fixed-Income Instruments and TVM

Fixed-income instruments are contracts where an issuer borrows money and promises future cash flows — interest, principal, or both. The discount rate for these instruments is the interest rate, and the implied return is commonly called the yield-to-maturity (YTM).

Cash flows from fixed-income instruments follow one of three patterns:

Discount (Zero-Coupon) Bonds

The simplest case: one single cash flow at maturity. The investor pays PV today and receives the face value (FV) at maturity. The difference is the implied interest earned.

For example, a 20-year government zero-coupon bond with a 6.70% YTM and face value of 100 would trade at 100 / (1.067)²⁰ = 27.33 today. As time passes and the bond approaches maturity, PV naturally rises toward FV — this is called “accretion” of the discount.

A critical relationship to remember: when rates rise, the PV of the bond falls. When rates fall, PV rises. This inverse relationship between price and yield is fundamental to fixed-income analysis.

Coupon Bonds

Most bonds pay periodic interest (coupons) plus return the principal at maturity. The price of a coupon bond is the sum of the present values of all its cash flows:

Key pricing rules for coupon bonds:

Perpetual Bonds

A perpetual bond (or perpetuity) has no maturity date and pays a fixed coupon forever. As the number of periods approaches infinity, the pricing formula simplifies dramatically:

Annuity Instruments (Mortgages & Amortizing Loans)

Annuities are instruments where each payment includes both interest and principal repayment. Mortgages are the most common example. The periodic payment formula is:

Although each payment is identical, the composition shifts over time: early payments are mostly interest, while later payments are mostly principal. This is the standard amortization schedule structure.

Equity Instruments and TVM

Unlike fixed-income instruments, equities have no maturity date and their cash flows (dividends) are discretionary. Valuing equities with TVM requires assumptions about future dividend patterns.

Constant Dividend Model (Zero Growth)

If a stock pays a fixed dividend D indefinitely with no growth, it’s just a perpetuity:

Gordon Growth Model (Constant Growth)

The more common approach assumes dividends grow at a constant rate g forever. This is the Gordon Growth Model, also known as the dividend discount model (DDM):

where D_t+1 is the expected dividend next period, r is the required return, and g is the constant growth rate. The model only works when r > g.

A higher growth rate g reduces the denominator (r − g), which increases the stock price. This sensitivity means small changes in growth assumptions can have outsized impacts on valuation — a frequent exam theme.

Two-Stage Dividend Growth Model

Many companies experience a period of high growth followed by a period of slower, sustainable growth. The two-stage model handles this by splitting the valuation into two parts:

The first term sums the present value of dividends during the high-growth phase. The second term is the present value of the “terminal value” — the stock’s price at the point where growth stabilizes, calculated using the Gordon Growth Model.

Implied Return and Implied Growth

Sometimes you know the price and want to solve for the return or growth rate embedded in that price. This is the concept of implied return.

Implied Return on Discount Bonds

Implied Return on Coupon Bonds

For coupon bonds, the implied return is the YTM — the discount rate that equates the bond’s price to the present value of all promised cash flows. This is solved iteratively (or with a financial calculator / spreadsheet RATE function).

Implied Return and Growth for Stocks

Rearranging the Gordon Growth Model gives the implied return on a stock:

In plain English: the implied return equals the expected dividend yield plus the expected growth rate.

Alternatively, if you know the required return but want the implied growth rate:

Forward Price-to-Earnings Ratio

The relationship between P/E ratios and TVM connects valuation multiples to fundamentals. The forward P/E ratio can be expressed as:

This shows that a higher payout ratio or higher growth increases the P/E ratio, while a higher required return decreases it. This is why high-growth companies trade at higher P/E multiples — it’s TVM at work.

Cash Flow Additivity and No-Arbitrage Pricing

The cash flow additivity principle states that the present value of any set of future cash flows equals the sum of the present values of those individual cash flows, as long as they occur at the same point in time. This sounds simple, but its implications are powerful.

The key consequence: if two investment strategies produce identical future cash flows, they must have the same present value. If they don’t, an arbitrage opportunity exists — a riskless profit with no initial investment.

Implied Forward Interest Rates

Cash flow additivity lets us derive forward rates from spot rates. If a 1-year bond yields r₁ and a 2-year bond yields r₂, the implied 1-year forward rate starting in one year (F_1,1) must satisfy:

Solving: F_1,1 = (1 + r₂)² / (1 + r₁) − 1

If the actual forward rate differs from this implied rate, investors can construct arbitrage strategies by borrowing at the cheaper rate and lending at the more expensive one.

Forward Exchange Rates

The same no-arbitrage logic applies to currencies. If you can invest domestically or convert to a foreign currency, invest abroad, and convert back, both strategies must yield the same result. The forward exchange rate enforces this equivalence using the interest rate differential between the two currencies.

When the domestic interest rate exceeds the foreign rate, the domestic currency is expected to depreciate on a forward basis. This is the foundation of covered interest rate parity.

Option Pricing via Cash Flow Additivity

The no-arbitrage framework also extends to options. By constructing a replicating portfolio of the underlying asset and risk-free borrowing that matches the option’s payoffs in every state of the world, you can determine the option’s fair price. This is the basis of the binomial option pricing model.

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