Sequences, Series and the Time Value of Money

1Introduction

A payment has a date as well as an amount: NT$1 next year is worth less than NT$1 today, because today's money earns interest. Discounting at a constant rate multiplies each payment by the same factor per period, so level or steadily growing payments form a geometric series. The sum of that series prices deposits, mortgages, bonds and dividend streams. Add one more idea, no-arbitrage, and futures can be priced too.

Sections 2–6 build the geometric sum into annuities, perpetuities, the Gordon model, bonds and loans. Section 7 adds no-arbitrage and the futures fair value used above; Section 8 resolves the case, and Section 9 collects the code.

2Sequences and Geometric Series

A deposit earning simple interest grows by the same amount each year; one earning compound interest grows by the same factor. The first is an arithmetic sequence, the second a geometric one.

NT$100 at 2% simple interest is worth 100, 102, 104, … after 0, 1, 2, … years: an arithmetic sequence with $d=2$. At 2% compound interest it is worth 100, 102, 104.04, …: a geometric sequence with $x=1.02$. The discount factors $v,v^2,v^3,\dots$ that convert future money into present money are geometric with ratio $v=1/(1+i)<1$.

Summing an arithmetic sequence

Write the sum forwards and backwards and add the two lines term by term:

Each pair $a_j+a_{n+1-j}=2a_1+(n-1)d=a_1+a_n$ has the same sum, and there are $n$ pairs, so

Section 6 uses (6.1) for equal-principal loans, whose payments fall arithmetically.

Summing a geometric sequence

Let ${\mathcal{S}}_n=a_1+a_{1}x+\cdots +a_1{x}^{n-1}$. Multiplying by $x$ shifts every term one place, $x{\mathcal{S}}_n=a_{1}x+\cdots +a_1{x}^n$, and subtracting cancels all but two terms: ${\mathcal{S}}_n-x{\mathcal{S}}_n=a_1-a_1{x}^n$.

Convergence follows from the gap between the full sum and a partial sum,

For $|x|<1$, $|x{|}^n=e^{n\ln |x|}\to 0$ because $\ln |x|<0$ (LM1), so the gap closes. For $|x|\ge 1$ the terms $a_1{x}^j$ do not shrink to zero. A series whose terms do not vanish cannot settle on a finite sum (Stewart, Clegg and Watson 2021).

By the gap formula, the fraction of an infinite geometric sum beyond its first $n$ terms is exactly $x^n$. For $x=v$ this is the share of a perpetuity's value paid after year $n$.

Every time-value problem is a recurrence

An account that earns $i$ per period and receives a net cash flow ${\mathrm{CF}}_t$ at the end of period $t$ evolves as $B_t=(1+i)B_{t-1}+{\mathrm{CF}}_t$. Deposits have ${\mathrm{CF}}_t>0$; withdrawals and loan payments have ${\mathrm{CF}}_t<0$. Unrolling the recursion from $B_0$ gives

as substituting (6.4) for $B_{n-1}$ into the recursion confirms. Level cash flows make the sum geometric with ratio $1+i$; flows growing at rate $\gamma$ make it geometric with ratio $(1+\gamma )/(1+i)$ after factoring.

3Present Value, Future Value and Annuities

One period at rate $i$ turns $\mathrm{PV}$ into $(1+i)\mathrm{PV}$, and $n$ periods compound to

The rate must match the period; LM1 converts quoted annual rates into periodic and effective rates.

Discounting multiplies each cash flow by a number, so a stream's present value is the sum of its parts' present values, $\mathrm{PV}={\sum }_t{\mathrm{CF}}_t{v}^t$. This cash-flow additivity lets us cut a complicated stream into simple pieces, value each piece and add. Section 7 turns the same principle into no-arbitrage pricing.

Ordinary annuities

An ordinary annuity pays $\mathrm{PMT}$ at the end of each of $n$ periods, at $t=1,\dots ,n$. Its present value is a geometric series with first term $\mathrm{PMT}v$ and ratio $v$, so by (6.2)

Since $1-v=i/(1+i)=iv$, the factor $v/(1-v)$ equals $1/i$, and

Compounding (6.5) forward $n$ periods gives the value on the date of the last payment:

We write $a_{n,i}$ and $s_{n,i}$ for the two fractions, the annuity factors.

Annuities due

An annuity due pays at the start of each period, at $t=0,\dots ,n-1$ (Exhibit 2). Every payment arrives one period earlier than in the ordinary annuity, so every value is multiplied by $1+i$:

Rent, insurance premiums and leases are usually paid in advance (annuities due); loan payments and bond coupons are paid in arrears (ordinary annuities).

Solving for the payment, the term or the rate

Each of (6.5)–(6.7) links a value, the payment $\mathrm{PMT}$, the rate $i$, the term $n$ and the timing convention. Given all but one, the last follows. The payment enters linearly. The term solves with logarithms: from (6.5), $v^n=1-\mathrm{PV}i/\mathrm{PMT}$, so

which requires $\mathrm{PMT}>\mathrm{PV}i$.

A payment no larger than the interest $\mathrm{PV}i$ never reduces the balance, so the payments can continue forever (Section 4). The rate has no closed form in general; Section 5 finds it numerically.

4Perpetuities, Growth and the Gordon Model

Let the number of payments grow without bound, or let the payments themselves grow. Both limits are geometric series, and together they value preferred shares, common stocks and the terminal stage of any long-lived cash flow.

Perpetuities

As $n\to \infty$ in (6.5), $(1+i{)}^{-n}\to 0$ and the value of a perpetuity appears:

A perpetual preferred share paying NT$2.40 a year is worth NT$60.00 at a required return of 4%. If the first payment comes at $t=m+1$ instead of $t=1$, the stream is worth $\mathrm{PMT}/i$ at $t=m$ and $v^m\mathrm{PMT}/i$ today. A first dividend in six years ($m=5$) gives NT$49.32. A perpetuity due, which also pays today, is worth $(1+i)\mathrm{PMT}/i=$ NT$62.40.

Growing annuities and perpetuities

Let the payment at $t$ be ${\mathrm{PMT}}_1(1+\gamma {)}^{t-1}$, growing at rate $\gamma$ per period. Factoring out ${\mathrm{PMT}}_1/(1+i)$,

a geometric series with ratio $x=(1+\gamma )/(1+i)$. Since $1-x=(i-\gamma )/(1+i)$, (6.2) gives

When $\gamma =i$ every term equals ${\mathrm{PMT}}_1/(1+i)$ and the sum is $n{\mathrm{PMT}}_1/(1+i)$. As $n\to \infty$, (6.3) applies if and only if $x<1$, that is $\gamma <i$:

If $\gamma \ge i$ the series diverges: payments that grow at least as fast as the discount rate have no finite value. As a check, ${\mathrm{PMT}}_1=1$, $i=5\%$, $\gamma =3\%$ and $n=20$ give 15.9648, both by (6.9) and by summing the 20 terms. With $\gamma =i$ the value is $20/1.05=19.0476$.

The Gordon growth model

A share entitles its holder to dividends $D_1,D_2,\dots$. If they grow at a constant rate $\gamma$ forever and investors require a return $k>\gamma$, (6.10) prices the share:

This is the Gordon growth model (Gordon 1959). Read backwards, a market price implies a required return or a growth rate. Solving (6.11) for $k$ gives the implied required return, dividend yield plus growth:

Solving instead for $\gamma$, $k{P}_0-\gamma P_0=D_0+\gamma D_0$ gives the implied growth rate

Dividing (6.11) by next year's earnings $E_1$ gives the justified forward P/E, $P_0/E_1=(D_1/E_1)/(k-\gamma )$: the payout ratio divided by $k-\gamma$. A 40% payout with $k=9\%$ and $\gamma =4\%$ justifies a forward P/E of 8.0.

Changing growth

A company may grow fast for some years and then mature. A two-stage model values the first $N$ dividends with (6.9) and the rest, growing at ${\gamma }_2$, with (6.11) applied at $t=N$:

What the price of 0050 implies

An index fund is a claim on the dividends of its basket, so the Gordon model can be read off its price. Exhibit 3 sets out the inputs for 0050 ten years apart.

Between 2016 and 2026 the yield fell from 3.98% to 1.39%. At a constant 10% required return the implied growth rose from 5.79% to 8.49% (Exhibit 3). Either investors came to expect faster dividend growth or they accepted a lower return: the model identifies only $k-\gamma =D_1/P_0$, never its two parts.

5Bond Pricing and Yield to Maturity

A bond's price is the present value of its promised cash flows, so bond pricing is annuity pricing. The only new problem is solving for the rate.

Three kinds of bond

A zero-coupon bond pays only its face value $M$ at maturity, so its price is $P=M(1+y{)}^{-n}$: it is a discount instrument. A coupon bond pays a coupon $C$ each period (coupon rate $C/M$) and $M$ at maturity: an ordinary annuity plus a single sum,

An amortizing bond pays a level amount $\mathrm{PMT}$ that covers interest and repays principal over its life. It is worth $\mathrm{PMT}{a}_{n,y}$: the loan of Section 6 seen from the lender's side.

Rearranging (6.14) shows when a bond trades above par. Because $yM{a}_{n,y}+M(1+y{)}^{-n}=M$, a bond whose coupon rate equals its yield is worth exactly par, and subtracting gives

Price exceeds par exactly when the coupon rate exceeds the yield, by the present value of the coupon surplus.

A coupon bond is also a portfolio of zero-coupon bonds, one per cash flow. By cash-flow additivity each cash flow must be discounted at its own spot rate $z_t$:

The yield to maturity $y$ is the single rate that reproduces this price in (6.14). It summarizes the spot curve, weighted by the bond's cash flows.

Yield to maturity as an implied return

Given a market price, the yield to maturity solves $P(y)=P_{\text{mkt}}$, a polynomial equation of degree $n$ in $1/(1+y)$. It has no closed-form solution in general but is well behaved. Every term of $P(y)$ falls as $y$ rises, so $P$ is continuous and strictly decreasing. It falls from ${\sum }_t{\mathrm{CF}}_t$ at $y=0$ towards 0 as $y\to \infty$, so any price in between has exactly one positive yield.

Bisection exploits this monotonicity. Start with a bracket $[y_{\text{lo}},y_{\text{hi}}]$ on which $P(y)-P_{\text{mkt}}$ changes sign, and repeatedly keep the half whose ends still differ in sign. Each step halves the bracket and with it the bound on the error.

The error itself need not fall: it rises from step 3 to step 4 in Exhibit 4. After $j$ halvings the midpoint lies within $(y_{\text{hi}}-y_{\text{lo}})/2^{j+1}$ of the yield. From a bracket 5 percentage points wide, $10^{-8}$ takes 22 halvings and 23 midpoints, since $2^{23}\ge 0.05/10^{-8}>2^{22}$. Newton's method, which also uses the slope $P^{\prime }(y)$, needs far fewer (LM4).

Is an IRR always unique? Write the NPV as a polynomial in $v=1/(1+y)$. Descartes' rule of signs, stated here without proof (Wang 2004), counts its positive roots. There are as many, counted with multiplicity, as the sign changes among its non-zero coefficients, or fewer by an even number.

Outflows followed by inflows, as in Example 6, change sign once: one IRR. The flows $-100$, $+230$ and $-132$ at $t=0,1,2$ change sign twice, and the IRR is both 10% and 20%. For such streams, report the NPV at a stated rate instead.

Pull to par

Holding the yield fixed, a bond's price still changes as time passes, because the remaining cash flows draw nearer. Exhibit 5 shows three 10-year bonds priced at a constant 2% yield, all converging to par at maturity. The premium bond (4% coupon) falls from 117.97 and the discount bond (0.5% coupon) rises from 86.53.

The par bond is at 100 on coupon dates; between them its clean price dips to 99.995. Futures converge too: at expiry a futures price meets its final settlement price (Section 7).

6Loans and Amortization

A level-payment loan is an ordinary annuity seen from the borrower's side. The lender hands over $B_0$ today in exchange for $n$ payments of $\mathrm{PMT}$, so $B_0=\mathrm{PMT}{a}_{n,i}$ and

Balance, interest and principal

Recurrence (6.4) with ${\mathrm{CF}}_t=-\mathrm{PMT}$ gives the balance after $t$ payments in two equivalent forms:

The first, retrospective, form accumulates the loan and subtracts the accumulated payments. The second, prospective, form says the balance is the present value of the payments still due. Substituting $B_0=\mathrm{PMT}{a}_{n,i}$ into the first form shows they agree:

Each payment splits into interest on the outstanding balance, $I_t=i{B}_{t-1}$, and principal $Q_t=\mathrm{PMT}-I_t$. The prospective form gives $i{B}_{t-1}=\mathrm{PMT}(1-v^{n-t+1})$, so

The principal portions form a geometric sequence with ratio $1+i$, from $\mathrm{PMT}{v}^n$ in the first payment to $\mathrm{PMT}v$ in the last. Interest portions fall as principal portions rise, and each pair sums to $\mathrm{PMT}$.

Principal exceeds interest, $Q_t>I_t$, exactly when $v^{n-t+1}>\frac{1}{2}$: when the $n-t+1$ payments remaining span less than the doubling time $\ln 2/\ln (1+i)$. At 2.2% a year money doubles in 378.4 months, longer than a 30-year term, so principal exceeds interest from the first payment. At 5% the doubling time is 166.7 months, and principal overtakes interest only at payment 195.

Equal principal and grace periods

Under equal-principal repayment the borrower repays $B_0/n$ each period plus interest on the remaining balance. The balance before payment $t$ is $B_0(1-(t-1)/n)$, so the payments fall by the constant amount $i{B}_0/n$: an arithmetic sequence. Total interest follows from (6.1):

A grace period (寬限期) of $m$ periods charges interest only, $i{B}_0$ per period, and then amortizes $B_0$ over the remaining $n-m$ periods with (6.16).

7Cash-Flow Additivity and No-Arbitrage

The law of one price says that two portfolios with identical future cash flows must cost the same today. Otherwise buying the cheaper and selling the dearer locks in a riskless profit, an arbitrage. With cash-flow additivity it prices anything that can be replicated; this section applies the pair three times.

Implied forward rates

An investor with a two-year horizon can buy a two-year zero at the spot rate $z_2$. Alternatively, she can buy a one-year zero at $z_1$ and agree today to reinvest for the second year at a forward rate $y_{1,2}$. Both strategies are riskless and pay off on the same date, so they must grow money equally: $(1+z_2{)}^2=(1+z_1)(1+y_{1,2})$. In general, the forward rate $y_{t_1,t_2}$ from year $t_1$ to year $t_2$ satisfies

With continuous compounding the relation is additive, $y_{t_1,t_2}=(t_2{z}_{t_2}-t_1{z}_{t_1})/(t_2-t_1)$. Taking logs of (6.20) and using $\ln (1+z_t)\approx z_t$ and $\ln (1+y_{t_1,t_2})\approx y_{t_1,t_2}$, which drop second-order terms (LM3), gives the same formula, approximately, for annual rates.

Forward exchange rates

Quote the exchange rate $S$ as TWD per USD. To hold TWD in $\tau$ years, invest TWD 1 at $i_{\text{TWD}}$. Alternatively, convert it into USD $1/S$, invest at $i_{\text{USD}}$ and sell forward at $F$. Both routes are riskless, so with simple annual money-market rates

covered interest parity; with continuous rates, $F=S{e}^{(r_{\text{TWD}}-r_{\text{USD}})\tau }$. We call $F-S$ the forward points and quote them in TWD per USD. The lower-rate currency trades at a forward premium: when TWD rates are below USD rates, a forward buys fewer TWD per USD than spot.

Index futures and the cost of carry

To replicate an index futures position, borrow $S{e}^{-q\tau }$ at the continuously compounded rate $r_f$ and buy $e^{-q\tau }$ units of the index basket. Reinvest the dividends, which arrive at the rate $q$, in the basket. At expiry the position has grown to exactly one unit, worth $S_T$, and the loan to $S{e}^{-q\tau }{e}^{r_f\tau }$. A futures contract bought at $F$ pays $S_T-F$, the same payoff, so the delivery price must be

If $F$ were higher, an arbitrageur would sell futures and buy the basket (cash-and-carry); if lower, buy futures and short the basket (reverse cash-and-carry). Futures are marked to market daily, but with a deterministic interest rate their fair value equals that of a forward (Hull 2022). With discrete dividends $D_j$ going ex $t_j$ years after the pricing date ($0<t_j<\tau$), and treated as received then, the same argument gives

The futures price is the index carried forward at $r_f$, minus the dividends carried forward to expiry.

The basis $F-S=S(e^{(r_f-q)\tau }-1)\approx S(r_f-q)\tau$, with a relative error of about $(r_f-q)\tau /2$, is negative whenever $q>r_f$: dividends outweigh interest. With illustrative values $S=20,000$, $r_f=2\%$, $q=3\%$ and $\tau =0.25$, $F=19,950.06$ and the basis is −49.94 points, against −50.00 to first order. At expiry TX is cash-settled against the average of the TAIEX values published in the last 30 minutes of trading (TAIFEX). $F$ converges to that average, not to the 13:30 close: the futures version of pull to par.

Reading the fair value backwards

Observed $F$ and $S$ pin down $r_f-q$, not $r_f$ and $q$ separately. Fixing one gives the other:

The first is the implied dividend yield, here measured over the contract's life as $q\tau$. The second is the implied financing rate, often called the implied repo rate. Two futures with expiries ${\tau }_1<{\tau }_2$ give, by the logic of (6.20), an implied forward dividend yield between the expiries: $q_{12}({\tau }_2-{\tau }_1)=r_f({\tau }_2-{\tau }_1)-\ln (F_2/F_1)$. It does not involve $S$ at all.

Frictions turn (6.22) into a band. Cash-and-carry borrows cash at a rate $r_b$. Reverse cash-and-carry must borrow every share in the basket, pay lending fees and usually earn less than $r_f$ on the short-sale proceeds. Its effective rate is some $r_{\ell }<r_b$.

Arbitrage then holds the futures price only within

The lower bound is the weak one, because shorting a whole index basket is harder than buying it.

8The TX Basis Through the 2025 Dividend Season

Equation (6.24) leaves one degree of freedom: choose $r_f$ to learn $q$, or $q$ to learn $r_f$. Deciding which reading fits needs the dividends themselves, measured outside the futures market, and TWSE effectively publishes them every day.

Measuring the dividends

Besides the price index $S$, TWSE computes a total-return TAIEX, $\mathrm{TR}$ (報酬指數), that reinvests cash dividends (TWSE 2023). On an ex-dividend date the price index falls by the dividends of the stocks going ex; the total-return index does not. If dividends worth a fraction $\delta$ of the previous close go ex, the ratio $\mathrm{TR}/S$ therefore rises by the factor $1/(1-\delta )$. Over the life of a contract,

which is the $q\tau$ of (6.22) under proportional dividends, counted on their ex-dates.

TSMC's NT$5 dividend went ex on 11 December 2025, inside the December contract's life, but was paid on 8 January 2026 (TSMC 2025). On the ex-date the ratio $\mathrm{TR}/S$ rose by 14.21 bp in log terms, so $\delta =1-e^{-0.00142149}=14.205$ bp of the previous close, 28,400.73: 40.34 index points. The day before, the press had estimated the points removed by that day's ex-dividend stocks at the same 40.34, almost all from TSMC (money.udn.com).

Exhibit 8 accumulates the measure over 2025. Dividends going ex during the year came to $q\tau =2.93\%$ of the index, 2.44% of it (83%) between June and September. July alone contributed 1.08%.

Carry, dividends and a residual

With the dividends measured, (6.22) becomes an accounting identity. Let $F_{\text{fair}}=S{e}^{(r_f-q_{\text{real}})\tau }$ be the fair value given the dividends that actually went ex, and $\epsilon =F-F_{\text{fair}}$ the residual. Then

Carry is the interest on the index value at the 2% proxy rate. The dividend term applies the realized dividend yield to the index on the pricing date and carries it to expiry. The residual is everything a frictionless, perfect-foresight model leaves unexplained; a negative residual means the futures were cheap.

Dividends explain 223.3 of the July contract's 458.1-point discount (Exhibit 9); carry pushes 23.5 points the other way, leaving a residual of −258.3. For December, carry adds 215 points and dividends explain 422 of 880, leaving a residual of −673: the residual grows with maturity. Read as financing rates, with dividends fixed at their realized values, the contracts priced money at −20.3% to −4.4% a year. Nobody lends at such rates.

The dividend column values the realized yields at the 27 June index level. TAIEX rose 22% by 17 December, so the points actually removed on the December contract's ex-dates were larger: 440.7, or 444.0 carried to expiry. Measured in points, the December residual would be −651 instead of −673.

Is the residual an artifact of one day? Partly. 27 June was the sample's extreme session: the December contract's discount, residual and implied financing rate were the most negative of all 146. TAIEX also closed at its high of the day.

By 30 June the December residual was −370 points. Over the nine sessions from 23 June to 3 July it averaged −461, and it was negative on 131 of the 146 sessions.

Back months add a second kind of noise: they trade thinly. On 27 June the July contract traded 60,785 lots but December only 49 (Exhibit 1). Over the sample, December traded a median of 14.5 lots a day.

Without a trade in the final minute, TAIFEX sets the settlement by rule. It can sit far from the last trade: 256 points for December on 11 July, when one lot traded. On 30 June the June 2026 contract did not trade at all, yet settled at 21,501. A residual that grows with maturity on one day is therefore weak evidence.

A cleaner test avoids the index altogether: calendar spreads compare two settlement prices. On 27 June they implied forward dividends of 0.98% from the July to the August expiry, 0.67% to September and 1.11% to December. Just 0.44%, 0.27% and 0.16% went ex. Only the first spread rests on two actively traded contracts.

The season, month by month

Dividends explain the shape of the season (Exhibit 10). In June they account for 135 points of the 192-point average discount, with carry offsetting 18 and the residual adding 75. In April, when almost nothing went ex, the residual of −84 exceeded the whole 65-point discount, because carry pushed the other way. The residual stayed between −84 and −68 until June, then narrowed, turning positive in September (+4) and reaching +38 in October.

The December contract, whose life spans the whole season, shows the pattern most clearly (Exhibit 11). Its residual averaged −191.5 points in April, −450.4 in June, −213.4 in August and +30.9 in October. As an implied financing rate it moved from −2.15% in June to +2.82% in October.

Exhibit 12 shows the whole curve on three days. On 1 April contracts expiring in September or later sat about 270 points below the spring contracts, pricing the coming dividends. On 27 June every contract was deep below the index, and the curve kept falling with maturity well beyond the dividend season. By 31 October it was above the index for every contract except September 2026, whose life includes the 2026 season.

Why the residual can persist

The no-arbitrage band of Section 7 is asymmetric. A rich TX is arbitraged with a stock purchase and a futures sale. A cheap one needs a short sale of every stock in the index, at an effective rate $r_{\ell }$ possibly far below 2%. Yet even at $r_f=0$ the December contract sat below fair value on 43 of the 44 sessions in June and July (Example 9).

The 2025 pattern is consistent with hedging pressure against that weak lower bound, a hypothesis rather than a measured cause. The near-month residual was deepest just after the April tariff shock (−298 points on 8 April). Holders of Taiwan stocks then had reason to sell futures rather than shares. The longer contracts' residuals were deepest in May and June.

Near-month and December residuals both turned positive in a rising October market. MacKinlay and Ramaswamy (1988) found that S&P 500 futures mispricing grows, on average, with time to maturity.

For a basis trader the residual is the opportunity, but not a riskless one. Dividends must be forecast and the stock leg must track TAIEX. The residual can widen before expiry forces convergence (LM25 models it as a mean-reverting spread).

9Time Value of Money in Python

Three habits keep TVM code honest. Write one function per formula, with its equation number in the docstring. Check the bracket before any numerical search. Use Polars expressions, not Python loops, for market data.

def annuity_pv(pmt, i, n, due=False):
    """PV of n level payments PMT at periodic rate i, eq. (6.5); due: (6.7)."""
    pv = pmt * n if i == 0 else pmt * (1 - (1 + i) ** -n) / i
    return pv * (1 + i) if due else pv

def bond_price(y, c, n, M=100.0):
    """Annual-coupon bond with coupon rate c and face value M, eq. (6.14)."""
    return annuity_pv(c * M, y, n) + M * (1 + y) ** -n

def bisect(fn, lo, hi, tol=1e-8):
    """Root of fn on a sign-changing [lo, hi]; each pass halves the bracket."""
    f_lo = fn(lo)
    if f_lo * fn(hi) > 0:
        raise ValueError("root not bracketed")
    while (hi - lo) / 2 > tol:
        mid = (lo + hi) / 2
        f_mid = fn(mid)
        if f_lo * f_mid <= 0:
            hi = mid
        else:
            lo, f_lo = mid, f_mid
    return (lo + hi) / 2

ytm = bisect(lambda y: bond_price(y, 0.015, 10) - 97.20, 0.0, 0.05)
print(f"{ytm:.6%}")                                     # 1.808601%
print(f"{annuity_pv(1_000, 0.03, 10, due=True):,.2f}")  # 8,786.11