Model 2 — Borrow Rate / Advance Rate Trade-off
Research Question
How do we optimize Return on Equity (ROE) given the Option 2 negotiation variables Advance Rate (a) and Charged Borrow Rate (r), assuming a constant Net Productive Asset Return (y)?
Bank offer assumption: a — Advance Rate (a) and a — Charged Borrow Rate (r).
Parameters & Controls
Set Option 2 with the first two negotiation variables. Option 1 is the fixed bank quote below.
Derived Model Parameters
1. Advance Rate & Leverage
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1. Advance Rate & Leverage
When you increase the Advance Rate (a), the lender funds a larger portion of each asset. Your Return on Equity (ROE) climbs because leverage multiplies your net asset earnings.
Why advance rates are capped below 100%: At a 100% advance rate, the borrower puts up zero equity capital. Without borrower equity as a first-loss cushion, the lender absorbs 100% of the financial risk if asset returns drop or defaults spike. Lenders enforce an advance rate cap so the borrower maintains skin in the game to protect lender capital.
2. ROE & Borrow Rate
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2. Borrow Rate & The Value Creation Limit (r = y)
This chart varies the Charged Borrow Rate (r) while holding advance rate constant. Return on equity declines as debt funding costs increase. Here, r is the rate charged by lenders, and y is the net annual return of the productive asset.
The limit line r = y (—) marks the economic usefulness cap of borrowing. At r = y, debt cost matches asset earnings, so leverage creates zero excess gain (ROE = y). Above r = y borrowing destroys equity value at every advance rate, so the chart and the slider both stop there.
O1 is your bank, at its quoted rate of 18% and its 40% advance rate. Everything on this chart is in one currency, so no conversion and no hedging cost enters here.
3. Return Surface
Charts 1 and 2 combined. Drag to rotate.
3. The Two Variables Together
Chart 1 moves the Advance Rate (a). Chart 2 moves the Charged Borrow Rate (r). This surface is both at once, with Return on Equity (ROE) as the height.
The surface has no summit. It is a fan that pivots about the line r = y, drawn as a laterite dash. Along that line the return on equity equals y at every advance rate. Below the line, more advance rate always raises the return. Above the line, more advance rate always lowers it. There is no interior best point to find, only the cap.
Two points sit on the surface:
• O1 (Bank): —
• O2 (The offer): —
Comparing Two Offers
—How to compare option 2 with option 1
Option 1 gives you a small amount of cheap debt. Option 2 gives you more debt at a higher price. They cannot be compared by looking at either number alone. A lower charged borrow rate is not better if it comes with a smaller advance rate, and a higher advance rate is not better if the rate that comes with it is too high. There are three ways to settle it, and all three always give the same answer.
Test A — compare the two returns on equity. Put each offer through ROE = (y − a × r) / (1 − a) and take the higher one. It is the direct answer, but it tells you nothing about how much room you had to spare.
Test B — the break-even borrow rate. This is the one to negotiate with. Ask a different question: at option 2's advance rate, what is the highest charged borrow rate I could pay and still end up exactly level with the bank? That is the break-even borrow rate on the card, and it comes from r = (y − ROEbank × (1 − a)) / a. Everything the lender asks below that number is a gain; everything above it is a loss. The gap between the break-even borrow rate and what they actually charge is your headroom, and it is how much the rate could rise before the offer stops being worth taking.
Test C — the price per point of advance rate. Option 2 buys you extra advance rate, and the extra charged borrow rate is the price. Divide one by the other: (r₂ − r₁) / (a₂ − a₁). That is the price per point on the card. Beside it sits the most the lender could have charged per point and still left you level with the bank. Accept while the price they charge is below it.
Worked example. Your bank offers 40% at 18%, which returns 79.7% on equity. A second lender offers 70% at 30%. That returns 113.3%, so option 2 wins by 33.7 points (test A). At a 70% advance rate you could have paid up to 44.4% and still matched the bank, so you have 14.4 points of headroom (test B). They are charging 0.40 points of rate per point of advance rate, against a break-even price of 0.88, so they are selling advance rate at less than half what it is worth to you (test C). Take it, and note that you had room to accept far worse.
Do not use the MRS for this. The exchange rate above it is a derivative. It measures the value of one extra percentage point at the point you are standing on. Moving from 40% to 70% is a jump of thirty points, and the exchange rate changes all the way along it. For a jump between two term sheets, use the break-even rate. Use the MRS to judge a small concession inside a negotiation.
Why This Model Has No Peak
You would expect a model like this to hand you one best combination of advance rate and charged borrow rate. It does not, and there is a reason. Understanding it changes what you argue about in the room, so this box works through why no peak exists and what to bargain over instead.
Three quantities follow from that one formula. Each answers a different question.
∂ROE/∂a = (y − r) / (1 − a)²
The advance-rate sensitivity. How many percentage points of return on equity you gain from one extra percentage point of advance rate, holding the charged borrow rate still.
∂ROE/∂r = −a / (1 − a)
The borrow-rate sensitivity. How many percentage points of return on equity you lose when the charged borrow rate rises by one percentage point, holding the advance rate still.
MRS = (y − r) / (a · (1 − a))
The exchange rate between the two, and the ratio of the first to the second. How many extra percentage points of charged borrow rate you could accept in return for one extra percentage point of advance rate, and be no better and no worse off.
1. The return-on-equity formula contains no best point. Take the advance-rate sensitivity above. It is the partial derivative of return on equity with respect to the advance rate a, holding the charged borrow rate r still. Its denominator (1 − a)² is a square, so it is always positive. Its numerator (y − r) is positive whenever the net return your productive assets earn, y, is greater than the charged borrow rate you pay, r. So for as long as your productive assets earn more than your debt costs, every increase in the advance rate raises your return on equity. The curve never turns back down, so it never has a peak. The formula says one thing only: take every increase in the advance rate you can get. A page that shows "more capital against cheaper capital" as a balance inside this formula is showing a trade-off that is not in it.
2. The trade-off is in the lender's price list, not in your formula. A lender who agrees to fund a larger share of each productive asset will charge a higher rate for doing so. That relationship is the lender's price ladder: its list of paired offers, where each advance rate has a charged borrow rate. A best point somewhere in the middle of the range can only appear if that ladder rises steeply enough, and the test is the threshold slope shown on the left. The ladder must add more than that many percentage points of charged borrow rate for each extra percentage point of advance rate. Below the threshold, the best point is still the highest advance rate on offer. This model draws a flat ladder, and it has to. The only quote you hold is your bank's, at a 40% advance rate, where the receivables behind the loan still cover it more than twice over. At that coverage the bank is pricing almost no credit risk, which is why its floor is two points. One quote fixes where the ladder starts and says nothing about how steeply it rises, so drawing any slope through it would be inventing the number that decides the answer. Ask a lender to quote you a second advance rate, and you will have a real slope to test against the threshold.
3. What you actually negotiate is the exchange rate between the two. Divide the advance-rate sensitivity by the borrow-rate sensitivity and you get the MRS, the marginal rate of substitution. At your current point it is 1.54. Read it as a price: one extra percentage point of advance rate is worth about 1.54 extra percentage points on the charged borrow rate, and you would be no better and no worse off after the swap. That is the number to carry into the room. Accept any lender who sells you an extra percentage point of advance rate for less than the MRS. Stop when the price they charge per point equals it. Since the ladder here is flat, that moment never arrives, and the answer is the advance-rate cap.
Bringing In The Hedging Cost
The comparison above assumes both offers are quoted in the same currency. A foreign lender quotes dollars. That adds one step, and it is kept out of the comparison on purpose, because it is a separate question and it is where the most expensive mistakes happen.
1. Convert the foreign offer into shillings before you test it. Borrowing dollars at rUSD and buying a hedge costs you roughly rUSD + h per year in shillings. Run the three tests on that number. Never run them on the dollar headline.
2. A dollar rate that looks half the bank's is often exactly the same. Covered interest parity says a hedge should cost about the gap between the two risk-free rates, which is 11 points today. So a dollar lender at 7% costs 18% in shillings, identical to your bank. The apparent saving was never a saving. It was the currency.
3. Only the excess above parity is a genuine extra cost. That excess is the hedging wedge on the left. At a 12% hedge against 11 points of parity, borrowing abroad costs you one extra point, not twelve. Set the hedging cost slider below parity and the wedge reads zero, which is why the lender's rate does not fall down there.
4. Declining to hedge does not avoid the cost. Unhedged, the shilling is expected to weaken against the dollar by roughly the same 11 points. You would swap a known fee for variance, not for a saving. The one real argument for going unhedged is that you think the market's expectation is wrong.
Worked example. A dollar lender offers 70% advance at 20% USD. Hedged at parity that is 31% in shillings. Your break-even rate at a 70% advance rate is still 44.4%, so you have 13.4 points of headroom and the offer beats your bank. Now compare it with the 30% shilling offer, which gave 14.4 points. The dollar lender looked far cheaper on its headline and is in fact slightly worse.
What this box does not model. It assumes the hedge is available in the size and tenor you need, for the whole life of the facility, at one fixed price. In frontier currencies that is the weakest assumption on this page. A hedge available for one year against a three-year facility leaves you exposed for two, and no number here shows that.