# Why does the nominal interest rate equal 25% in this problem, instead of 16.4%?

My reasoning: The nominal interest rate = the real interest rate + the expected inflation rate. We need to calculate the nominal interest rate. We know the expected inflation rate, so the only problem is to calculate the real interest rate. In order to calculate the real interest rate we need to see how much money received is more than money given. And due to inflation we need to adjust the initial sum to inflation. 110 - 100*1.03=7, this is inflation-adjusted amount of money by which Daenerys will become richer. In order to calculate the real value we need do following: 7/110=0.0(63), which is about 6.4%. Thus the nominal rate equals 6.4+10=16.4

Here is the offical answer, but I'm afraid it doesn't make sense:

Keep in mind that nominal interest rate=Profit/Initial Amount.

The exact relationship that links real interest rate, nominal interest rate and inflation is $$(1+R_{nominal})=(1+R_{real})(1+\pi)$$ where $$\pi$$ is the inflation rate (Known as Fisher's identity)

$$R_{nominal}=R_{real}+\pi$$ is an approximation that works best when those variables are relatively small.

• "Keep in mind that nominal interest rate=Profit/Initial Amount". Thanks! Not only it works with the example in the OP, but it also works this problem that I posted in comments where the lender was Ygritte. – user161005 Apr 8 '20 at 16:30
• A more general way of finding $i$ is by setting the present values equal, here that would mean 20=25/(1+i). But if they devised a different scheme, say Grey Worm has to pay 15 in one and two years, then the interest rate would be calculated by solving the equation $$20=\frac{15}{1+i}+\frac{15}{(1+i)^2}$$ – actuarialboi9 Apr 8 '20 at 19:29

Have you considered that the nominal interest rate is the rate of interest without adjusting for inflation? So repaying \$25 on \$20 comes to an interest of 25%, no inflation adjustment is needed.

• "Have you considered that the nominal interest rate is the rate of interest without adjusting for inflation?" But I didn't adjust it! I adjusted the initial sum of money. – user161005 Apr 7 '20 at 17:54
• By the way, I used the same logic to solve similiar problem and it seems like I was able to solve it correctly: "Ygritte loaned Mans \$100. Mans paid her back \$110 one year later. The annual rate of inflation was 3%, percent. What was the nominal interest rate that Ygritte earned on this loan?" My answer: At first we find the real interest rate, it's 7/110. Then we round it and get 7% of real interest rate. Then we add 3% of inflation. Thus we get 10% of the nominal interest rate. I used method from here: youtu.be/cNm196bVE5A – user161005 Apr 7 '20 at 18:05
• I just don't understand why this method gives correct answer for problem with Ygritte, but wrong with Daenerys. Problems seem identical – user161005 Apr 7 '20 at 18:09

The official answer explains it correctly. A nominal interest rate is calculated independent of inflation. \$25 = \$20 (1 + 0.25).

I cannot follow the logic of your calculations, as they do not correspond to how a real rate of interest is calculated. The real rate of interest is the nominal interest rate less inflation, or 15%.

• "The real rate of interest is the nominal interest rate less inflation". Not applicable if we don't know the nominal interest rate in the first place. Do you have formulas/ideas for cases when we don't know the nominal rate? – user161005 Apr 7 '20 at 17:53
• The nominal rate was calculated in the official answer, and my answer - 25%. Lent \$20, got back \$25. – Brian Romanchuk Apr 7 '20 at 18:34