Present Value Relations: Cash Flows, Compounding, Inflation, and Bonds

A dollar today and a dollar next year have the same currency label but different economic values. Present value gives us a common measuring date so that we can compare investments, loans, projects, and bonds. The essential habit is simple: draw the cash-flow timeline, identify the appropriate rate, and move every amount to the same date before adding it.

Learning goals

By the end, you should be able to:

• Translate an asset into dated cash flows and calculate its present value and NPV.
• Derive and apply perpetuity, growing-perpetuity, and annuity formulas.
• Convert quoted rates into rates that match the payment interval.
• Keep nominal and real valuations consistent.
• Price a bond with either a flat yield or maturity-specific spot rates, and explain the difference.

1. Start with cash flows and a timeline

From the investor’s perspective, represent an asset or project as

$$(CF_0,CF_1,CF_2,\ldots,CF_T).$$

An inflow is positive; an outflow is negative. A machine may require a purchase today, generate operating receipts, require maintenance, and have a resale value. Include all relevant incremental cash flows, including later costs. Accounting profit and cash flow are different quantities.

For a project costing 1,000 today and paying 600 at each of the next two year-ends:

Date (years)       0              1              2
                   |--------------|--------------|
Cash flow       -1,000           +600           +600

The timeline prevents two common mistakes: treating the first payment as immediate when it arrives a year later, and forgetting the final payment or resale proceeds.

Dated money as different “currencies”

Think of a dollar at date 0 and a dollar at date 1 as different dated commodities. An interest rate supplies the conversion factor. If one dollar invested today becomes 1.05 next year, then a certain 1.05 next year is worth one dollar today under that investment opportunity.

With a constant effective rate per period, $r$:

$$FV_T=PV_0(1+r)^T, \qquad PV_0=\frac{FV_T}{(1+r)^T}.$$

For example, 1,000 invested at 5% for three years becomes 1,157.625. Conversely, 1,157.625 due in three years has PV 1,000 at that rate.

The discount factor is $D(0,t)=(1+r)^{-t}$. It converts one unit paid at date $t$ into units at date 0. Discounting is the reverse of compounding, not a subtraction of $r$ from the future amount.

2. The opportunity cost of capital

The discount rate is the return available on an alternative investment with comparable risk, timing, and currency. It is the opportunity you give up by committing funds to this asset.

For certain cash flows, use appropriate default-free discount factors. For risky projects, a simplified discounted-cash-flow model uses expected cash flows and a risk-adjusted required return:

$$PV_0=\sum_{t=1}^{T}\frac{\mathbb{E}[CF_t]}{(1+r)^t}.$$

A single constant $r$ is an assumption, not a universal pricing law. Risks and market rates can vary across maturities. More general models use maturity-specific discount factors or state-dependent pricing. A risky promised payment should not automatically be treated as a certain payment.

Throughout the basic examples, assume equally spaced dates, end-of-period payments, a constant rate, and no taxes or transaction costs unless stated otherwise. Rates are written as decimals: 5% means 0.05. Cash flows and rates must use the same currency and inflation convention.

3. Net present value and the investment decision

Present value of future receipts is not yet profit. Subtract what you must pay to obtain those receipts:

$$\operatorname{NPV}_0=CF_0+\sum_{t=1}^{T}\frac{CF_t}{(1+r)^t}.$$

Usually $CF_0=-I_0$, where $I_0$ is the initial investment. The date-0 amount needs no discounting.

For an independent project with correctly estimated cash flows and opportunity cost:

• NPV > 0: the project adds value relative to the alternative investment.
• NPV < 0: the project destroys value relative to that alternative.
• NPV = 0: it earns exactly the required return in this model.

A positive NPV is not a guarantee that a risky project’s realized cash flows will be favorable. If projects are mutually exclusive, compare feasible alternatives by NPV; budget constraints can require selecting a combination of projects.

Worked example: a two-year project

Return to the timeline above and use $r=10\%$:

Date	Cash flow	Discount factor	Present value
0	-1,000	1	-1,000.00
1	600	$1/1.10$	545.45
2	600	$1/1.10^2$	495.87

$$\operatorname{NPV}_0=-1{,}000+\frac{600}{1.10}+\frac{600}{1.10^2} \approx 41.32.$$

The project creates about 41.32 of value today after allowing for the 10% opportunity cost. Its undiscounted surplus is 200, which overstates the value available today.

Why can we add present values?

Once all cash flows have been converted to the same date using a consistent pricing framework, valuation is additive:

$$PV(A+B)=PV(A)+PV(B).$$

If identical attainable cash-flow streams had different prices in a frictionless market, buying the cheaper stream and selling the more expensive one would create an arbitrage. This replication idea will explain bond pricing later.

4. Perpetuities: constant payments forever

An ordinary perpetuity pays $C$ at dates $1,2,3,\ldots$, with the first payment one full period away:

$$PV_0=\frac{C}{1+r}+\frac{C}{(1+r)^2}+\cdots=\frac{C}{r},\qquad r>0.$$

To derive this, let $q=1/(1+r)$. The geometric series is $Cq/(1-q)=C/r$.

A perpetual payment of 120 each year discounted at 6% is worth

$$PV_0=\frac{120}{0.06}=2{,}000.$$

The 120 is income paid out each year; the principal that supports those payments remains invested in the stylized replication. A perpetuity is an idealization, so the formula does not eliminate issuer risk or uncertainty about whether payments can continue.

Timing changes the answer

If the same stream includes an additional payment of 120 today, its value is $120+2{,}000=2{,}120$.

If the first payment is at date 4, the value $C/r$ sits at date 3, one period before the first payment. Therefore:

$$PV_0=\frac{C/r}{(1+r)^3}.$$

Always locate the date to which a shortcut assigns value.

5. Growing perpetuities

Let the first payment be $C_1$ at date 1, followed by $C_1(1+g)$ at date 2 and $C_1(1+g)^2$ at date 3. Then

$$PV_0=\sum_{t=1}^{\infty}\frac{C_1(1+g)^{t-1}}{(1+r)^t} =\frac{C_1}{r-g}.$$

The geometric ratio is $(1+g)/(1+r)$. With economically conventional $r>-1$ and $g>-1$, convergence requires $r>g$. If $r=g$, positive discounted payments do not shrink; there is no finite sum. If $g>r$, this positive-payment perpetuity also has no finite PV.

Worked example: next year’s payment versus today’s payment

Suppose the payment just made was $C_0=100$, expected growth is 2%, and the required return is 7%. Next year’s payment is

$$C_1=100(1.02)=102.$$

Thus

$$PV_0=\frac{102}{0.07-0.02}=2{,}040.$$

Using 100 in the numerator would incorrectly treat the most recent payment as the next payment. Using perpetual growth close to the discount rate makes the valuation extremely sensitive: at 3% growth, the same next payment of 102 would have PV 2,550. Long-run growth assumptions require care.

6. Annuities: a finite stream of equal payments

An ordinary annuity pays $C$ at dates $1$ through $T$:

$$PV_0=C\sum_{t=1}^{T}(1+r)^{-t} =\frac{C}{r}\left[1-\frac{1}{(1+r)^T}\right].$$

Derivation by subtracting two perpetuities

A perpetuity starting at date 1 has value $C/r$. Remove the payments from date $T+1$ onward. That removed perpetuity is worth $C/r$ at date $T$, so it is worth $(C/r)/(1+r)^T$ today. The difference leaves exactly the payments at dates 1 through $T$.

For $r=0$, use the limiting value $PV_0=CT$ rather than dividing by zero.

Worked example: three annual receipts

For $C=500$, $T=3$, and $r=5\%$:

$$PV_0=\frac{500}{0.05}\left[1-(1.05)^{-3}\right] \approx 1{,}361.62.$$

Directly discounting 500 at each of the three dates gives the same result. This is a useful way to audit a shortcut.

Annuity due and loan payments

An annuity due pays at dates $0$ through $T-1$. Each payment arrives one period earlier, so

$$PV_{\text{due}}=(1+r)PV_{\text{ordinary}}.$$

A fully amortizing loan with principal $L$, periodic rate $r$, and $T$ end-of-period payments satisfies

$$L=\frac{C}{r}[1-(1+r)^{-T}], \qquad C=\frac{Lr}{1-(1+r)^{-T}}.$$

A 10,000 loan with 1% monthly interest and 12 monthly payments requires approximately 888.49 per month. Here $T=12$ counts months and $r=0.01$ is monthly. Interest and principal allocations change over time even though the total payment is constant.

Extension: a growing annuity

For $T$ payments beginning at $C_1$ and growing at $g$:

$$PV_0=\frac{C_1}{r-g}\left[1-\left(\frac{1+g}{1+r}\right)^T\right],\qquad r\ne g.$$

Unlike the infinite case, a finite stream can be valued even when $g\ge r$. At $g=r$, each discounted payment equals $C_1/(1+r)$, so $PV_0=TC_1/(1+r)$.

7. Compounding: APR, EAR, and payment frequency

An APR here means a nominal annual rate quoted with a specified compounding frequency, excluding fees. Actual regulated APR disclosures may include costs and follow jurisdiction-specific conventions; inspect the product’s definition.

If APR is $a$ with $n$ compounding periods per year, the periodic rate is $a/n$. One unit becomes $(1+a/n)^n$ after a year, giving

$$EAR=\left(1+\frac{a}{n}\right)^n-1.$$

For a positive APR and more than one compounding period, EAR exceeds the numerical APR. With one annual compounding period, they are equal.

Worked example: 12% APR compounded monthly

The monthly rate is 1%, and

$$EAR=(1.01)^{12}-1\approx 12.6825\%.$$

The PV of 1,000 due in one year can be written in either equivalent form:

$$PV_0=\frac{1{,}000}{(1.01)^{12}} =\frac{1{,}000}{1.126825\ldots} \approx 887.45.$$

For 1,000 due in six months, use six monthly periods, giving approximately 942.05.

The rule is to match the rate to the cash-flow interval. Use EAR with annual periods, the effective monthly rate with monthly periods, and so on. “Always use EAR” is incomplete advice: discounting each monthly payment using a full annual rate would be wrong.

If you are given EAR rather than APR, the equivalent monthly rate is

$$r_m=(1+EAR)^{1/12}-1,$$

not $EAR/12$.

Continuous compounding

As compounding becomes arbitrarily frequent,

$$\lim_{n\to\infty}(1+a/n)^n=e^a.$$

Writing $\rho$ for the continuously compounded annual rate:

$$FV_t=PV_0e^{\rho t},\qquad PV_0=FV_te^{-\rho t},\qquad EAR=e^\rho-1.$$

Conversely, $\rho=\ln(1+EAR)$. A continuously compounded rate of 5% implies EAR approximately 5.1271%. Time $t$ must be in years when $\rho$ is annual.

8. Inflation: nominal money and real purchasing power

Nominal cash flows are measured in the money paid at each future date. Real cash flows are measured in the purchasing power of a chosen base date. Inflation changes the conversion between these units.

With deterministic annual inflation $i$ and consistent annual rates:

$$1+r_{\text{real}}=\frac{1+r_{\text{nominal}}}{1+i},$$

or

$$1+r_{\text{nominal}}=(1+r_{\text{real}})(1+i).$$

The approximation $r_{\text{real}}\approx r_{\text{nominal}}-i$ drops the cross-product term. Use the exact relation for calculations.

Worked example: two equivalent valuations

Suppose a project costs 900 today and pays a nominal 1,100 in two years. The nominal discount rate is 8%, and inflation is a deterministic 3% each year.

The real discount rate is

$$r_{\text{real}}=\frac{1.08}{1.03}-1\approx 4.8544\%.$$

The terminal receipt in today’s purchasing power is

$$CF_{2,\text{real}}=\frac{1{,}100}{1.03^2}\approx 1{,}036.86.$$

Both approaches give

$$PV_0=\frac{1{,}100}{1.08^2} =\frac{1{,}100/1.03^2}{(1.08/1.03)^2} \approx 943.07,$$

so NPV is approximately 43.07. Do not round the real rate before evaluating the second expression.

Discounting nominal 1,100 at the real rate would overvalue the receipt. Discounting the inflation-adjusted amount at the nominal rate would undervalue it.

When inflation is uncertain, this simple deterministic conversion is not a complete asset-pricing model: expected inflation, realized purchasing power, and inflation risk premia must be distinguished.

9. Bonds are packages of dated payments

A plain fixed-rate bond promises coupons and repayment of face value. Assume purchase immediately after a coupon date, no accrued interest, no default or embedded options, and annual coupons.

For face value $F$, annual coupon rate $c$, annual coupon $C=cF$, maturity $T$, and annual yield $y$:

$$P_0=\sum_{t=1}^{T}\frac{C}{(1+y)^t}+\frac{F}{(1+y)^T} =\frac{C}{y}[1-(1+y)^{-T}]+\frac{F}{(1+y)^T}.$$

The principal is paid in addition to the final coupon.

Date              0              1              2              3
                   |--------------|--------------|--------------|
Investor flow     -P             +C             +C            +(C+F)

Worked example: a coupon bond

Let $F=1{,}000$, $c=5\%$, $T=3$, and $y=6\%$. The coupons are 50:

$$P_0=\frac{50}{1.06}+\frac{50}{1.06^2}+\frac{1{,}050}{1.06^3} \approx 973.27.$$

The bond sells below face value because its coupon rate is below the required yield. Under the stated conventions, coupon rate equal to yield gives a par price; coupon rate above yield gives a premium price. For fixed positive cash flows, increasing the discount yield reduces price.

For semiannual coupons, use $2T$ periods and coupon $cF/2$. If the quoted yield is a nominal annual yield compounded semiannually, use $y/2$ per period. If it is an EAR, first convert to $(1+y)^{1/2}-1$.

Pure discount bonds and spot rates

A zero-coupon or pure discount bond pays only face value at maturity. Its price reveals the discount factor:

$$P_0^{\text{zero}}=FD(0,T),\qquad D(0,T)=\frac{1}{(1+s_T)^T},$$

where $s_T$ is the annual effective spot rate for maturity $T$.

A coupon bond can be replicated with zero-coupon bonds, one for each payment date. Consequently, with default-free payments and a consistent default-free curve,

$$P_0=\sum_{t=1}^{T}CF_tD(0,t) =\sum_{t=1}^{T}\frac{CF_t}{(1+s_t)^t}.$$

For a two-year bond paying 50 after one year and 1,050 after two years, let $s_1=4\%$ and $s_2=5\%$:

$$P_0=\frac{50}{1.04}+\frac{1{,}050}{1.05^2}\approx 1{,}000.46.$$

The two-year spot rate applies to the entire two-year discounting of the second payment; do not discount it as $1.04\times1.05$. The second year’s forward rate is a different object.

Yield to maturity versus the yield curve

Yield to maturity (YTM) is the single rate that makes the PV of a bond’s promised payments equal its observed price. A spot curve describes rates for individual maturities. A plotted yield curve must specify whether it shows spot rates, par yields, or yields on particular coupon bonds.

YTM summarizes a price; it does not imply every maturity has that same spot rate. Nor is it a guaranteed realized holding-period return: default, an early sale, and reinvestment rates affect outcomes. With positive coupons, equating YTM to a compounded terminal-wealth return requires reinvesting coupons at that yield and receiving the promised payments.

For defaultable bonds, discounting all promised payments with a default-free curve alone generally overstates value. Credit risk needs an appropriate model or adjustment.

10. A repeatable valuation workflow

1. Choose whose cash flows you are valuing and mark inflows and outflows.
2. Draw every payment date, including the initial cost and final principal.
3. Identify payment frequency, currency, risk, and nominal or real units.
4. Convert quoted rates into the required effective periodic rates, or obtain discount factors for each date.
5. Use direct discounting or a shortcut whose timing assumptions fit.
6. Add values at one common date and subtract the acquisition cost for NPV.
7. Check sensitivity to discount rates, growth, inflation, and uncertain cash flows.

Useful checks: an ordinary positive-rate annuity is worth less than its undiscounted payment total; its value rises when payments move earlier; a perpetuity with a later first payment is worth less; and nominal and real calculations agree when the conversion is consistent.

11. Practice questions

Try these before reading the answers. Unless specified otherwise, rates are effective per payment period and payments arrive at period-end.

1. An investment costs 800 today and pays 450 at the end of each of the next two years. At 8%, what is NPV?
2. A perpetuity pays 90 annually beginning one year from today. At 6%, what is its value? What if the first payment is at year 3?
3. Next year’s payment is 200 and perpetual growth is 3%. At a required return of 8%, find PV. What fails if the required return falls to 3%?
4. Find the PV of four annual payments of 300 at 5%. Then move all four payments one year earlier.
5. A nominal APR of 8% compounds quarterly. Find EAR and the PV of 1,000 due in 18 months.
6. If the nominal annual rate is 7% and deterministic annual inflation is 2%, find the real rate. Value a nominal 1,040.40 due in two years using both methods.
7. A two-year annual-coupon bond has face value 1,000, coupon rate 4%, and YTM 5%. Find its price and classify it as discount, par, or premium.
8. Explain why using a 12% EAR divided by 12 is incorrect for a monthly annuity.

12. Answers and reasoning

1. $NPV=-800+450/1.08+450/1.08^2\approx 2.47$. It adds a small positive value under the assumptions.
2. Ordinary perpetuity: $90/0.06=1{,}500$. First payment at year 3: value 1,500 at year 2, so PV is $1{,}500/1.06^2\approx1{,}334.99$.
3. $PV=200/(0.08-0.03)=4{,}000$. At $r=g=3\%$, the infinite series of positive discounted payments does not converge to a finite value.
4. Ordinary annuity: $(300/0.05)[1-1.05^{-4}]\approx1{,}063.79$. Annuity due: multiply the unrounded value by 1.05, giving approximately 1,116.97.
5. Quarterly rate is 2%; EAR is $1.02^4-1\approx8.2432\%$. Eighteen months contains six quarters: $PV=1{,}000/1.02^6\approx887.97$.
6. Real rate is $1.07/1.02-1\approx4.9020\%$. Nominal PV is $1{,}040.40/1.07^2\approx908.73$. The real payment is $1{,}040.40/1.02^2=1{,}000$; discounting it at the unrounded real rate gives the same PV.
7. $P=40/1.05+1{,}040/1.05^2\approx981.41$. It is a discount bond because the coupon rate is below YTM.
8. Division ignores the compounding already included in EAR. The equivalent monthly rate is $1.12^{1/12}-1\approx0.9489\%$, rather than 1%.

Further reading

• Andrew W. Lo, MIT OpenCourseWare, Present Value Relations, lecture slides 1–36: original lecture framework for dated cash flows, discounting, compounding, and inflation.
• MIT OpenCourseWare, Present Value Relations recitation notes: supplementary practice on recurring cash flows and interest-rate conventions.
• Andrew W. Lo, MIT OpenCourseWare, Fixed-Income Securities, lecture slides 1–65: bonds, discount factors, and term structure.
• MIT OpenCourseWare, Fixed-Income Securities recitation: spot rates, forward rates, YTM, and bond-pricing exercises.

Feynman study lab

Explain it simply

Money at different dates must be translated to one date before comparison.

Concrete example. Receiving 110 next year has present value 100 at 10% annual interest.

Teach it back without looking

Close the lesson and explain the idea aloud to a beginner in two minutes. Define each technical term you use, walk through the example, and state one assumption or limitation. Then answer:

Why is a promised 110 next year not automatically better than 105 today?

Check your explanation after you try

At that discount rate, 105 today exceeds the future payment’s value of 100. The appropriate rate also depends on risk and alternatives.

Find the gap and try again

If you hesitated, relied on a formula you could not explain, or missed the self-check, write down the exact gap. Revisit the relevant section of this lesson and work through its example. Close the notes, explain it again in simpler language, then change one number or assumption and predict what changes. Recalculate or check the lesson to test your prediction. Try once more tomorrow from memory.