Lesson 17 of 25 · 7 min

Why a higher discount rate compresses a long-duration asset more

The same rate rise removes four percent from a cash flow two years out and thirty-six percent from one twenty years out.

Two assets can face an identical rate move and lose wildly different amounts. The reason is not risk, quality or liquidity. It is the exponent.

A cash flow arriving in a given number of years is worth one divided by the discount factor raised to that many powers. Raising the rate divides by a slightly larger number once per year of waiting. Over two years that happens twice. Over twenty it happens twenty times, and the small annual penalty accumulates into a large one.

present value of $1 = 1 ÷ (1 + rate)^years

present value of $1: What a dollar arriving in the future is worth today.
rate: The discount rate applied, as a decimal.
years: How long until the dollar arrives. The exponent, and the source of the whole effect.

At 6.0 percent, a dollar in two years is worth $0.889996 and a dollar in twenty years is worth $0.311805. Raise the rate to 8.4 percent and those become $0.851023 and $0.199257. The near cash flow lost 4.379 percent; the distant one lost 36.096 percent. The ratio of the two losses is 8.24, from one rate change applied to both.

This is what long duration means for an equity. A business whose value is mostly cash arriving in the next few years behaves like the two-year column. A business whose value is almost entirely a terminal figure many years out behaves like the twenty-year column. The share price does not label which it is.

So two pledged portfolios of identical market value, at the same lender, under the same advance rate schedule, can lose very different lending value in the same tightening. The schedule sorts collateral by asset class and liquidity, as Track 8 lesson 2 described. It does not sort by where in time the cash flows sit, and that is the axis this lesson is about.

Key terms

Present value factor: What a future dollar is worth today at a given rate. Falls with the rate and falls much faster the further out the dollar is.
Long-duration asset: One whose value comes mostly from cash flows far in the future. Disproportionately sensitive to the discount rate for that reason alone.
Terminal value: The portion of an asset's worth attributed to everything beyond the explicitly forecast years. Often the majority, and always the most rate-sensitive part.

Worked example

The exponent does all of the work

One dollar, two arrival dates, one rate change from 6.0 to 8.4 percent.

Dollar in two years, at 6.0%: $0.889996
Dollar in two years, at 8.4%: $0.851023
Dollar in twenty years, at 6.0%: $0.311805
Dollar in twenty years, at 8.4%: $0.199257
Ratio of the two falls: 8.24
Difference in the rate applied: None

One rate change, two assets, eight times the damage on one of them. Nothing about credit, liquidity or quality entered the calculation.

Check yourself

A dollar arriving in ten years is discounted at 5.5 percent, and the rate then rises to 7.9 percent. By what percentage does its present value fall?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.