Investment decisions: NPV, IRR and bonds

Businesses constantly decide whether to spend money now in the hope of more money later: buying a machine, opening a store, launching a product, upgrading software or buying another company. These are capital investment decisions, and they are among the most important a business makes, because they commit large sums for years. The compound interest and annuity tools from earlier lessons give a disciplined way to evaluate them: convert every future cash flow to its present value at a rate that reflects the cost of the money and the risk, and see whether the investment creates value. This lesson covers net present value, internal rate of return, payback, and the valuation of bonds, which applies the same ideas to one of the most widely traded investments in the world.

Cash flows and the discount rate

An investment is described by its cash flows: the money paid out and received at each date. By convention, the initial outlay is at time 0 and later cash flows are placed at the end of each year. Only cash matters, not accounting profit, and only incremental cash flows count: the extra cash that happens because the investment is made. Typical cash flows include the purchase price and installation, extra revenue or cost savings, extra operating costs, any extra working capital (which is usually recovered at the end), and the salvage value when the asset is sold or scrapped.

The discount rate is the return the business requires on an investment of that risk, often its cost of capital: the blended rate it pays for its debt and the return its owners expect on their equity. Riskier projects deserve higher discount rates. Choosing the discount rate is a judgment, and it has a large effect on the answer.

Net present value

The net present value (NPV) of an investment is the sum of the present values of all its cash flows, including the negative initial outlay. It measures, in today's dollars, how much value the investment adds above the required return.

The decision rule is simple. If NPV is positive, the investment earns more than the required return and should be accepted. If it is negative, it earns less and should be rejected. If it is zero, it earns exactly the required return.

Internal rate of return

The internal rate of return (IRR) is the discount rate that makes the NPV exactly zero. It is the investment's own rate of return, the compound annual rate that its cash flows earn. If the IRR is above the required return, the NPV is positive and the investment is acceptable.

There is no formula for the IRR when there are more than two cash flows. It is found by trial and error, narrowing in on the rate where NPV changes sign, or with the spreadsheet IRR function or a financial calculator.

When NPV and IRR disagree

For a single, ordinary investment (an outlay followed by inflows), NPV and IRR always give the same accept or reject decision. They can rank mutually exclusive projects differently, however, when the projects differ in size or in the timing of their cash flows. When they conflict, NPV is the better guide, because it measures the value added in dollars at the business's actual required return. IRR implicitly assumes that cash flows can be reinvested at the IRR itself, which may be unrealistic.

Payback and the profitability index

The payback period is the time it takes for the cumulative cash flows to recover the initial outlay. It is easy to understand and is a rough measure of risk and liquidity, but it ignores the time value of money and everything that happens after payback. The discounted payback period fixes the first flaw by using present values. The profitability index (PI), the present value of future cash flows divided by the initial outlay, shows the value created per dollar invested and is useful when capital is limited.

Bonds

A bond is a long-term loan sold to investors. The issuer, a government or corporation, promises to pay interest, called coupons, at a fixed rate on the face value (usually $1,000) at regular intervals, typically every six months, and to repay the face value at maturity. Once issued, bonds trade in the market, and their prices move as interest rates change.

US Treasury notes and bonds, municipal bonds and most corporate bonds pay semi-annual coupons. Bond prices are quoted as a percentage of face value; a price of 98.50 means $985.00 per $1,000 of face value. Treasury prices are traditionally quoted in 32nds of a point.

The price of a bond is the present value of its cash flows at the market's required yield: an ordinary annuity of coupons plus a single payment of the face value at maturity.

Here C is the coupon per period (face value × coupon rate ÷ 2 for semi-annual coupons), i the market yield per period, n the number of periods to maturity and F the face value. When the market yield is above the coupon rate, the bond sells at a discount (below face value); when it is below, the bond sells at a premium; when they are equal, it sells at par.

Longer bonds are more sensitive to changes in interest rates. A 2-year bond with the same 4 % coupon would be priced at $981.19 at 5 % and $1,019.27 at 3 %, a much smaller swing than the 10-year bond. This interest rate risk is one reason long-term bonds usually offer higher yields.

# NPV, IRR (by bisection) and bond prices
def npv(r, cfs): return sum(c / (1 + r) ** t for t, c in enumerate(cfs))
def irr(cfs, lo=-0.9, hi=1.0):
    for _ in range(100):
        mid = (lo + hi) / 2
        lo, hi = (mid, hi) if npv(mid, cfs) > 0 else (lo, mid)
    return mid
van = [-60000, 18000, 20000, 22000, 20000, 23000]
print(f"Van: NPV at 10% {npv(.10, van):,.2f}, IRR {irr(van):.2%}")
X = [-100000, 80000, 40000, 5000, 0]
Y = [-100000, 0, 10000, 50000, 110000]
print(f"X: NPV {npv(.10, X):,.2f}, IRR {irr(X):.2%};  Y: NPV {npv(.10, Y):,.2f}, IRR {irr(Y):.2%}")
def bond(face, coupon, y, years):
    c, i, n = face * coupon / 2, y / 2, years * 2
    return c * (1 - (1 + i) ** -n) / i + face * (1 + i) ** -n
for y in (0.03, 0.04, 0.05):
    print(f"10-year 4% bond at {y:.0%}: {bond(1000, .04, y, 10):,.2f}")

Van: NPV at 10% 17,362.95, IRR 20.37%
X: NPV 9,541.70, IRR 17.62%;  Y: NPV 20,961.68, IRR 16.07%
10-year 4% bond at 3%: 1,085.84
10-year 4% bond at 4%: 1,000.00
10-year 4% bond at 5%: 922.05