Finance-Theory · Unit 4 · Video 5 · Interactive Practice

Real Returns: Match the Discount Rate to the Cashflow

IKey Formulas

Formula	Name	What you need
$(1 + r_{\text{nominal}})^k = \dfrac{W_{t+k}}{W_t}$	Nominal return	Wealth at two dates
$\widetilde{W}_{t+k} \equiv \dfrac{W_{t+k}}{(1+\pi)^k}$	Real wealth	Nominal wealth and inflation
$r_{\text{real}} = \dfrac{1 + r_{\text{nominal}}}{1 + \pi} - 1 \;\approx\; r_{\text{nominal}} - \pi$	Real return — exact, then the rule of thumb	Both rates
$\text{PV} = \sum_t \dfrac{C_t^{\text{real}}}{(1 + r_{\text{real}})^{t}} = \sum_t \dfrac{C_t^{\text{nominal}}}{(1 + r_{\text{nominal}})^{t}}$	The matching rule	Cashflows labelled real or nominal

Key Insight: $(1 + r_{\text{nominal}}) = (1 + r_{\text{real}})(1 + \pi)$ — inflation belongs on exactly one side of that product, which is why discounting a real cashflow at a nominal rate takes it out twice.

IICounting Dollars, Counting Baskets

More dollars next year need not mean more of anything next year.

Inflation π 10%

IIIDivide, or Subtract?

The rule of thumb $r_{\text{nominal}} - \pi$ misses the exact real return by how much, and where?

Nominal return r 15%

💡 The two curves meet at $\pi = 0$ and again at $\pi = r_{\text{nominal}}$ , because the shortcut's error is exactly $\dfrac{(r_{\text{nominal}} - \pi)\,\pi}{1 + \pi}$ .

IVThe Present Value of a Career

One career, valued four ways: two pairings obey the rule and two cross it.

Cashflows Discount rate

💡 The two matched routes agree term by term, not merely in total: multiplying a real cashflow by $(1+\pi)^{t}$ and its discount factor by the same $(1+\pi)^{t}$ leaves every ratio unchanged.

VQuiz Questions

Problem 1 · Deflating a Nominal Return

Given: a portfolio returns $8\%$ over the year while inflation runs at $3\%$ — find the exact real return.

✅ Correct!

\dfrac{1.08}{1.03} - 1 = 0.048544

, so

4.85\%

— a shade under the

5\%

the rule of thumb reports.

❌ That multiplies instead of dividing.

1.08 \times 1.03 - 1 = 11.24\%

compounds inflation onto the return; real wealth is nominal wealth divided by

1 + \pi

❌ Right numerator, wrong deflator.

\dfrac{0.05}{0.97}

divides by

1 - \pi

. Prices rose, so the deflator is

1 + \pi = 1.03

❌ That is the rule of thumb, not the exact figure.

8\% - 3\%

subtracts; the exact form divides by

1 + \pi

, giving

\dfrac{0.05}{1.03} = 4.85\%

Show solution

Deflate the nominal return by the accumulated inflation:

r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + \pi} - 1 = \frac{1.08}{1.03} - 1 = 1.048544 - 1 = 4.85\%

Rearranged, the same thing is $\dfrac{r_{\text{nominal}} - \pi}{1 + \pi} = \dfrac{0.05}{1.03} = 4.85\%$ : the rule of thumb's numerator, but carried over $1 + \pi$ .

The shortcut's error here is $\dfrac{(0.08 - 0.03)(0.03)}{1.03} = 0.146$ percentage points — small, which is exactly why the shortcut survives in practice.

Problem 2 · Which Rate Belongs in the Denominator

Given: a pension forecast quoted in today's purchasing power, a market interest rate of $6\%$ and inflation of $2.5\%$ — find the rate that belongs in the discount factor.

✅ Correct! The forecast is real, so the rate must be real:

\dfrac{1.06}{1.025} - 1 = 3.41\%

❌ That crosses the pair. The forecast already has inflation stripped out of it. A

6\%

market rate still contains inflation, so using it removes inflation a second time and understates the present value.

❌ Inflation is not a discount rate. It converts between real and nominal units. The time value of money is still sitting inside the

6\%

and has to be discounted for.

❌ That compounds inflation in rather than out. The real rate is the market rate deflated,

\dfrac{1.06}{1.025} - 1

, not

1.06 \times 1.025 - 1

Show solution

The cashflows are stated in constant purchasing power, so they are real, and the rule sends you to the real rate:

r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + \pi} - 1 = \frac{1.06}{1.025} - 1 = 3.41\%

The rule of thumb would say $6\% - 2.5\% = 3.5\%$ , close but $0.085$ percentage points high.

The alternative is equally valid and must give the same answer: inflate the forecast into actual dollars by $(1.025)^{t}$ and discount at $6\%$ . What is never valid is one of each.

Problem 3 · One Year of the Career

Given: interest rates of $5\%$ , inflation of $2\%$ , and next year's salary forecast at $102,000 stated in today's purchasing power — find the real discount rate and then the present value of that single cashflow.

What is the real rate?

What is the present value of that cashflow?

✅ Correct! The real rate is

\dfrac{1.05}{1.02} - 1 = 2.94118\%

, and $102,000 divided by

1.0294118

is $99,086.

❌ Not the real rate. Deflate the market rate:

\dfrac{1 + 0.05}{1 + 0.02} - 1

. Subtracting gives

3.00\%

, multiplying gives

7.10\%

, and dividing by

1 + r

instead of

1 + \pi

gives

2.86\%

— none of them is that ratio.

❌ Check the divisor. A real cashflow one period away is divided by

1 + r_{\text{real}}

— not by

1 + \pi

, and not by the rounded

3\%

❌ That undoes the growth rather than discounting it. $102,000 divided by

1.02

is simply this year's salary; the cashflow still has to be moved back one year in time.

❌ That crosses the pair. $102,000 is already a real figure, so the

5\%

nominal rate removes inflation a second time — $97,143 is $1,943 too low.

Show solution

Step 1 — the real rate.

r_{\text{real}} = \frac{1.05}{1.02} - 1 = 0.0294118 \quad\text{i.e. } 2.94118\%

The printed $2.94\%$ is a rounding; keep the ratio itself for the arithmetic.

Step 2 — discount the real cashflow at the real rate. $102,000 ÷ $1.0294118$ = $99,085.71, which is $99,086 to the nearest dollar.

Check by the other route. In actual dollars next year's salary is $100,000 × $1.02$ × $1.02$ = $104,040, and $104,040 ÷ $1.05$ = $99,085.71 — the same number, because $(1 + r_{\text{nominal}}) = (1 + r_{\text{real}})(1 + \pi)$ .

Problem 4 · Three Years of Purchasing Power

Given: a fund returning $9\%$ a year for three years while inflation runs at $4\%$ a year — find the factor by which your purchasing power grows over the three years.

✅ Correct!

\left(\dfrac{1.09}{1.04}\right)^{3} = 1.1513

: three years of

9\%

against

4\%

inflation buys

15.13\%

more.

❌ That compounds the rule of thumb.

1.05^{3} = 1.1576

uses

9\% - 4\% = 5\%

. The exact real return is

\dfrac{1.09}{1.04} - 1 = 4.808\%

, and the shortcut's error compounds right along with the rate.

❌ That is the dollar count, not the basket count.

1.09^{3} = 1.2950

ignores that the price level also rose, by

1.04^{3} = 1.1249

; the ratio

\dfrac{1.2950}{1.1249}

is the answer.

❌ That is one year of it.

\dfrac{1.09}{1.04} = 1.0481

is the single-period real factor; purchasing power compounds, so raise it to the third power.

Show solution

Real wealth divides nominal wealth by the price level, so over three years the two factors divide:

(1 + r_{\text{real}})^{3} = \frac{(1.09)^{3}}{(1.04)^{3}} = \frac{1.295029}{1.124864} = 1.151276

Purchasing power grows $15.13\%$ , against the $15.76\%$ the rule of thumb would give and the $29.50\%$ the dollar count shows.

Per year the real return is $\dfrac{1.09}{1.04} - 1 = 4.808\%$ , and $1.04808^{3} = 1.1513$ — the same figure, because a ratio raised to a power is the power of the ratio.

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