Finance-Theory · Unit 14 · Video 4 · Interactive Practice

Real Rates and the Risk-Return Tradeoff, 1946–2001

IKey Formulas

Formula	Name	From the 1946–2001 table (monthly)
$r_{\text{real}} \approx r_f - \text{inflation}$	Real rate (approximation)	$0.38\% - 0.32\% = 0.06\%$ per month, six basis points
$r_{\text{real}} = \dfrac{1 + r_f}{1 + \text{inflation}} - 1$	Real rate (exact)	$\dfrac{1.0038}{1.0032} - 1 \approx 0.0598\%$ per month, the same 6 bp once rounded
$\text{annual} \approx 12 \times \text{monthly}$	Simple annualization (not compounded)	$0.06\% \times 12 \approx 0.72\%$ per year
$R = \dfrac{P_{\text{end}} - P_{\text{start}}}{P_{\text{start}}}$	Total return over a holding month	What the table's bill, note and stock rows record; a chart of yields to maturity records something else

Key Insight: Two facts come out of one table. The asset treated as risk-free stayed only slightly ahead of inflation, and average return rises with standard deviation: T-bill ( $\sigma = 0.24\%$ , average $0.38\%$ ), T-note, the two stock indexes, then Motorola ( $\sigma = 10.02\%$ , average $1.66\%$ ).

IIThe Real Rate from Two Table Entries

The real rate is roughly the gap between the T-bill rate and inflation, both per month.

T-bill rate (% per month) 0.38 Inflation (% per month) 0.32

IIIRisk and Reward Down the Table

Moving from T-bills to a single stock, higher standard deviation comes with a higher average return and wider extreme months.

IVWhy a T-Bill’s $\sigma$ Is Above Zero

A bill's return over one holding month depends on where yields go that month, not only on the yield quoted at purchase.

Yield at purchase 11-month yield one month later (% a year) 7.00

💡 Illustrative yields; the bill is priced as $100/(1+y)^t$ per 100 of face value, with $t$ in years. The rest of the real T-bill's $\sigma = 0.24\%$ comes from the level of rates itself moving over 1946–2001, which spread its monthly returns from $0.03\%$ to $1.34\%$ .

VQuiz Questions

Problem 1 · A Real Rate, Monthly and Annual

Given: Illustrative figures: over some period a one-year T-bill returned $0.34\%$ per month on average and inflation averaged $0.25\%$ per month. Find the approximate real rate per month, and its simple annualization.

Approximate real rate per month

Simple annualization of that real rate

✅ Correct!

0.34\% - 0.25\% = 0.09\%

a month, nine basis points, and

0.09\% \times 12 = 1.08\%

a year, simple rather than compounded.

❌ Inflation is subtracted, not added. Rising prices eat into what the bill earns, so the real rate is the nominal rate minus inflation.

❌ The sign is reversed. The real rate is nominal minus inflation; here the bill earned more than inflation, so the real rate is positive.

❌ That is the nominal rate. The real rate removes inflation from it.

❌ Still a monthly figure. A monthly rate becomes a simple annual rate on multiplying by 12.

❌ That annualizes the nominal rate.

0.34\% \times 12 = 4.08\%

; annualize the real rate instead.

❌ That annualizes inflation.

0.25\% \times 12 = 3.00\%

; the question asks for the real rate.

❌ Check the monthly real rate. Subtract inflation from the T-bill rate.

❌ Check the annualization. Multiply the monthly real rate by 12.

Show solution

The real rate is approximately the nominal rate minus inflation:

r_{\text{real}} \approx 0.34\% - 0.25\% = 0.09\% \text{ per month} = 9 \text{ bp}

Simple annualization multiplies by 12:

0.09\% \times 12 = 1.08\% \text{ per year}

(The exact monthly figure, $1.0034/1.0025 - 1 \approx 0.0898\%$ , rounds to the same nine basis points.)

Problem 2 · The Exact Real Rate

Given: Illustrative figures: a Treasury bill yields $4.0\%$ a year and inflation runs at $3.1\%$ a year. Find the exact real rate per year, $\frac{1 + r_f}{1 + \text{inflation}} - 1$ .

✅ Correct!

1.040/1.031 - 1 \approx 0.873\%

, slightly below the approximation's

0.90\%

: the exact rate is that

0.9

-point gap divided by

1.031

❌ That is the approximation.

4.0\% - 3.1\% = 0.9\%

is the subtraction; the exact rate divides the gross returns

1.040

and

1.031

❌ The ratio is upside down. Divide one plus the nominal rate by one plus inflation:

1.040/1.031

, not

1.031/1.040

❌ Divide gross returns, not the rates.

4.0/3.1 - 1 \approx 29\%

drops the ones; the formula uses

1 + r_f

and

1 + \text{inflation}

❌ Not quite. Form the gross returns

1.040

and

1.031

, divide, and subtract 1.

Show solution

r_{\text{real}} = \frac{1 + 0.040}{1 + 0.031} - 1 = \frac{1.040}{1.031} - 1 \approx 1.00873 - 1 = 0.873\% \text{ per year}

The approximation gives $4.0\% - 3.1\% = 0.90\%$ . The two differ by about $0.03$ points here; with the table's much smaller monthly rates, $0.38\%$ and $0.32\%$ , the gap is only $0.0002$ points ( $0.0598\%$ against $0.06\%$ ).

Problem 3 · Worst Months Against the 99% Band

Given: The 1946–2001 table (monthly, in percent): the ten-year T-note averaged $0.46\%$ with standard deviation $2.63\%$ , and its worst month was $-7.73\%$ . For a normal distribution, $99\%$ of outcomes lie within $\mu \pm 2.58\sigma$ . Find how many standard deviations below its average the T-note's worst month lies, and which of the five investments in the table is the only one whose worst month lies inside its own $99\%$ band.

T-note's worst month, in standard deviations from its average

The only worst month inside its own 99% band

✅ Correct!

(-7.73 - 0.46)/2.63 \approx -3.11

, beyond

-2.58

. Only the T-bill's worst month,

(0.03 - 0.38)/0.24 \approx -1.46

standard deviations, stays inside its band; the VW index's worst month is about

5.56

standard deviations below its average.

❌ Measure from the average, not from zero.

-7.73/2.63 \approx -2.94

leaves out the

0.46\%

average; subtract it first.

❌ The average is subtracted, not added.

(-7.73 + 0.46)/2.63 \approx -2.76

; the distance below the average is

-7.73 - 0.46 = -8.19

points.

❌ That is the gap in percentage points.

-7.73\% - 0.46\% = -8.19

points; divide by

\sigma = 2.63\%

to count standard deviations.

❌ The T-note's worst month is outside. It lies about

3.11

standard deviations below the average, beyond the band's

2.58

❌ Motorola's wide range does not put it inside. Its standard deviation is large,

10.02\%

, but

(-33.49 - 1.66)/10.02 \approx -3.51

, still beyond

-2.58

❌ One of them is inside. The T-bill's worst month,

0.03\%

, is

(0.03 - 0.38)/0.24 \approx -1.46

standard deviations from its average.

❌ Check the standard-deviation count. Subtract the average from the worst month, then divide by

\sigma

❌ Check each row. Compute (worst month

-

average)

/\sigma

and compare it with

-2.58

Show solution

Count standard deviations from the average, $(\text{worst} - \mu)/\sigma$ , for each row:

Investment	Worst month	$(\text{worst} - \mu)/\sigma$	Inside $\mu \pm 2.58\sigma$ ?
T-bill (1 yr)	$0.03\%$	$(0.03 - 0.38)/0.24 \approx -1.46$	yes
T-note (10 yr)	$-7.73\%$	$(-7.73 - 0.46)/2.63 \approx -3.11$	no
VW stock index	$-22.49\%$	$(-22.49 - 1.01)/4.23 \approx -5.56$	no
EW stock index	$-27.09\%$	$(-27.09 - 1.18)/5.30 \approx -5.33$	no
Motorola	$-33.49\%$	$(-33.49 - 1.66)/10.02 \approx -3.51$	no

Only the T-bill's worst month is inside its $99\%$ band. One month outside the band is no surprise by itself: over $56 \times 12 = 672$ months a normal distribution puts about $1\%$ of them, roughly $7$ , outside it. Worst months $5.3$ to $5.6$ standard deviations below the average, as for the two stock indexes, are another matter: they are the fat left tail that makes normality only an approximation.

Problem 4 · A Bill's Holding-Month Return

Given: Illustrative figures: you buy a one-year bill paying 100 at maturity at a quoted yield of $4.0\%$ a year, so its price is $100/1.04$ . One month later the now 11-month bill yields $5.0\%$ a year and is priced at $100/1.05^{11/12}$ . Find your return over that holding month.

✅ Correct! The price fell from about

96.15

to about

95.63

, a return of about

-0.55\%

, even though the bill was bought at a positive quoted yield.

❌ That is the quoted yield turned monthly.

1.04^{1/12} - 1 \approx 0.33\%

is what you earn only if the yield stays at

4.0\%

; here it rose, so the price fell.

❌ The new yield is not your return.

5.0\%/12 \approx 0.42\%

is what a buyer at the new price expects to earn from now on; your return compares the two prices.

❌ The change in yield is not the return. A one-point rise in the yield moves an 11-month bill's price by roughly

\tfrac{11}{12}

of a percent, and you also earned a month of interest; compute the two prices.

❌ Not quite. Compute both prices, then

(P_{\text{end}} - P_{\text{start}})/P_{\text{start}}

Show solution

P_{\text{start}} = \frac{100}{1.04} \approx 96.154, \qquad P_{\text{end}} = \frac{100}{1.05^{11/12}} \approx 95.626

R = \frac{95.626 - 96.154}{96.154} \approx -0.55\% \text{ over the month}

Had the yield stayed at $4.0\%$ , the bill would have been worth $100/1.04^{11/12} \approx 96.469$ , a return of $1.04^{1/12} - 1 \approx 0.33\%$ . The quoted yield is fixed at purchase; the holding-month return depends on where yields move, which is why the T-bill row of the table has a standard deviation above zero.

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Real Rates and the Risk-Return Tradeoff, 1946–2001

IKey Formulas

IIThe Real Rate from Two Table Entries

IIIRisk and Reward Down the Table

IVWhy a T-Bill’s σ\sigmaσ Is Above Zero

VQuiz Questions

IVWhy a T-Bill’s $\sigma$ Is Above Zero