QM 4 The Time Value of Money in Finance

Discounting cash flows to value bonds and stocks, and pricing forwards and options by no arbitrage. Check-yourself items are study aids.

Fixed-income instruments are debt: an issuer takes capital now and promises future repayment. Their discount rate is the yield-to-maturity (YTM), the return earned if the instrument is held to maturity. Cash flows follow one of three patterns.

Discount (zero-coupon) instruments

A zero-coupon bond pays a single amount at maturity, so its price is just that amount discounted back.

Example 1 · Worked

A single principal cash flow of INR100 is payable in 20 years on an Indian government strip, and the required return is 6.70 percent (annual compounding).

What should the investor pay per INR100 of principal, and what happens if r immediately rises to 7.70 percent?

Solution. Price is INR100 divided by (1.067) to the 20th power, which gives INR27.33. If r jumps to 7.70 percent, the price falls to INR100 divided by (1.077) to the 20th power, which is INR22.68. A one percentage point (100 basis point) rise in the yield cuts the price by INR4.65, which is the inverse price-yield relationship in action.

When rates go negative

Germany, back in July 2016, was the first eurozone government to sell a 10-year bond at a yield below zero. A zero-coupon bond priced at a required return of minus 0.05 percent costs EUR100.50 per EUR100 of principal: the investor pays more than the principal returned at maturity. Investors accept negative rates for reasons such as safety in uncertain times, regulatory requirements, or expectations about monetary policy, prioritizing protection of capital over growth.

Coupon instruments

A coupon bond pays periodic interest plus principal at maturity. Discounting each coupon and the final principal, and simplifying the coupon stream as a geometric series, gives a compact price formula.

When the required return equals the coupon rate, the bond prices at par. A seven-year Greek government bond with a 2 percent annual coupon priced at a YTM of 2.00 percent is worth exactly EUR100 per EUR100 of principal. If the coupon rate exceeds the required return the bond prices above par (a premium bond), and if the coupon rate is below the required return it prices below par (a discount bond).

Yields are quoted on an annual basis, so a bond paying semiannually is valued with half the annual coupon and half the annual yield over twice the number of periods. A 20-year Indian government bond issued at par with a 6.70 percent annualized coupon pays INR3.35 every six months at a 3.35 percent periodic rate; if the annualized yield immediately rises to 7.70 percent (3.85 percent per period over 40 periods), the price falls to INR89.88.

A perpetual bond has no maturity date and never repays principal. Letting maturity run to infinity collapses the coupon-bond formula to the value of a perpetuity, PMT divided by r. KB Financial issued perpetual bonds paying a 3.30 percent annual coupon in quarterly instalments of KRW8.25 per KRW1,000 of face value; at a required return of 3.40 percent (0.85 percent per quarter), the price is KRW8.25 divided by 0.0085, which is KRW970.59.

Level-payment (amortizing) loans

An amortizing loan, such as a fixed-rate mortgage, pays equal instalments that blend interest and principal. Rearranging the annuity relationship solves for the level payment.

Example 2 · Worked

A 30-year mortgage funds 80 percent of a USD1,000,000 purchase, so the loan is USD800,000, at an annual rate of 5.25 percent paid monthly.

Find the monthly payment and the first month’s split between interest and principal.

Solution. The monthly rate is 5.25 percent divided by 12, or 0.4375 percent, over 360 periods. The payment is USD4,417.63. Month 1 interest is 0.4375 percent of USD800,000, which is USD3,500.00, so principal repaid is USD4,417.63 minus USD3,500.00, which is USD917.63. As the balance falls, the interest portion shrinks and the principal portion grows, even though the total payment stays fixed.

Equity is a fractional claim on a company with no maturity date. One common approach values a share by discounting expected future dividends at the required return, r. Three dividend patterns are used.

Constant dividends

A share expected to pay a fixed dividend forever is a perpetuity.

Shipline PLC pays GBP1.50 a year indefinitely. At a required return of 15 percent the share is worth GBP1.50 divided by 0.15, which is GBP10.00.

Constant dividend growth

If the next dividend then grows at a constant rate g forever, with r greater than g, the perpetuity becomes the constant-growth (Gordon) model.

Changing growth

Firms that grow fast then mature are valued with a two-stage model: discount each dividend through the high-growth phase of n periods, then add the value of the constant-growth perpetuity that begins afterward, itself discounted back to today.

Example 3 · Worked

Shipline PLC pays a GBP1.50 annual dividend and the required return is 15 percent. An analyst now assumes the dividend grows 6 percent a year for three years, then 2 percent a year forever.

What is the share worth under this two-stage assumption?

Solution. The three high-growth dividends are GBP1.5900, GBP1.6854, and GBP1.7865, whose present values at 15 percent are 1.3826, 1.2744, and 1.1747, summing to GBP3.8317. The Year 4 dividend is 1.7865 times 1.02, which is GBP1.8222, so the terminal value at the end of Year 3 is 1.8222 divided by (0.15 minus 0.02), and discounting it three years gives about GBP9.22. The share is worth 3.8317 plus 9.2163, which is 13.0480, or about GBP13.05, against GBP10.00 under constant dividends and GBP17.67 under constant 6 percent growth: the growth assumption drives the answer.

Check yourself

A preferred share pays a fixed USD4 dividend forever and the required return is 8 percent. What is it worth?

USD50. A constant dividend is a perpetuity, so the value is USD4 divided by 0.08, which is USD50.

The earlier sections went from cash flows and a rate to a price. Here we reverse it: given the market price and the cash flows, we solve for the rate or the growth the market is implying, which tells us whether an instrument looks correctly valued.

Implied return on a bond

For a discount bond, rearranging the price formula isolates the periodic return over its life.

Example 4 · Worked

A German 10-year zero was issued at EUR100.50 per EUR100 (a YTM of minus 0.05 percent). Six years later it trades at EUR95.72.

What was the realized annual return over the six years, and what return will a new buyer earn over the remaining four years?

Solution. Over six years the return is (95.72 divided by 100.50) to the power one-sixth, minus 1, which is minus 0.81 percent. For the remaining four years the buyer earns (100 divided by 95.72) to the power one-quarter, minus 1, which is 1.10 percent. Compounding the two periods over the full ten years returns to the original minus 0.05 percent YTM, absent transaction costs.

For a coupon bond the YTM is the single internal rate of return that equates all discounted cash flows to the price, solved iteratively. When a seven-year 2 percent Greek bond fell to EUR93.091 after two years, the YTM that solves the five remaining cash flows is 3.532 percent.

Implied return and growth on a stock

Rearranging the constant-growth model splits a stock’s implied return into its expected dividend yield plus its growth, and lets us solve for either one given the other.

Example 5 · Worked

Coca-Cola trades at USD63.00 with an expected next-year dividend of USD1.76.

If dividends are expected to grow 4 percent forever, what return is implied? If instead investors should expect a 7 percent return, what growth is implied?

Solution. Implied return is USD1.76 divided by USD63.00, plus 0.04, which is 6.79 percent. Implied growth is 0.07 minus USD1.76 divided by USD63.00, which is 4.21 percent. Assuming a higher return forces a higher growth rate to justify the same USD63.00 price.

Analysts often compare prices through the price-to-earnings ratio rather than in currency terms. Dividing the constant-growth model by earnings shows the forward P/E rising with the expected payout ratio and growth, and falling with the required return.

Check yourself

A stock trades at USD40 with an expected next-year dividend of USD2, and investors require a 9 percent return. What growth rate is implied?

4 percent. Implied growth is the required return minus the expected dividend yield: 0.09 minus USD2 divided by USD40, which is 0.09 minus 0.05, or 0.04.

Cash flow additivity says that a package of future cash flows is worth, today, the total of the present values of its individual pieces. This principle underpins no arbitrage: prices must line up so that no riskless profit is available. Two strategies with identical future cash flows must have identical present values, or a trader could buy the cheaper and sell the dearer for free money.

Implied forward interest rates

Compare investing for two years at the two-year rate against rolling two one-year investments. Equating the two locks the forward rate.

Example 6 · Worked

The one-year rate is 2.50 percent and the two-year rate is 3.50 percent.

What is the one-year forward rate starting in one year, and why can it not be, say, 5 percent?

Solution. The forward rate is (1.035) squared divided by 1.025, minus 1, which is 4.51 percent. If it were 5 percent, an investor could borrow GBP100 for two years at 3.50 percent (owing GBP107.12) and lend at 2.50 percent then 5 percent (receiving GBP107.63), pocketing a riskless GBP0.51 with no money down. Competition forces the forward rate to the no-arbitrage level of 4.51 percent.

Forward exchange rates

The same logic prices currency forwards. Investing in the home currency must match converting, investing abroad, and converting back at the forward rate, so the forward equals the spot adjusted by the interest-rate differential (continuous compounding shown, with the rate quoted as price currency per base currency).

With a spot of JPY134.40 per USD, a six-month JPY rate of 0.05 percent, and a six-month USD rate of 2.00 percent, the no-arbitrage six-month forward is JPY133.096 per USD. Any other forward would let an investor earn a riskless profit switching between the two currencies.

Option values

A one-period binomial tree prices an option by building a replicating portfolio: buy the hedge ratio of shares against a short option so the portfolio pays the same amount in both states, making it risk free, then discount that certain payoff at the risk-free rate.

Example 7 · Worked

An asset priced at CNY40 either rises 40 percent to CNY56 or falls 20 percent to CNY32. A call to buy at CNY50 is worth CNY6 in the up state and CNY0 in the down state. The one-period rate is 5 percent.

Find the no-arbitrage call price.

Solution. The hedge ratio is (6 minus 0) divided by (56 minus 32), which is 6 divided by 24, or 0.25. Holding 0.25 shares against the short call pays 0.25 times 56 minus 6, which is CNY8, in the up state, and 0.25 times 32 minus 0, which is CNY8, in the down state: a certain CNY8. Setting 0.25 times 40 minus the call price equal to CNY8 divided by 1.05 gives a call price of CNY2.38.

Check yourself

Using the same tree, a put to sell at CNY50 is worth CNY0 up and CNY18 down. With a hedge of 0.75 shares held alongside the put, the portfolio pays CNY42 in both states. What is the put worth?

CNY10. Setting 0.75 times 40 plus the put price equal to CNY42 divided by 1.05 gives the put price as CNY42 divided by 1.05, minus CNY30, which is CNY10.

The same no-arbitrage logic extends to real options: the flexibility to defer, expand, contract, or abandon a project. The value of that flexibility is the difference the choice makes to the project’s worth, net of the present value of the extra investment it requires.

Overview

Discounting: the foundation

Valuing fixed-income instruments

Discount (zero-coupon) instruments

Coupon instruments

Level-payment (amortizing) loans

Valuing equity instruments

Constant dividends

Constant dividend growth

Changing growth

Implied return and implied growth

Implied return on a bond

Implied return and growth on a stock

Cash-flow additivity and no arbitrage

Implied forward interest rates

Forward exchange rates

Option values