A bird in the hand is worth two in the bush (depending on the interest rate).
In a previous post we covered interest rates and showed how money invested now can, with time, grow to a larger amount, its “future value”. In this post we’ll turn that concept on its head and go through how to calculate the “present value” — what an amount in the future is worth today. Present Value is used to calculate loans, to value discount securities, to price shares and to figure out whether a large capital investment is a viable project to undertake. More broadly, future and present values fall under the banner of “the time value of money” which is one of the fundamentals of finance: people naturally want to use their money now and need to be compensated with more money for postponing the use of it until later. There are some calculations in this post and I’ve chosen to leave out the formulas for the sake of simplicity.
If we continue from previous examples in our 5% interest rate world, $100 invested today for one year will return us $105 at the end of the period. $105 is therefore the future value of $100 (at 5% interest) and, conversely, $100 is the present value of $105 (at 5% interest). Extending the theory a rational investor should be indifferent to receiving $100 now or receiving $105 in one year’s time. The interest of $5 is compensation for giving up the opportunity of spending $100 today and for any inflation which would erode the spending power of the $100. Another name for the present value is the discounted value; the interest rate used to calculate present value is often referred to as the discount rate. Whether you’d prefer the $100 now or could hold out for the $105 is a personal matter ands it’s related to your ability to put off current consumption for future benefit, a bit like the Stanford Marshmallow Experiment.
Present Value of a Bond
A one-period example is very simplistic. In the real world present value is often calculated over much longer periods. In an earlier example we looked at cashflows from an investment in a bond which paid 5.2% interest twice a year and repaid the principal when it matured. Finding the present value of all these cashflows is how the price of a bond — how much an investor should pay for it — is calculated. Let’s take a closer look…
In the original example the bond’s face value was $1 million, its coupon rate was 5.2% and the discount rate used to find the bond’s present value was also 5.2%. The discount rate (or what is also called the yield to maturity) is a remarkably sensitive number and affects the bond’s present value greatly.
Interest | Principal | PV of Interest | PV of Principal | |
January | -$1,000,000 | -$1,000,000 | ||
July | $26,000 | $25,341 | ||
January | $26,000 | $24,699 | ||
July | $26,000 | $24,073 | ||
January | $26,000 | $23,463 | ||
July | $26,000 | $22,868 | ||
January | $26,000 | $22,289 | ||
July | $26,000 | $21,724 | ||
January | $26,000 | $21,174 | ||
July | $26,000 | $20,637 | ||
January | $26,000 | $1,000,000 | $20,114 | $773,618 |
Total | $1,260,000 | Total | $1,000,000 |
TABLE 4.1: FUTURE AND PRESENT VALUES OF 5 YEAR, 5.2% BOND WITH YTM 5.2%
Looking at Table 4.1, above, you can see that for a bond with a face value of $1 million and a 5.2% coupon payment twice a year will pay $26,000 every six months (unless it defaults, but let’s assume that this bond has an impeccable credit quality). Each January and July the investor in the bond will receive their repayment. But these repayments are in the future so in order to calculate the present value of these cashflows we need to apply the discount rate to each one of them in turn and then add them together to find the total present value. When interest rates are above zero, a future cashflow will be worth less today, and the further into the future it is the lower that value will be for a cashflow of the same size (see Figure 4.1, below). This is shown in the second and fourth columns: as we go further into the future the present value of the $26,000 coupon payment become less. This is a good illustration of the power of compounding. Comparing the present and future values you can see the value falling each six months. The repayment of the $1 million face value at the end of five years has a present value of $773,618.
FIGURE 4.1: PRESENT VALUES GET SMALLER AS TIME TO RECEIPT INCREASES
These present values represent the amounts which, if invested today at the discount rate used, would grow to the future value amounts. Another point to notice is that, when the coupon rate and the discount rate are the same, the present value of the cashflows will always equal the bond’s face value when calculated on the first day of the bond’s life.
Discount Rate: 5.0% | Discount Rate: 5.4% | |||
PV Interest | PV Principal | PV Interest | PV Principal | |
July | $25,366 | $25,316 | ||
January | $24,747 | $24,651 | ||
July | $24,144 | $24,003 | ||
January | $23,555 | $23,372 | ||
July | $22,980 | $22,757 | ||
January | $22,420 | $22,159 | ||
July | $21,873 | $21,576 | ||
January | $21,339 | $21,009 | ||
July | $20,819 | $20,457 | ||
January | $20,311 | $781,198 | $19,919 | $766,118 |
Total | $1,008,752 | Total | $991,338 |
TABLE 4.2: PRESENT VALUES OF 5 YEAR, 5.2% BOND WITH DIFFERENT DISCOUNT RATES
As the discount rate changes so does the present value of future cashflows. Table 4.2, above, gives a good example of this. In the table the bond cashflows when discounted at 5.2% in Table 4.1 can be compared to the cashflows of the same bond when discounted at 5.0% and 5.4%. It’s clear that the higher the discount rate, the lower the present value; the opposite also applies. The total present value of the bond — which is the fair price of it — moves down as discount rates go up. Table 4.3, below, summarises these results.
Discount Rate | 5.0% | 5.2% | 5.4% |
Bond Price | $1,008,752 | $1,000,000 |
$991,338 |
TABLE 4.3: FAIR VALUES OF A 5 YEAR, 5.2% BOND WITH DIFFERENT DISCOUNT RATES
Just as a bond can be valued by discounting its future cashflows any cashflow-generating investment of project can be valued the same way. Importantly, finding the present value of a stream of future cashflows allows different investment options to be compared on the same basis. An investment in a bond can be compared to an investment in a company’s shares, a new piece of machinery or a major multi-year infrastructure project. Present value is a great leveller and measurement tool.
Present Value of a Share
We’ll explore the subject of equity valuations in depth in a later post but, for now, let’s take a quick look at a simple valuation of one company share. A share in a company gives an investor ownership in that company and an entitlement to receive any dividends paid. If the company pays a regular, fixed dividend and can maintain that payment indefinitely into the future then the share can be valued as the present value of those dividends. There’s some room for debate as to the correct discount rate to use, but that’s an argument for another day.
If a company can pay a dividend of $0.20 per share indefinitely into the future then, at a 5% discount rate, the present value of those dividends — and therefore the share price — is $4.00. At 6% the share price is $3.34; at 4% the share price is $5.00. You can see that the inverse relationship between the discount rate and the present value also applies in this case.
Present Value of a Project Investment
I like this technique. You can’t hide from the implications of a project valuation but it does involve quite a number of assumptions and can be manipulated by the unscrupulous. That’s a warning to anyone listening to a government touting the benefits of any big investment outlay. This technique, called net present value (NPV) analysis, is a valuable tool for evaluating investment opportunities.
Consider a theoretical toll road project detailed in Table 4.4, below. The initial cost is $1 million. It will take five years to build the road after which the tolls will return $100,000 per year growing at 6% annually. The project will have a working life of ten years after which the toll will be lifted and the road will have no toll charge. Total tolls are $1,318,079 which look like they cover the cost of the road, but they are not present values. In PV terms these tolls are worth only $997,083. Subtracting the initial investment of $1 million leaves a negative value.
This means that the project does not recover its costs; it loses money. However here’s where the manipulation can enter the calculations.
Cashflows | PV of Cashflows | |
Year 1 | $0 | $0 |
Year 2 | $0 | $0 |
Year 3 | $0 | $0 |
Year 4 | $0 | $0 |
Year 5 | $0 | $0 |
Year 6 | $100,000 | $85,727 |
Year 7 | $106,000 | $88,567 |
Year 8 | $112,360 | $91,502 |
Year 9 | $119,102 | $94,535 |
Year 10 | $126,248 | $97,667 |
Year 11 | $133,823 | $100,904 |
Year 12 | $141,852 | $104,248 |
Year 13 | $150,363 | $107,702 |
Year 14 | $159,385 | $111,271 |
Year 15 | $168,948 | $114,959 |
Total PV of Tolls | $997,083 | |
Cost | $1,000,000 | |
NPV | -$2,917 |
TABLE 4.4: NET PRESENT VALUE OF A PROJECT
The discount rate used was 5.2%. At this discount rate the project will lose money. There’s an opportunity to find a discount rate that makes the project profitable. In this case that discount rate is not too different from 5.2%: using 5.14% the project is slightly profitable and it will make $229. The cashflows are also able to be re-estimated since they are based on assumptions about the future. Beware next time you hear that a large government project has undergone a rigorous cost/benefit analysis.
Despite these shortcomings a net present value analysis performed with appropriate assumptions is a sound methodology for comparing investment projects.
This post has covered many of the processes involved in finding the present value of a future cashflow and a reader may be able to think of other examples of when it may be applied. It’s a fundamen tal of finance and will be used in most financial calculations, particularly those concerning investment. We will expand on these techniques and re-use them in later posts. Enjoy the marshmallows.
Categories: Interest rates, Investment Concepts, Level: Intermediate