The easy arithmetic and the impossible choice
In the first guide of this rung you saw why insurers and pension funds invest at all: they are promise machines, taking money in today and paying it out years later, and the assumed investment earnings are baked into the price of every product. Now we confront the question that turns that whole arrangement into a number on a balance sheet. An insurer owes 1,000 in ten years. What is that promise worth *today*? You already know the mechanics from interest theory — you discount it, dividing by growth factors that strip out the interest the money would earn in the meantime. The trouble is not the dividing. The trouble is deciding what to divide by.
Watch how much the rate alone moves the answer. That 1,000 due in ten years is worth about 744 if you discount at 3%, but only about 558 at 6%. Same promise, same ten years, same arithmetic — yet the second figure is roughly a quarter smaller. Nothing about the obligation changed; you simply assumed your money works harder before the bill comes due. This is the heart of [[discounting-liabilities|discounting a liability]]: the cash flows are usually given to you by the actuarial model, but the discount rate is a *judgement*, and it is the single most powerful lever in the entire valuation.
Promise: pay 1,000 in 10 years. What is it worth today? at 3%: 1000 / 1.03^10 = 1000 / 1.344 = 744 at 6%: 1000 / 1.06^10 = 1000 / 1.791 = 558 Same promise. The rate alone moved the value by ~25%.
Three philosophies of the rate
So which rate is right? There is no single answer, because the rate is really answering a *question*, and different stakeholders ask different questions. Three broad philosophies compete. The first is the risk-free rate: discount using yields you could lock in safely today, the rates on government bonds. The logic is sober — a promise is a debt, and the cleanest measure of a debt is what it would cost to fund it with assets that carry essentially no risk of not paying. This gives the lowest rate, and therefore the largest, most conservative liability value.
The second is the expected-asset-return rate: discount using the rate the assets actually backing the promise are expected to earn. If a pension fund holds equities and corporate bonds expected to return 7%, why not value the liabilities at 7%? This is the traditional approach, and its appeal is intuitive — but watch the sleight of hand. It quietly assumes the risky assets *will* deliver their expected return, and bakes that hope straight into the value of a promise that is itself certain. A higher rate makes the liability look smaller, which means less money has to be set aside today. The institution is, in effect, valuing a guaranteed obligation as if it were as risky as its stock portfolio.
The third, and the direction modern standards have moved toward, is the market-consistent rate: value the liability at whatever the market would charge to take it off your hands. In practice this usually means discounting along the risk-free curve, perhaps with a small, defensible adjustment for the illiquidity of the promise. The key idea is that the value of an obligation should not depend on which assets you happen to hold against it. A fixed promise to pay 1,000 in ten years is worth the same whether you back it with cash or with a casino chip — and a sound valuation says so. This connects directly to asset-liability management, where assets and liabilities must be measured on the same footing before you can manage them together.
How a pension fools itself
The stakes become vivid in pensions, where the discount rate is the subject of decades of argument. Consider a plan that owes its members 100 million dollars' worth of pensions, spread over the coming forty years. Discount those promises at an optimistic 7% — the expected return on its mixed portfolio — and the present value of the liability might come out around 30 million. Discount the very same promises at a market-consistent 4% and the liability balloons to perhaps 45 million. At 7% the plan can declare itself comfortably funded; at 4% the same plan is staring at a large deficit. Neither number is arithmetically wrong. They are answers to different questions.
But the questions are not equally honest. Choosing a high rate *because* it makes the liability disappear is one of the classic ways an institution fools itself — and pushes the bill to a future generation. This is exactly the trap built into the investment return assumption you met last time: the temptation to assume the markets will be generous, precisely because that assumption makes today's promises look cheap. Underfunded public pensions around the world are, in large part, a monument to optimistic discount rates that the markets then declined to honour. The promise was real and fixed; only the assumed return was hopeful.
The term structure, revisited
There is a deeper problem hiding under the phrase "the discount rate" — the word *the*. So far we have spoken as if one number discounts every cash flow, but you already know from interest theory that this is a convenient fiction. Lending money for one year and for thirty years earn different rates; the relationship between term and rate is the term structure of interest rates, and plotted out it is the yield curve. Most days it slopes gently upward — longer money pays a little more — but it can flatten or even invert. A single flat rate blurs all of that together, and when the curve is steep, it quietly misstates the value of a liability spread across many dates.
The market-consistent answer is to stop using one rate and instead discount each cash flow at the rate appropriate to *its own* maturity — using the spot rate for each date. Next year's claim is discounted at the 1-year spot rate; the claim due in thirty years is discounted at the 30-year spot rate. Add the pieces up and you have a value that tracks the actual market that will one day fund the promise, not a value smeared by a single chosen number. This is no longer optional in much of the world: modern standards build the discount curve straight from observed market rates, precisely so the value of the obligation moves with the market rather than with an actuary's mood.
And notice what this means for the assets standing behind the promise. Once the liability is valued along the curve, it becomes obvious which maturities of bonds best back which parts of it — short bonds for near claims, long bonds for distant ones. The shape of the curve, not just its level, now matters: if long rates rise while short rates do not, a long liability becomes cheaper to fund while a portfolio of short bonds barely moves. That mismatch is the seed of interest-rate risk, and discounting along the curve is what makes it visible in the first place. The link to bonds and equities is no accident: a bond is a liability turned around, money you receive on known dates, which is exactly why bonds are the natural matching asset.
Living with the most consequential number
If the choice of rate is unavoidably a judgement, how does an honest actuary discharge it? Not by finding the one true rate — there isn't one — but by being transparent and disciplined about the one chosen. The professional habits matter more than the number itself.
- State the rate and its basis openly — is it risk-free, market-consistent, or an expected asset return? Each answers a different question, and the reader deserves to know which question you answered.
- Justify it against reality — ground an expected-return rate in the actual asset mix and sober long-term market evidence, not in the number that makes today's funding look comfortable.
- Show the sensitivity — report how much the liability moves if the rate is a point higher or lower, so no one mistakes a single fragile figure for a fact about the future.
- Revisit it regularly — the curve moves, the asset mix drifts, and a rate that was defensible five years ago may no longer be. The assumption is a living thing, not a monument.
Step back and the lesson is bigger than any formula. The discounting arithmetic is exact; the rate you feed it is the most consequential judgement in the room, and choosing it well is as much an act of integrity as of mathematics. With the value of the liability now pinned down — and pinned down along the whole curve — the next guide asks the natural follow-on question: how much does that value *swing* when rates move, and how can the assets be arranged so the swings cancel? That is the world of duration, convexity, and the duration gap, the subject we turn to next.