Money has a time value
By the end of this lecture you will be able to
- Move any cash flow forward or back in time, and convert between APR, effective and continuously compounded rates
- Value annuities, annuities due, perpetuities and growing streams in one line, including streams that start later
- Use NPV and IRR to accept or reject an investment, and explain where IRR, MIRR and payback mislead
- Handle irregular dates with XNPV and XIRR, and keep real and nominal figures consistent
- Estimate returns in your head: the rule of 72, doubling and tripling times, and the multiple-to-IRR grid
Every valuation you will ever build rests on one idea: a dollar today is worth more than a dollar tomorrow. This lecture makes that idea precise. By the end you will be able to move any cash flow through time, value a whole stream of payments in one line, judge an investment with NPV and IRR, and do the return arithmetic that interviewers expect you to do in your head.
I start every first-year class with the same question. Would you rather have $100 today or $100 in a year? Everyone picks today. The interesting part is why, because the reasons tell us how to price the difference.
1. Why a dollar today beats a dollar tomorrow
There are three reasons, and each one maps onto something we will use later.
- Opportunity cost. If you hold the dollar today, you can invest it and earn a return. Waiting means giving that return up. This is the most important reason, and it is why we call the discount rate an opportunity cost of capital.
- Inflation. Prices tend to rise, so a dollar next year buys less than a dollar today.
- Risk. A promise of money in the future may not be kept. The further away and the less certain the payment, the less it is worth to you now.
The interest rate is simply the price of time. If the market rate is 5%, then $100 today and $105 in a year are, for an investor who can lend at 5%, the same thing. Finance is mostly the art of making cash flows at different dates comparable.
One convention before we start. Today is time 0, and we count periods forward, usually years. Unless we say otherwise, a cash flow "in year 3" arrives at the end of year 3, three full periods from now. Draw a timeline for every problem, with each cash flow under its date and a sign: money you pay out is negative, money you receive is positive. Most errors in this lecture are timing errors, and a timeline catches them.
2. Compounding: moving money forward
Put $1,000 in an account paying 8% a year. After one year you have $1,080. In the second year you earn 8% on $1,080, not on $1,000, so interest itself starts earning interest. That is compounding. Each year multiplies the balance by \(1+r\), so after \(n\) years:
Worked example
$1,000 at 8% for 10 years grows to \(1{,}000 \times 1.08^{10} = \$2{,}158.92\). Simple interest, 8% of the original $1,000 each year, would have given you only $1,800. The extra $358.92 is interest on interest. Leave it for thirty years and it becomes $10,063: the rate never changes, but the base it applies to keeps growing.
A useful shortcut is the rule of 72: money doubles in roughly \(72 / r\) years, with \(r\) in percent. The exact doubling time solves \((1+r)^n = 2\):
That would suggest a rule of 69. We use 72 for two reasons: \(\ln(1+r)\) is a little smaller than \(r\), which makes the true doubling time a little longer, and 72 divides neatly by 2, 3, 4, 6, 8, 9 and 12. The rule is almost exact around 8% and drifts as you move away:
| Rate | Rule of 72, years | Exact, years |
|---|---|---|
| 4% | 18.0 | 17.7 |
| 8% | 9.0 | 9.0 |
| 12% | 6.0 | 6.1 |
| 20% | 3.6 | 3.8 |
The same logic with \(\ln 3 = 1.10\) gives a rule of about 114 for tripling: at 8%, money triples in about 14 years.
3. Compounding frequency: APR, EAR and continuous compounding
Rates are usually quoted per year but often paid more frequently. A loan "at 12% a year, compounded monthly" charges 1% a month. The quoted 12% is a nominal rate, the annual percentage rate (APR): the periodic rate times the number of periods. What you actually pay over a year is the effective annual rate (EAR), because each month's interest is itself charged interest in later months:
| Compounding of a 12% APR | Periodic rate | EAR |
|---|---|---|
| Annual | 12% | 12.00% |
| Semi-annual | 6% | 12.36% |
| Quarterly | 3% | 12.55% |
| Monthly | 1% | 12.68% |
| Daily | 0.0329% | 12.75% |
| Continuous | 12.75% |
More frequent compounding raises the EAR, but with sharply diminishing effect. As \(m\) goes to infinity, \((1+r/m)^m\) tends to \(e^r\), and we get continuous compounding:
At 5% for 10 years, $1,000 grows to $1,628.89 with annual compounding and to \(1{,}000 \times e^{0.5} = \$1{,}648.72\) with continuous compounding; an 8% effective rate is a 7.70% continuous rate. Option pricing uses continuous rates because they add across periods: two years at 5% is simply 10%. Banking models rarely need them, but you should be able to convert.
Where students go wrong
Comparing rates quoted on different bases. A US consumer-loan APR is a nominal rate, while the EU and UK APRC already is an effective rate, fees included. US bond yields are semi-annual: a 6% yield means 3% every six months, an effective 6.09%. In a model, match the rate to the period of the cash flows: if your annual rate is effective, as a WACC is, the monthly rate is \((1+r)^{1/12}-1\), slightly less than \(r/12\).
4. Discounting: moving money back
Discounting is compounding run backwards. If a payment \(FV\) arrives in \(n\) years and your opportunity cost is \(r\), its value today is:
Worked example
A client will receive $10,000 in five years. With a 7% discount rate, it is worth \(10{,}000 / 1.07^5 = \$7{,}129.86\) today. Put $7,129.86 in an investment earning 7% and you would have exactly $10,000 in five years. That is what "worth" means here.
The term \(1/(1+r)^n\) is the discount factor: the value today of $1 received in year \(n\). At 7%, $1 in five years is worth 71.3 cents. A spreadsheet model puts a row of discount factors under the cash flows and multiplies; mechanically, that is all a DCF does.
Notice the two levers. A higher discount rate lowers the present value. A later payment lowers it too. Every valuation debate you will sit through is, underneath, an argument about one of those two things: the size and timing of cash flows, or the rate used to discount them. Distance makes the rate matter more. At 8%, $1,000 due in 30 years is worth $99 today; at 10%, $57. Two points on the rate cut the value of that distant payment by 42%.
Pause and think: interest rates rise by one point overnight. What happens to the value today of a fixed payment due in ten years?
It falls, and by more than a payment due in one year would. The longer the wait, the more times the higher rate compounds against you. This is why long-dated assets, like growth stocks and long bonds, fall hardest when rates rise. The next lecture puts a number on this sensitivity: duration.
5. Streams of cash flows
Most assets pay more than once. The value of a stream is the sum of the present values of each payment, because present values are all in today's dollars and can be added:
A handful of patterns come up so often that they have their own formulas. Each is a shortcut for that sum.
A perpetuity pays \(C\) at the end of every year, forever. Its value is remarkably simple:
The intuition: put \(C/r\) in the bank at rate \(r\) and it pays interest of exactly \(C\) every year without touching the principal. A $1 million endowment invested at 5% pays $50,000 a year forever. A mature business that neither grows nor shrinks is a decent approximation.
An annuity pays \(C\) every year for \(N\) years: a mortgage, a lease, a bond's coupons. The cleanest way to value it is as the difference between two perpetuities. A perpetuity starting next year is worth \(C/r\). Take away a perpetuity whose first payment is in year \(N+1\); it is worth \(C/r\) at year \(N\), so \(C/r \times 1/(1+r)^N\) today. What is left is exactly \(N\) payments:
Worked example
$500 a year for 10 years at 6% is worth \(500 \times 7.3601 = \$3{,}680.04\). You receive $5,000 in total, but only $3,680 of value, because most of it arrives late. Valued at the end of year 10 instead, the stream is worth \(500 \times (1.06^{10}-1)/0.06 = \$6{,}590.40\), which is simply \(3{,}680.04 \times 1.06^{10}\).
An annuity due pays at the start of each period instead of the end: rent, many leases and insurance premiums are paid in advance. Every payment arrives one period earlier, so the whole stream is worth \(1+r\) times more:
Worked example
A five-year lease costs $24,000 a year and the discount rate is 6%. Paid at the end of each year, the payments are worth \(24{,}000 \times 4.2124 = \$101{,}096.73\). Paid in advance, they are worth \(101{,}096.73 \times 1.06 = \$107{,}162.53\). Same rent, $6,066 more value to the landlord, because each payment arrives a year sooner.
Loans. A loan repaid in equal instalments is an annuity seen from the lender's side: solve the annuity formula for \(C\) and you have the payment. Each payment first covers interest on the balance outstanding; the rest repays principal. Early on, interest takes most of the payment. Later, principal does.
Worked example
$10,000 borrowed over three years at 8%: the payment is \(10{,}000 \times 0.08 / (1-1.08^{-3}) = \$3{,}880.34\) a year.
| Year | Opening balance | Interest at 8% | Principal repaid | Closing balance |
|---|---|---|---|---|
| 1 | 10,000.00 | 800.00 | 3,080.34 | 6,919.66 |
| 2 | 6,919.66 | 553.57 | 3,326.76 | 3,592.90 |
| 3 | 3,592.90 | 287.43 | 3,592.90 | 0.00 |
The same mechanics sit inside the debt schedule of every LBO model in lecture 12, although most leveraged loans amortise far more slowly than this one.
A growing perpetuity pays \(C_1\) next year and grows at \(g\) every year after. This is the formula behind the terminal value of almost every DCF you will build, so learn it cold:
Read \(r-g\) as a net discount rate: each year the cash flow is pushed down by \(r\) and pulled up by \(g\). Rearranged, the formula gives the return implied by a price, \(r = C_1/P + g\), the cash yield plus growth, which analysts use to back out the cost of equity the market is pricing in.
Worked example
A company will pay a $2 dividend next year, growing 3% a year forever. Investors want 8%. The share is worth \(2 / (0.08-0.03) = \$40\). Now push growth to 4%: \(2 / 0.04 = \$50\). One point of growth added 25% to the value. Keep that sensitivity in mind when we reach terminal values.
A growing annuity pays \(C_1\) next year, growing at \(g\), for \(N\) years. It is a growing perpetuity minus a growing perpetuity that starts after year \(N\):
Because the stream is finite, this formula still works when \(g > r\). When \(g = r\) it divides by zero, but the answer is easy: every payment has the same present value, \(C_1/(1+r)\), so the total is \(N \times C_1/(1+r)\).
Worked example
A mine has ten years of reserves. Next year's royalty is $10m, rising 3% a year with prices, and the discount rate is 8%. The royalty is worth \(10/0.05 \times [1 - (1.03/1.08)^{10}]\), which is \(200 \times 0.3775 = \$75.50\text{m}\). A level $10m for ten years would be worth $67.10m, and a growing perpetuity $200m. The finite life removes more than 60% of the perpetuity's value, which is why mines are valued over their reserve life rather than with a terminal value.
Streams that start later. Value them in two steps. The formula gives a value one period before the first payment; discount that value back to today.
Worked example
$100 a year forever, starting in year 4, at 10%. The perpetuity formula gives \(100/0.10 = \$1{,}000\) at year 3, one period before the first payment. Today that is \(1{,}000/1.1^3 = \$751.31\). Discounting by four years instead, a classic slip, gives $683.01. This is exactly the terminal value step in a DCF: the growing perpetuity formula applied to year 6's cash flow gives a value at year 5, which you discount by five years, not six.
Where students go wrong
Two mistakes, every year. First, the growing perpetuity uses next year's cash flow, \(C_1\), not this year's. If you have this year's \(C_0\), multiply by \(1+g\) first. Second, the formula breaks when \(g\) approaches \(r\): the value explodes towards infinity. A growth rate above the discount rate forever is not a valuation, it is a typo. In a DCF, long-run growth is usually set near long-run inflation or nominal GDP growth, because no company can outgrow the economy forever.
6. NPV and IRR: making decisions
We can now judge an investment. The net present value is the present value of everything it pays, minus what it costs today:
The rule is simple. A positive NPV means the investment earns more than your opportunity cost, so it creates value. A negative NPV destroys value. When a company makes an acquisition, the board is asking exactly this question, with a lot more noise around it.
The internal rate of return is the discount rate that makes NPV equal to zero. It is the return the investment earns on its own terms. You accept if the IRR beats your cost of capital.
Worked example
A project costs $1,000 and pays $400 a year for three years. At a 10% cost of capital, the inflows are worth $363.64 + $330.58 + $300.53 = $994.74, so NPV is −$5.26. Reject. The IRR is 9.70%, just below 10%, which tells the same story from another angle.
Plot NPV against the discount rate and you get the project's NPV profile. For a conventional project, with money out first and money in later, the curve slopes down. At 0% the NPV is just the sum of the cash flows, $200 here, and it falls as the rate rises, crossing zero at the IRR. For a single conventional project, NPV and IRR therefore give the same decision. They can disagree in four situations, and an interviewer who asks "why might NPV and IRR disagree?" wants you to name them.
Scale. IRR is a percentage; NPV is money. Project A turns $10 into $15 in a year: an IRR of 50% and, at 10%, an NPV of $3.64. Project B turns $1 million into $1.2 million: an IRR of 20% and an NPV of $90,909. If you can only do one, you want B. Shareholders spend dollars, not percentages.
Timing. Two projects of the same size can rank differently depending on the rate. Both of the projects below cost $1,000. A pays $800, $300 and $100 over three years. B pays nothing until a $1,400 lump in year 3. A has the higher IRR, 14.0% against 11.9%. But at an 8% cost of capital B is worth more: an NPV of $111 against $77. The profiles cross at about 10.1%, the crossover rate. Below it, B's larger but later cash wins; above it, A's earlier cash wins. For mutually exclusive projects, pick the higher NPV at your actual cost of capital.
Multiple IRRs. The IRR is the root of a polynomial, and a polynomial can have several roots. That happens when cash flows change sign more than once, the typical case being a mine with large closure and rehabilitation costs at the end. Take −$1,000 today, +$2,500 in year 1 and −$1,540 in year 2. NPV is zero at both 10% and 40%: two IRRs, and neither means anything on its own. NPV is negative below 10%, positive between 10% and 40%, and negative again above. At a 5% cost of capital the project destroys value, with an NPV of −$15.87, because the closure bill is barely discounted. Descartes' rule of signs gives the maximum number of IRRs: one per change of sign in the cash flows. Some streams have no IRR at all.
The reinvestment assumption. An IRR is the rate that sets NPV to zero, nothing more. Reading it as "the annual return I earn on my money" carries a hidden condition: that each interim cash flow is reinvested at the IRR itself until the end. NPV implicitly values interim cash at the cost of capital, a more realistic opportunity rate. That is why IRR flatters high-return projects with early cash flows.
The modified IRR (MIRR) fixes this. Compound every inflow to the final year at a reinvestment rate, usually the cost of capital; discount every outflow to today at a finance rate; then find the single rate that links the two:
Worked example
A project costs $1,000 and pays $500 a year for three years. Its IRR is 23.4%. Reinvested at 10%, the inflows grow to \(500 \times 1.1^2 + 500 \times 1.1 + 500 = \$1{,}655\) by year 3, so \(MIRR = (1{,}655/1{,}000)^{1/3} - 1 = 18.3\%\). Still a good project, but the headline drops by five points once reinvestment is realistic. MIRR also has a single value, so it sidesteps the multiple-IRR problem. In Excel it is =MIRR(values, finance rate, reinvestment rate).
Private equity lives on IRR, so these issues return in lecture 12. A dividend recapitalisation, which returns cash to the fund early, raises the IRR far more than the multiple of money: the timing effect at work. That is why funds report both numbers.
Pause and think: a project has an IRR of 18% and your cost of capital is 12%. Can you conclude that its NPV is positive?
For a conventional project, with one outflow followed by inflows, yes: the NPV profile slopes down and crosses zero only at 18%, so it is positive at 12%. With cash flows that change sign more than once, no. There may be several IRRs, and you have to compute the NPV at 12% directly.
7. Payback period
The payback period is the time it takes for cumulative cash inflows to repay the initial investment. Mining boards and lenders still ask for it because it speaks to risk: the sooner you have your money back, the less time there is for something to go wrong.
Worked example
A mine costs $500m and generates $150m a year for eight years. Simple payback is \(500/150 = 3.3\) years. The discounted payback uses present values instead: at 8%, cumulative discounted cash is still −$3.2m after year 4 and turns positive early in year 5, a payback of about 4.0 years. The NPV is $362m and the IRR 25.0%.
Payback has two serious flaws. It ignores the time value of money, which the discounted version fixes, and it ignores everything after the cutoff, which nothing fixes. Project C costs $100 and returns $60 in each of the next two years: payback 1.7 years, NPV at 10% only $4.13. Project D costs $100 and returns $20, $30, $50 and then $200: payback 3 years, NPV $117. A payback rule picks C. Use payback as a risk indicator next to NPV, never instead of it.
8. Irregular dates: XNPV and XIRR
Real cash flows do not arrive on neat anniversaries. Acquisitions close mid-quarter and private equity funds call capital whenever they find a deal. When dates are irregular, discount each cash flow by its exact distance from the start, in years:
This is what Excel's XNPV and XIRR do: actual days divided by 365, with everything discounted to the first date in the list.
Worked example
You invest €1,000 on 1 March 2026 and receive €300 on 15 August 2026, €400 on 31 December 2026 and €500 on 30 September 2027. Those dates are 167, 305 and 578 days after the start: 0.458, 0.836 and 1.584 years. At 10% the XNPV is €86.53 and the XIRR is 19.3%. Pretending the receipts arrive at the ends of years 1, 2 and 3 gives an NPV of −€21.04 and the wrong decision.
A related convention appears in every DCF: the mid-year convention. A company earns cash throughout the year, not on 31 December, so analysts discount year \(t\)'s cash flow by \(t-0.5\) years. At 9%, that raises the value of every cash flow by \(1.09^{0.5}-1 = 4.4\%\). We use it in the DCF lectures.
Where students go wrong
Excel's NPV function assumes the first cash flow in its range arrives one period from now. Typing =NPV(10%, −1000, 400, 400, 400) discounts the initial outlay by a year and returns −$4.78 instead of −$5.26. The correct formula is =−1000 + NPV(10%, 400, 400, 400). XNPV has no such quirk, because it discounts to the first date you give it.
9. Real and nominal
A nominal rate includes inflation. A real rate strips it out. They are linked by the Fisher relation:
For small numbers, the real rate is roughly the nominal rate minus inflation. At 7% nominal and 2.5% inflation the shortcut gives 4.5%, and the exact answer is \(1.07/1.025 - 1 = 4.39\%\). The gap widens as inflation rises, which matters in high-inflation countries.
The rule that matters in practice is consistency. Discount nominal cash flows, which include price increases, at a nominal rate. Discount real cash flows, in today's money, at a real rate. Done consistently, the two give the same answer.
Worked example
A project earns $100 a year for three years in today's money. Inflation is 3% and the real discount rate 5%, so the consistent nominal rate is \(1.05 \times 1.03 - 1 = 8.15\%\). In nominal terms the cash flows are $103.00, $106.09 and $109.27. Real flows at the real rate are worth $272.32; nominal flows at the nominal rate, also $272.32. Mix them and you go wrong: real flows at the nominal rate give $257.01, too low, because inflation is removed twice; nominal flows at the real rate give $288.72, too high.
Consistency is not always enough, because some items are fixed in money terms: tax depreciation is based on historical cost, and debt service and many contracts are set in money. In today's money these items shrink: $100 of tax depreciation in year 5 is worth only $86.26 today at 3% inflation, so higher inflation raises the real tax bill. When such items are material, model in nominal terms. Mining models are often run in real terms with these items adjusted, and the mining course on this site works that way.
10. Mental maths for interviews
Interviewers, especially in private equity and leveraged finance, expect you to estimate returns without a calculator. A few anchors cover most questions.
Multiples of money and IRR. Doubling your money in five years is an IRR of about 15% (exactly 14.9%). Tripling in five years is about 25% (24.6%). Doubling in three years is about 26%. The grid below solves \(MOIC^{1/n} - 1\), where MOIC is the multiple of invested capital:
| Multiple of money | 3 years | 4 years | 5 years | 7 years |
|---|---|---|---|---|
| 1.5x | 14.5% | 10.7% | 8.4% | 6.0% |
| 2.0x | 26.0% | 18.9% | 14.9% | 10.4% |
| 2.5x | 35.7% | 25.7% | 20.1% | 14.0% |
| 3.0x | 44.2% | 31.6% | 24.6% | 17.0% |
Powers to remember. \(1.1^5 \approx 1.61\), \(1.08^5 \approx 1.47\), \(1.05^{10} \approx 1.63\), \(1.07^{10} \approx 1.97\), \(1.15^5 \approx 2.01\), \(1.2^4 \approx 2.07\), \(1.25^3 \approx 1.95\). With these you can test any guess. If someone claims that 3x in five years is a 20% IRR, you know \(1.2^5 \approx 2.49\), so the true answer must be higher.
Perpetuities as multiples. \(1/(r-g)\) is a multiple. At 10% with no growth, a perpetuity is worth 10 times its cash flow; at 9% with 3% growth, 16.7 times. Valuation multiples and discount rates are two views of the same thing, a link we build in lectures 8 and 10.
In the interview
“Why do we discount future cash flows?”
Because money has a time value. A dollar received later is worth less than a dollar today since today's dollar could be invested to earn a return, and because future cash is less certain. The discount rate reflects that opportunity cost and risk. Discounting puts every cash flow in today's dollars so we can add them up.
“Would you rather have $100 today or $110 in a year?”
It depends on the rate I can earn at similar risk. The break-even is 10%: above it, I take the $100 and invest it; below it, I wait. A strong answer states the break-even and then adds risk: $110 promised by a shaky counterparty is worth less than $110 from a government.
“What happens to a company's valuation when interest rates rise?”
All else equal it falls. Higher rates raise the discount rate, so future cash flows are worth less today. Companies whose value sits far in the future, such as high-growth businesses, are hit hardest because their cash flows have the longest duration. Higher rates can also raise the company's own borrowing costs and slow the economy.
“When can NPV and IRR give different answers?”
Name the four cases: scale, because IRR ignores the size of the investment; timing, when mutually exclusive projects have NPV profiles that cross; non-conventional cash flows, which can produce several IRRs or none; and the reinvestment assumption, since IRR is only the realised return if interim cash earns the IRR. Then give the rule: trust NPV, and use MIRR if you need a percentage.
“A fund doubles its money in five years. What is the IRR? And if it triples?”
About 15% and about 25%. Show how you know: the rule of 72 gives 72/5, about 14.4% for doubling, and \(1.25^5 \approx 3.05\) confirms the tripling figure. Interviewers care less about the decimal than about seeing you check your own estimate.
“Why does the growing perpetuity formula need r to be greater than g?”
If g is at least r, each discounted payment is as large as the one before or larger, so the sum never converges. Economically, nothing can grow faster than its discount rate forever, so terminal growth should sit near long-run inflation or nominal GDP growth.
Key takeaways
- A dollar today is worth more than a dollar later because of opportunity cost, inflation and risk. Draw a timeline, with signs, for every problem.
- FV = PV(1+r)^n moves money forward; PV = FV/(1+r)^n moves it back. The rule of 72 gives doubling times; about 114 gives tripling times.
- EAR = (1 + APR/m)^m − 1. Continuous compounding is the limit, with r_c = ln(1 + EAR). Compare rates only on the same basis.
- Perpetuity C/r, annuity C(1 − (1+r)^−N)/r, annuity due × (1+r), growing perpetuity C₁/(r − g), growing annuity for finite growing streams. A formula gives a value one period before the first payment.
- Positive NPV creates value. IRR can mislead on scale, timing, sign changes and reinvestment; trust NPV, and use MIRR if you need a rate. Payback ignores everything after the cutoff.
- With irregular dates, discount by exact year fractions (XNPV, XIRR). Excel’s NPV function discounts its first cash flow by a full period.
- Discount nominal cash flows at nominal rates and real ones at real rates; items fixed in money terms need nominal modelling.
- 2x in 5 years ≈ 15%, 3x in 5 years ≈ 25%, 2x in 3 years ≈ 26%. A perpetuity is a multiple: 1/(r − g).
Problem set
1Two warm-ups. (a) You invest $2,500 at 5% a year for 8 years. How much do you have at the end? (b) A pension promises $1,000,000 in 20 years. What is it worth today at 6%?
(a) Since \(1.05^8 = 1.4775\), you have \(2{,}500 \times 1.4775 = \$3{,}693.64\). Of the $1,193.64 gained, $1,000 is simple interest (8 × $125) and $193.64 is interest on interest.
(b) Since \(1.06^{20} = 3.2071\), it is worth \(1{,}000{,}000 / 3.2071 = \$311{,}805\). Less than a third of the headline amount: this is why long-dated promises are cheap to make and expensive to fund late.
2A credit card charges 18% APR, compounded monthly. What is the effective annual rate? What continuously compounded rate is equivalent? Is a loan at 19% compounded annually cheaper or dearer?
The monthly rate is \(18\%/12 = 1.5\%\), so \(EAR = 1.015^{12} - 1 = 19.56\%\).
The equivalent continuous rate is \(\ln(1.1956) = 17.87\%\), which is also \(12 \times \ln 1.015\).
The 19% annual loan is cheaper: its effective rate is 19.00% against 19.56%. The card's headline looks lower only because it is quoted on a different basis.
3You borrow $200,000 over 25 years at 4%. (a) What is the annual payment if you pay at the end of each year? (b) What if each payment is due at the start of the year? (c) What is the monthly payment if the 4% is an APR with monthly payments, and how much interest do you pay in total?
(a) Solve the annuity formula for \(C\): \(C = 200{,}000 \times 0.04 / (1-1.04^{-25})\). Since \(1.04^{25} = 2.6658\), the denominator is \(1 - 0.3751 = 0.6249\), and \(C = 8{,}000 / 0.6249 = \$12{,}802\) a year.
(b) An annuity due is worth \(1+r\) times an ordinary annuity with the same payment, so the payment needed falls by that factor: \(12{,}802 / 1.04 = \$12{,}310\).
(c) The monthly rate is \(0.04/12 = 0.333\%\) over 300 months. The numerator is \(200{,}000 \times 0.003333 = 666.67\) and the denominator \(1 - 1.003333^{-300} = 0.6315\), so \(C = \$1{,}055.67\). You pay \(300 \times 1{,}055.67 = \$316{,}702\), of which $116,702 is interest, against $120,060 with annual payments. Monthly payments repay principal sooner, so less interest builds up, even though 4% compounded monthly is an effective 4.07%.
4A share will pay $3 next year. Dividends grow 4% a year and investors require 9%. What is it worth? What if growth is 5%? If the share actually trades at $50 with 4% growth, what return is the market pricing in?
\(3 / (0.09-0.04) = \$60\). At 5% growth, \(3 / 0.04 = \$75\), a 25% jump from one point of growth. The closer \(g\) gets to \(r\), the more sensitive the value becomes.
At $50, the implied return is \(r = D_1/P + g = 3/50 + 4\% = 10\%\). Either the market demands a point more than you assumed or, at your 9%, it is pricing growth of only \(9\% - 6\% = 3\%\).
5A royalty pays $5m next year, rising 2% a year for 15 years, then stops. At a 9% discount rate, what is it worth? Compare it with a level $5m a year for 15 years and with a growing perpetuity.
Growing annuity: \(PV = \dfrac{5}{0.09-0.02}\left[1-\left(\dfrac{1.02}{1.09}\right)^{15}\right]\). Since \((1.02/1.09)^{15} = 0.3695\), \(PV = 71.43 \times 0.6305 = \$45.04\text{m}\).
A level $5m for 15 years is worth \(5 \times 8.0607 = \$40.30\text{m}\). A growing perpetuity is worth \(5/0.07 = \$71.43\text{m}\). Growth adds about 12% over the level stream, and the finite life costs 37% of the perpetuity value.
6A DCF forecasts free cash flow of $50m in year 6, growing 2.5% a year forever after. The discount rate is 9%. What is the terminal value at the end of year 5, and what is it worth today? What does the common timing mistake give?
The growing perpetuity formula, applied to the first cash flow of the stream (year 6), gives a value one period earlier, at year 5: \(TV_5 = 50/(0.09-0.025) = \$769.23\text{m}\).
Today: \(769.23 / 1.09^5 = 769.23 / 1.5386 = \$499.95\text{m}\).
Discounting by six years, as if the value sat at year 6, gives $458.67m, 8% too low. If you are given year 5's cash flow instead, grow it one year first: \(TV_5 = FCF_5 \times (1+g)/(r-g)\).
7A sponsor invests $250m of equity and sells its stake after five years for $600m. Estimate the IRR in your head, then check it. How do the multiple and the IRR change if the company also pays the sponsor a $50m dividend at the end of year 2?
The multiple is \(600/250 = 2.4\text{x}\) over five years. Anchor: 2.5x in five years is about 20%, so the answer is a little lower, about 19%. Check: \(2.4^{1/5} - 1 = 19.1\%\).
With the dividend, the sponsor receives $650m in total, a 2.6x multiple. The IRR solves \(-250 + 50/(1+r)^2 + 600/(1+r)^5 = 0\), giving 22.6%. Had the extra $50m come at exit instead, 2.6x in five years would be 21.1%. Early cash lifts the IRR more than the same cash later, which is the logic of a dividend recap.
8An open-pit extension costs $100m today, generates $230m next year and requires $132m of rehabilitation spending in year 2. Find the IRRs. Should you accept at a cost of capital of 8%? At 12%? What does the MIRR say?
Write \(x = 1+r\) and set NPV to zero: \(-100x^2 + 230x - 132 = 0\), so \(x = (230 \pm \sqrt{52{,}900 - 52{,}800})/200\), that is \((230 \pm 10)/200\). The IRRs are 10% and 20%, and NPV is positive only between them.
At 8%: \(-100 + 230/1.08 - 132/1.08^2\), which is \(-100 + 212.96 - 113.17 = -\$0.21\text{m}\). Reject. At 12%: \(-100 + 205.36 - 105.23 = +\$0.13\text{m}\). Accept.
MIRR at 8%, using 8% as both finance and reinvestment rate: outflows are worth \(100 + 132/1.08^2 = 213.17\) today, and the inflow grows to \(230 \times 1.08 = 248.40\) by year 2. \(MIRR = (248.40/213.17)^{1/2} - 1 = 7.95\%\), below 8%: reject. At 12% it is \((257.60/205.23)^{1/2} - 1 = 12.03\%\): accept. MIRR gives one answer and agrees with NPV. Note also how little value is at stake: two impressive-looking IRRs on a project worth almost nothing.
9You invest $5m on 15 January 2026, receive $1.5m on 30 June 2026 and $4.2m on 31 March 2027. What is the XIRR? What IRR do you get by wrongly treating the receipts as year-1 and year-2 cash flows?
The receipts come 166 and 440 days after the start, or 0.4548 and 1.2055 years. Solve \(-5 + 1.5/(1+r)^{0.4548} + 4.2/(1+r)^{1.2055} = 0\). At 14%: \(1.5/1.0614 + 4.2/1.1711\), which is \(1.4133 + 3.5863 = 4.9996\), almost exactly 5. The XIRR is 14.0%.
Treating the receipts as arriving at the ends of years 1 and 2 gives \(-5 + 1.5/(1+r) + 4.2/(1+r)^2 = 0\), an IRR of 7.9%. Same money, roughly half the return, because the dates were wrong.
10A mine plan shows cash flows of $100m a year for ten years in today’s money. The company’s discount rate is 8% nominal and expected inflation is 2.5%. What real rate should you use? What error do you make if you discount the plan at 8%?
The real rate is \(1.08/1.025 - 1 = 5.37\%\), not the 5.5% of the shortcut. At 5.37% the plan is worth $758.6m.
At 8% it is worth $671.0m, 11.5% too low: the cash flows already exclude inflation, and the nominal rate removes it a second time. Inflating the cash flows at 2.5% a year and discounting at 8% gives the same $758.6m as the real calculation, as it must.